Large values of the remainder term of the prime number theorem
Pintz, Janos
In the lecture we prove a lower estimate for the average of the absolute value of the remainder term of the prime number theorem which depends in an explicit way on a given zero of the Riemann Zeta Function. The estimate is only interesting if this hypothetical zero lies off the critical line which naturally implies the falsity of the Riemann Hypothesis. (If the Riemann Hypothesis is true, stronger results areobtainable by other metods.) The first explicit results in this direction were proved by Turán and Knapowski in the 1950s, answering a problem of Littlewood from the year 1937. They used the power sum method of Turán. Our present approach does not use Turán’s method and gives sharper results.
 So, thank you for the invitation and I will then switch to full screen mode, which means I will not see myself, but this is not a big problem. And the topic, what I am talking about is large values of the remainder term of the prime number CRM and I will immediately see what I mean exactly by these large values. First, we have to remember the explicit formula for primes and then delta x is the remainder term of the prime number CRM and we see that this depends on zeros of the zeta function and which I will always denote by beta plus i gamma and we can restrict our attention to zeros with height less than x and then we see that the oscillation, what I can say caused alone by a zero row is x to the beta divided by the modulus of row, which means that asymptotically it is x to the beta divided by gamma. Now little would raise the problem, which perhaps is not clear why he raised it, prove an explicit oscillation result for the remainder term, if we suppose the existence of a hypothetical zero with real part at least one half and for simplicity with imaginary part positive. Now the reason why he raised this problem was that earlier the best result was a consequence of the fragment Lindelof CRM and it had two drawbacks. One was that the bound was just x to the beta minus epsilon, so smaller than the oscillation caused by the zero and secondly it were even this weaker bound was ineffective. So that means if we even knew the existence of a concrete zero or a hypothetical zero, we could not see, could not say from what point on and what size is this oscillation. And this problem was then, this is due to the possible interference of the set of zeros, which means that if I consider some of complex numbers, then I cannot be sure that the sum of these complex numbers can be estimated from below by any term of this sum. So that means if we have positive real numbers, then actually we can estimate from below the sum by any term of it, but if we have complex numbers, then this is naturally not the case. Now in order to settle this case, this was, I don't know whether for which particular problem, but actually in order to settle the problem of interference of complex zeros, Turan developed his method Turan's power sum method and this was the most frequent tool used in these investigations as well. Now just to remark that if we consider zeros on the critical line, which we know that there are a lot of zeros there, then we get an even larger oscillation than the one caused by any single zero itself. Maybe we can gain a factor three times the diluted logarithm of x, which I will denote by log 3x and even this oscillation occurs in most plus and minus direction, in most direction, which actually disproved type of conjecture of Riemann, which appeared in his work, where he showed that at least the second form of the remainder term, which appears here, this pi x minus li x is oscillating in both directions. Riemann showed that pi x is always less than li x and this was just in the same year 1914 and it proved that this oscillates in both directions, which is a size even bigger than the oscillation caused by a single zero. Interestingly, in the same year, Lehmann showed that at least for up to 10 million, pi x is less than li x and now we know that at least until 10 to 20 really pi x is less than li x. But we will not concentrate on sign changes now, so we will concentrate just on the size of the absolute value of the remainder term and in this case, it is not so critical which form of the remainder term we consider. So we will work with this delta x, which contains as you see the prime powers as well and therefore it doesn't appear this negative effect, which makes pi x less than li x usually. Now if we define with dy the average of the modulus of delta x and with sy, so phi, the maximum values of the modulus of the remainder term for x less than y, then to run succeeded in 1950 to solve the problem and the lower bound is really bigger than x y to better not minus epsilon. So there is a concrete explicit function and it is also explicitly given independence of the row from which point this estimate is true. Now some eight years later or nine years later, now we will prove the same for the average of the remainder term. The method in both worlds, the most critical part is to run spores some methods I mentioned or edited complex numbers if we have complex numbers and there are some might be much smaller than the largest term for example. And the two round work the other method which up to some extent helped to face with this situation. So in some sense with a factor what we will see later he succeeded to neglect this inter possible interference of zeros. So the result of Turan which was used in both his proof and Knapov's proof was the so-called second mean theorem of Turan's power some method that if we consider power some of n complex numbers, we use a normalization that the largest of this number has modulus one. And then if we consider we cannot state naturally that for every new this value should be large, but what we can state and this is what his second main theorem states that if we consider at least n values and consecutive values of this exponent, then apart from a very significant factor which is much less than one, we will reach the bound one and we have still a factor depending on these coefficients bj. Now in the present problem we cannot we don't need any coefficients. So that means we have a special case of this Turan second mean theorem which can be written in this case so that the new power some of n complex numbers taken if you take the maximum value for n consecutive values. And then this is a function which depends exponentially on the number of variables and it depends somewhat less drastically on the beginning of this interval. Now one can easily see that if we consider for example nth root of unity, then we have n minus one consecutive values always which are exactly zero and then we have one which is large. So that means in some sense this result is optimal and we don't know whether the lower bound is optimal or not. I used not exactly the same result but a similar argument that means it was more generalization of Dirichlet's approximation theorem to show that if we consider oscillation then this oscillation is really apart from a factor one minus epsilon, the oscillation is x to the better not divided by rho not in modulus. So that means the oscillation caused by a zero occurs really from time to time and this seemed to settle so to say to be precise almost completely the oscillation problem of delta x. And here this function c2 of rho not and epsilon was effective as well. The proof used Turan's method in not Turan's power sums but it was similar to the way Turan and Kanapovsky handled their proof their results but the difference was that I used another approximation theorem and I used still the general framework of Turan's result. Now immediately there is a question that what is the optimal oscillation result because we know that we have two zeros to conjugate zeros of the zeta function always and therefore I thought also that one minus epsilon could be replaced by two minus epsilon and this would be really the case if rho not was isolated if we would not have nearby other zeros and then the question was up to some extent or almost completely really settled by Silad Reves in 83 he showed that we don't know what is the optimal constant for the zeta function naturally if the Riemann hypothesis is true then the optimal constant is infinity. We remember that we have then an oscillation of size log log x times the log log x times the term of any zero on the critical line and so then this question is interesting just when Riemann hypothesis is false and so therefore it is very hard to say even how to say mathematically precisely what we mean by the optimal constant. Now he proved that the optimal constant is pi over 2 which meant the pi over 2 the result which I stated was true this pi over 2 minus epsilon instead of one minus epsilon on one hand and on the other hand if we consider a larger by the class of functions which include the zeta function as well then at least for this larger class of functions there are some functions for which pi over 2 is really the optimal constant. We could formulate it also that there could appear such intricate configuration of zeros of the zeta function so that the oscillation would be really not exceed pi half minus epsilon times the expected and proved value x to the rho naught over rho naught. Now I proved also several results on these other methods both for the average and the maximum value thereby improving the earlier results of Turan and Knapovsky also Schrager Puchta showed such an improvement and what our goal is that naturally different forms of these CRMs had different conditions concerning the dependence of y from gamma naught that means from if rho naught is given and then from which point on can we assert that the average or the maximum value is large then a lower bound as a function of y and gamma naught that means how large is really the average or the maximum dependent on gamma naught. A certain problem is that whether this large values occur just somewhere between one and y we can state that the average or the maximum is large or we can say that they occur up to some extent near to y so that means near to y means unfortunately not near on the normal scale but near on the logarithmic scale so that means that the logger if for an interval of the form a y to y and this beginning of the interval a y should be at least on the logarithmic scale near to log y. Now again we can consider effective or ineffective estimate and another method of mine gave a result which produced a lower bound which depended on the derivative on the absolute value of the derivative at the place rho naught which for concrete zeros which we know is really completely okay but it would be already somewhat disturbing so it would make the CRM let's say a little bit ineffective if we speak about a hypothetical zero and this could cause problems in some applications. Now I mentioned already that the really difficult case is Bendy and the really interesting case is Bendy rho naught is not on the critical line that means the Riemann hypothesis is really false. Now I mentioned just some results in this direction I proved that actually we can always reach that not only we have values as large as y to the better not divided by y to the better not divided by rho naught or x to the better not divided by rho naught sometimes but also that the average is at least y to the better not multiplied by some constant depending on the imaginary part of the zero and this has a corollary that on the Riemann hypothesis the average of the remainder term in modulus is at least square root y or the average is square root x. Now Krama showed already exactly 100 years ago that on the other hand the average of the remainder term in absolute value is at most constant times square root y so we know exactly apart from a constant the order of magnitude of the average order of the remainder term supposing the Riemann hypothesis this is this estimate here and on the other hand this is still unclear the maximum value of the remainder term between these two bounds. Now this actually served as an introduction for the CRM for which I will also sketch the proof namely that if we have a concrete zero of the zeta function and we know that y is enough large depending on an explicit way from the imaginary part of the zeta I just mentioned that already I mentioned that but what emphasize that all these constants c will be explicit constants. In this case we can localize we can consider the average of the remainder term in modulus and localize in the average in the logarithm scale from a beginning point near to y until y and we get for it a lower estimate which is just with a factor log log y square smaller x minus log log y square smaller than the expected value and concerning the localization I lose again the same log log y square quantity. Now the strategy is that we consider weighted mean value that means as I mentioned already concentrated to interval a y y where log y is near to log y we already specified that actually this a y would be this quantity so log a y is less than log y this quantity log log y square. Now this weighted mean value can be transferred into a sum of residues containing the zeros of the zeta function and up to this point the strategy is same as by Turan, Knapovsky and in my earlier works. The novelty is that the weight is chosen so that only zeros very near from row not should get non negligible weights and in this way we can avoid both Turan's method and both the loss which is caused by using Turan's second mean theorem. This means this quantity what we have here this is actually a loss compared with the expected value and naturally due to this averaging process and due to this procedure of getting rid of other zeros near to row not we have also a loss in the procedure but this loss is just a function x minus log log y squared various in the works of Turan and Knapovsky for example they had a loss of log y divided by log log y and we have instead log log y squared so we have a much smaller loss than in the procedure of Turan and Knapovsky. Now in the present method we will what I said we will get rid of zeros not very near to the critical given zero row not so that means we will consider the construct of weight function which is asymptotically the same for a small group by group I mean just the set so there is no group operation here so for a small set R of zeta zeros near to row not it is asymptotically the same naturally these zeros are very near to row not then at the distance basically this C naught we can say it's five so at the distance at most five for all the other zeros the weight function will be just zero and in this way those residues will not appear at all you know and due to other part of the weight function the weights there will be not zero but negligible the weights for the residues if we are more far than five from this point now the in fact the original CRM was formulated in such a way that the upper bound was by now we consider in on the logarithmic scale an interval symmetric to by and this will be our estimate and here what we should possibly keep in mind is that script L is log log y squared lambda is log y and L is log log y so capital L is log log y lambda is log y and script L is log log y squared and script L is which on in the exponential scale appears both in the loss and the both in the localization and this will be the exact form in which I will prove our result and the beginning is a procedure what was observed by Turan that if we have a zero and we have near to that zero near means just that the on the logger on the the height is not too far from it but the zero is little bit more on to the right then we take this other zero instead of our original zero and so with step by step but the number of steps is at least at the most just log log y or we can log y we can go further to another zero what we call an extreme right hand zero and this extreme right hand zero has the property that its real part is at least as large as the real part of the original zero so in this oscillation CRM which has gained by it the imaginary part is can be somewhat larger but this is not decisive and still we have a control about it the original imaginary part was at most square root of lambda and by this procedure it will be just lambda squared and lambda is just log y so that means it will be still near to log y log y squared as at most but on the other hand more on the right and this is an important point of view that for this zero we have already if we go a little bit far to the right with a quantity one over log y then we will have a large so if we go a little bit more to the right then already in the how to say vicinity but not vicinity but in a rather large height so that means three times log y higher or three times log y lower there is no zero as at all so that means in some direction and at least we solved the isolation of row not but we are very far from saying that the row is an isolated zero so this procedure is just how to say routine used always by Turan and Knobsky and also myself in these works and this is just a nearly trivial starting point for the procedure and if so that means our final zero is this gamma wave K beta wave K row K wave this is what we substitute the original row not by this zero but then comes the idea is that we take a very small epsilon and then we consider zeros which can be reached from one from the other by a chain of zeta zeros so that neighboring zeta zeros are at a distance at most epsilon from each other but this epsilon is not a small absolute constant but this epsilon is something which is very small so first we consider already zeros which could be reached from each other by a chain of zeros at most with a distance at most as you see one over log y and this is even in the denominator multiplied so that means this one over log log y to the third power so these zeros are very near to each other if in this group there would be just one zero then this wave could be right called an isolated zero already and then we could so to say go further with this procedure but what we know just is that it might be so that means definitely the starting set of zeros was contained at most log log y that means L zeros this is by the result of back line and this was later we can so to say nest into each other smaller and smaller groups of zeta zeros so that they would be very near to each other but any zero outside that small set would be much more far from these zeros than those zeros are from each other so that means this whole we repeat the same procedure and we take all these sets which of zeros which could be reached so equivalence classes of the zeros which could be reached from other by a chain with a smaller epsilon to smaller means by a factor log log y third smaller and if we always consider and continue our work from the smallest set then this whole procedure must finish quite soon so the number of step should be just log L that means much much smaller and in this way in this way we will get very soon just really not an isolated but a set of zeros what we will call our by our this the property that they are very near to each other any other zero is already much further from it much further means still it may be a small distance but much further with a power with a factor log log y to the second power and then we consider an arbitrary zero from this set our and we will have so that means basically the same extreme right hand property will be true for this zero maybe with a little bit changed parameters and after which we consider polynomial we construct such a polynomial which is so to say since the residues will occur at the races robin minus row prime if so that means in this way we will get such a polynomial that if we consider any other zero which is not in this small group but at most the distance five from it then we will have a property that will be interesting in our case so that means if we are not in this small set of of equivalent zeros then the polynomial is zero and we have so this is one thing for those zeros which are in this small group this polynomial at the place row minus row one is asymptotically the same exactly one one plus or the one over log log y and we know bound for the value of this polynomial at different places a roughly speaking e to the l or s absolute value to the l so on in an exponential scale we have this l which is log log y squared l squared ten times or forty times i don't know how a hundred times i don't know this is the value of l and this will be equal to 40 times l will be the actual loss in the procedure and so this l will have an important role but that means we have a control over the size of the polynomial and here comes really the weighted mean value of delta we consider this delta x divided by x to one plus row one times weight on the value y over x so that means this is also in some sense on the logarithmic scale and the weight function is defined together with this gs function and the gs function has three factors the first factor is relatively small if we are let's say at a distance at least five and at most log y from the zero then if we are at least a distance log y from the zero then this factor will be very small that means negligible and with this ps we have a control over its increase along the imaginary axis or along any vertical line and we have this property that the row minus row one will be just for those zeros for which the value is naturally very near to one then for those zeros will be zero and so I go back to this that the polynomial will be zero at this points which are very near no the polynomial will be zero for those zeros which are not very near to the distinguished row one in this gamma less than gamma one less than five and the polynomial will be at the places row minus row one nearly equal to each other so that means the residues will be nearly equal to each other in this small group for the small group of zeros. Now one can show relatively easily that this weight function which is defined by an integral what we can do that if a is large then we shift the line of integration onto the left if a is small we shift the line of integration on into the right and if a is so to say in between that means not too small not too large then we integrate on the line on the imaginary line itself and then due to this factor which is sigma square minus t square in absolute value this will be very small if we are far at least the distance log y from the zero p s takes care to eliminate the residues at the zeros which are near to row one but not very near to row one and this factor will take care for others but now considering just the weight it is still this property is not needed just the way how quickly the polynomial p might increase and what we get will be that we will get either a very small quantity lambda is just log y so this would be the same as y to minus a for example and this will be very small if a is large this will be very small if a is small and in between we have again the same e to the l which is up to some extent lost in the procedure and this is the way how we show this and so that means it is a technical question now if we know this weight function then we can already estimate the weight of mean value of our original u y which was defined earlier in this way so that means delta x divided by x to one plus row one multiplied by these weight functions and then this weight function has the property that it localizes the whole integral just for the range y times e to the minus l to y times e to the 2l apart from a loss of factor e to the l and so that means that we can really forget about the tails of this integration and we get a localized average of the delta and this is exactly this d y what we will in this way we can connect the localized average with the full average of u y now if we go further to v y then this v y can be which is a weight of mean value of the remainder term can be expressed by integral and this integral is can be proved easily by partial summation we get the logarithmic derivative of zeta there and this means that the original mean value as we see here if we change the integration in this way then we get an integration for the function h s plus row one so we get an integral here in this way and we see that the residues occur at the places then s plus row one is equal to row that means if s is equal to row minus row one and this g s is small if row one is more far than five already from the original point row one this p is equal to zero that means g is also equal to zero if row is not in R but I forgot to write here it's here if so it's a distance is at least at most five in this case this polynomial p takes care for eliminating the residues at those zeros which are not in R so these are the zeros which are at the distance at most five maybe quite near to row one but not very near in the sense that not in this small group are so in this place is the function is analytic and this means that we get actually type of we can call it a power sum but I'm not quite precise here because I should write here that row is not equal in R plus oh no no that's okay that for all the zeros which are not equal to not in this size very near to R and but fortunately let me see oh not in R yes so that means okay yeah please continue Janosch Christian should ask the question after your talk yes that's thank you thank you so that means in this way we have the residues for all the zeros of the zeta function which are not near not very near to the original one and in this way we will get so we can call it a power sum but the value of this to evaluate this power sum is very easy excuse me for a moment yeah yeah so UV is yeah so UV is contains zeros which are so this is not not correct at all not equal to R yeah this is not correct we have zeros which are very near to R plus we have zeros which are at least the distance from R so that means these are not equal to R not in R these are aranials maybe the question refer to this so we are some zeros which are so if we consider the zeros which are so to say at the distance at least log y then we get y to minus two as an upper bound here if we consider zeros which are at at least five from it then again due to another part of the g function we get again something relatively small not so small like here but relatively small now what appears still still what remains still is that for those zeros which are in R we get asymptotically one for for each zeros and so we can evaluate easily the power sum for R and as I mentioned the zeros which are not in R but at a distance at most five they don't appear at all and in this way we get for this power sum we get exactly the lower estimate and we get by this lower estimate we can prove the original CRM as well and thank you for your attention
 So, thank you for the invitation and I will then switch to full screen mode, which means  I will not see myself, but this is not a big problem.  And the topic, what I am talking about is large values of the remainder term of the  prime number CRM and I will immediately see what I mean exactly by these large values.  First, we have to remember the explicit formula for primes and then delta x is the remainder  term of the prime number CRM and we see that this depends on zeros of the zeta function  and which I will always denote by beta plus i gamma and we can restrict our attention  to zeros with height less than x and then we see that the oscillation, what I can say  caused alone by a zero row is x to the beta divided by the modulus of row, which means  that asymptotically it is x to the beta divided by gamma.  Now little would raise the problem, which perhaps is not clear why he raised it, prove  an explicit oscillation result for the remainder term, if we suppose the existence of a hypothetical  zero with real part at least one half and for simplicity with imaginary part positive.  Now the reason why he raised this problem was that earlier the best result was a consequence  of the fragment Lindelof CRM and it had two drawbacks.  One was that the bound was just x to the beta minus epsilon, so smaller than the oscillation  caused by the zero and secondly it were even this weaker bound was ineffective.  So that means if we even knew the existence of a concrete zero or a hypothetical zero,  we could not see, could not say from what point on and what size is this oscillation.  And this problem was then, this is due to the possible interference of the set of zeros,  which means that if I consider some of complex numbers, then I cannot be sure that the sum  of these complex numbers can be estimated from below by any term of this sum.  So that means if we have positive real numbers, then actually we can estimate from below the  sum by any term of it, but if we have complex numbers, then this is naturally not the case.  Now in order to settle this case, this was, I don't know whether for which particular  problem, but actually in order to settle the problem of interference of complex zeros, Turan  developed his method Turan's power sum method and this was the most frequent tool used in  these investigations as well.  Now just to remark that if we consider zeros on the critical line, which we know that there  are a lot of zeros there, then we get an even larger oscillation than the one caused by  any single zero itself.  Maybe we can gain a factor three times the diluted logarithm of x, which I will denote  by log 3x and even this oscillation occurs in most plus and minus direction, in most  direction, which actually disproved type of conjecture of Riemann, which appeared in  his work, where he showed that at least the second form of the remainder term, which appears  here, this pi x minus li x is oscillating in both directions.  Riemann showed that pi x is always less than li x and this was just in the same year 1914  and it proved that this oscillates in both directions, which is a size even bigger than  the oscillation caused by a single zero.  Interestingly, in the same year, Lehmann showed that at least for up to 10 million,  pi x is less than li x and now we know that at least until 10 to 20 really pi x is less  than li x.  But we will not concentrate on sign changes now, so we will concentrate just on the size  of the absolute value of the remainder term and in this case, it is not so critical which  form of the remainder term we consider.  So we will work with this delta x, which contains as you see the prime powers as well and therefore  it doesn't appear this negative effect, which makes pi x less than li x usually.  Now if we define with dy the average of the modulus of delta x and with sy, so phi, the  maximum values of the modulus of the remainder term for x less than y, then to run succeeded  in 1950 to solve the problem and the lower bound is really bigger than x y to better  not minus epsilon.  So there is a concrete explicit function and it is also explicitly given independence  of the row from which point this estimate is true.  Now some eight years later or nine years later, now we will prove the same for the average  of the remainder term.  The method in both worlds, the most critical part is to run spores some methods I mentioned  or edited complex numbers if we have complex numbers and there are some might be much smaller  than the largest term for example.  And the two round work the other method which up to some extent helped to face with this  situation.  So in some sense with a factor what we will see later he succeeded to neglect this inter  possible interference of zeros.  So the result of Turan which was used in both his proof and Knapov's proof was the so-called  second mean theorem of Turan's power some method that if we consider power some of n  complex numbers, we use a normalization that the largest of this number has modulus one.  And then if we consider we cannot state naturally that for every new this value should be large,  but what we can state and this is what his second main theorem states that if we consider  at least n values and consecutive values of this exponent, then apart from a very significant  factor which is much less than one, we will reach the bound one and we have still a factor  depending on these coefficients bj.  Now in the present problem we cannot we don't need any coefficients.  So that means we have a special case of this Turan second mean theorem which can be written  in this case so that the new power some of n complex numbers taken if you take the maximum  value for n consecutive values.  And then this is a function which depends exponentially on the number of variables and  it depends somewhat less drastically on the beginning of this interval.  Now one can easily see that if we consider for example nth root of unity, then we have  n minus one consecutive values always which are exactly zero and then we have one which  is large.  So that means in some sense this result is optimal and we don't know whether the lower  bound is optimal or not.  I used not exactly the same result but a similar argument that means it was more generalization  of Dirichlet's approximation theorem to show that if we consider oscillation then this  oscillation is really apart from a factor one minus epsilon, the oscillation is x to  the better not divided by rho not in modulus.  So that means the oscillation caused by a zero occurs really from time to time and this  seemed to settle so to say to be precise almost completely the oscillation problem of delta  x.  And here this function c2 of rho not and epsilon was effective as well.  The proof used Turan's method in not Turan's power sums but it was similar to the way Turan  and Kanapovsky handled their proof their results but the difference was that I used  another approximation theorem and I used still the general framework of Turan's result.  Now immediately there is a question that what is the optimal oscillation result because  we know that we have two zeros to conjugate zeros of the zeta function always and therefore  I thought also that one minus epsilon could be replaced by two minus epsilon and this  would be really the case if rho not was isolated if we would not have nearby other zeros and  then the question was up to some extent or almost completely really settled by Silad  Reves in 83 he showed that we don't know what is the optimal constant for the zeta function  naturally if the Riemann hypothesis is true then the optimal constant is infinity.  We remember that we have then an oscillation of size log log x times the log log x times  the term of any zero on the critical line and so then this question is interesting just  when Riemann hypothesis is false and so therefore it is very hard to say even how to say mathematically  precisely what we mean by the optimal constant.  Now he proved that the optimal constant is pi over 2 which meant the pi over 2 the result  which I stated was true this pi over 2 minus epsilon instead of one minus epsilon on one  hand and on the other hand if we consider a larger by the class of functions which include  the zeta function as well then at least for this larger class of functions there are some  functions for which pi over 2 is really the optimal constant.  We could formulate it also that there could appear such intricate configuration of zeros  of the zeta function so that the oscillation would be really not exceed pi half minus epsilon  times the expected and proved value x to the rho naught over rho naught.  Now I proved also several results on these other methods both for the average and the  maximum value thereby improving the earlier results of Turan and Knapovsky also Schrager  Puchta showed such an improvement and what our goal is that naturally different forms  of these CRMs had different conditions concerning the dependence of y from gamma naught that  means from if rho naught is given and then from which point on can we assert that the  average or the maximum value is large then a lower bound as a function of y and gamma  naught that means how large is really the average or the maximum dependent on gamma  naught.  A certain problem is that whether this large values occur just somewhere between one and  y we can state that the average or the maximum is large or we can say that they occur up  to some extent near to y so that means near to y means unfortunately not near on the normal  scale but near on the logarithmic scale so that means that the logger if for an interval  of the form a y to y and this beginning of the interval a y should be at least on the  logarithmic scale near to log y.  Now again we can consider effective or ineffective estimate and another method of mine gave a  result which produced a lower bound which depended on the derivative on the absolute  value of the derivative at the place rho naught which for concrete zeros which we know is  really completely okay but it would be already somewhat disturbing so it would make the CRM  let's say a little bit ineffective if we speak about a hypothetical zero and this could cause  problems in some applications.  Now I mentioned already that the really difficult case is Bendy and the really interesting case  is Bendy rho naught is not on the critical line that means the Riemann hypothesis is  really false.  Now I mentioned just some results in this direction I proved that actually we can always  reach that not only we have values as large as y to the better not divided by y to the  better not divided by rho naught or x to the better not divided by rho naught sometimes  but also that the average is at least y to the better not multiplied by some constant  depending on the imaginary part of the zero and this has a corollary that on the Riemann  hypothesis the average of the remainder term in modulus is at least square root y or the  average is square root x.  Now Krama showed already exactly 100 years ago that on the other hand the average of  the remainder term in absolute value is at most constant times square root y so we know  exactly apart from a constant the order of magnitude of the average order of the remainder  term supposing the Riemann hypothesis this is this estimate here and on the other hand  this is still unclear the maximum value of the remainder term between these two bounds.  Now this actually served as an introduction for the CRM for which I will also sketch the  proof namely that if we have a concrete zero of the zeta function and we know that y is  enough large depending on an explicit way from the imaginary part of the zeta I just  mentioned that already I mentioned that but what emphasize that all these constants c  will be explicit constants.  In this case we can localize we can consider the average of the remainder term in modulus  and localize in the average in the logarithm scale from a beginning point near to y until  y and we get for it a lower estimate which is just with a factor log log y square smaller  x minus log log y square smaller than the expected value and concerning the localization  I lose again the same log log y square quantity.  Now the strategy is that we consider weighted mean value that means as I mentioned already  concentrated to interval a y y where log y is near to log y we already specified that  actually this a y would be this quantity so log a y is less than log y this quantity log  log y square.  Now this weighted mean value can be transferred into a sum of residues containing the zeros  of the zeta function and up to this point the strategy is same as by Turan, Knapovsky  and in my earlier works.  The novelty is that the weight is chosen so that only zeros very near from row not should  get non negligible weights and in this way we can avoid both Turan's method and both  the loss which is caused by using Turan's second mean theorem.  This means this quantity what we have here this is actually a loss compared with the  expected value and naturally due to this averaging process and due to this procedure of getting  rid of other zeros near to row not we have also a loss in the procedure but this loss  is just a function x minus log log y squared various in the works of Turan and Knapovsky  for example they had a loss of log y divided by log log y and we have instead log log y  squared so we have a much smaller loss than in the procedure of Turan and Knapovsky.  Now in the present method we will what I said we will get rid of zeros not very near to  the critical given zero row not so that means we will consider the construct of weight function  which is asymptotically the same for a small group by group I mean just the set so there  is no group operation here so for a small set R of zeta zeros near to row not it is  asymptotically the same naturally these zeros are very near to row not then at the distance  basically this C naught we can say it's five so at the distance at most five for all the  other zeros the weight function will be just zero and in this way those residues will not  appear at all you know and due to other part of the weight function the weights there will  be not zero but negligible the weights for the residues if we are more far than five  from this point now the in fact the original CRM was formulated in such a way that the  upper bound was by now we consider in on the logarithmic scale an interval symmetric to  by and this will be our estimate and here what we should possibly keep in mind is that  script L is log log y squared lambda is log y and L is log log y so capital L is log log  y lambda is log y and script L is log log y squared and script L is which on in the  exponential scale appears both in the loss and the both in the localization and this  will be the exact form in which I will prove our result and the beginning is a procedure  what was observed by Turan that if we have a zero and we have near to that zero near  means just that the on the logger on the the height is not too far from it but the zero  is little bit more on to the right then we take this other zero instead of our original  zero and so with step by step but the number of steps is at least at the most just log  log y or we can log y we can go further to another zero what we call an extreme right  hand zero and this extreme right hand zero has the property that its real part is at  least as large as the real part of the original zero so in this oscillation CRM which has  gained by it the imaginary part is can be somewhat larger but this is not decisive and  still we have a control about it the original imaginary part was at most square root of  lambda and by this procedure it will be just lambda squared and lambda is just log y so  that means it will be still near to log y log y squared as at most but on the other hand  more on the right and this is an important point of view that for this zero we have already  if we go a little bit far to the right with a quantity one over log y then we will have  a large so if we go a little bit more to the right then already in the how to say vicinity  but not vicinity but in a rather large height so that means three times log y higher or  three times log y lower there is no zero as at all so that means in some direction and  at least we solved the isolation of row not but we are very far from saying that the row  is an isolated zero so this procedure is just how to say routine used always by Turan and  Knobsky and also myself in these works and this is just a nearly trivial starting point  for the procedure and if so that means our final zero is this gamma wave K beta wave  K row K wave this is what we substitute the original row not by this zero but then comes  the idea is that we take a very small epsilon and then we consider zeros which can be reached  from one from the other by a chain of zeta zeros so that neighboring zeta zeros are at  a distance at most epsilon from each other but this epsilon is not a small absolute  constant but this epsilon is something which is very small so first we consider already  zeros which could be reached from each other by a chain of zeros at most with a distance  at most as you see one over log y and this is even in the denominator multiplied so  that means this one over log log y to the third power so these zeros are very near to  each other if in this group there would be just one zero then this wave could be right  called an isolated zero already and then we could so to say go further with this procedure  but what we know just is that it might be so that means definitely the starting set of  zeros was contained at most log log y that means L zeros this is by the result of back  line and this was later we can so to say nest into each other smaller and smaller groups  of zeta zeros so that they would be very near to each other but any zero outside that small  set would be much more far from these zeros than those zeros are from each other so that  means this whole we repeat the same procedure and we take all these sets which of zeros  which could be reached so equivalence classes of the zeros which could be reached from other  by a chain with a smaller epsilon to smaller means by a factor log log y third smaller  and if we always consider and continue our work from the smallest set then this whole  procedure must finish quite soon so the number of step should be just log L that means much  much smaller and in this way in this way we will get very soon just really not an isolated  but a set of zeros what we will call our by our this the property that they are very near  to each other any other zero is already much further from it much further means still it  may be a small distance but much further with a power with a factor log log y to the second  power and then we consider an arbitrary zero from this set our and we will have so that  means basically the same extreme right hand property will be true for this zero maybe  with a little bit changed parameters and after which we consider polynomial we construct  such a polynomial which is so to say since the residues will occur at the races robin  minus row prime if so that means in this way we will get such a polynomial that if we consider  any other zero which is not in this small group but at most the distance five from it  then we will have a property that will be interesting in our case so that means if we  are not in this small set of of equivalent zeros then the polynomial is zero and we have  so this is one thing for those zeros which are in this small group this polynomial at  the place row minus row one is asymptotically the same exactly one one plus or the one over  log log y and we know bound for the value of this polynomial at different places a roughly  speaking e to the l or s absolute value to the l so on in an exponential scale we have  this l which is log log y squared l squared ten times or forty times i don't know how  a hundred times i don't know this is the value of l and this will be equal to 40 times l  will be the actual loss in the procedure and so this l will have an important role but  that means we have a control over the size of the polynomial and here comes really the  weighted mean value of delta we consider this delta x divided by x to one plus row one times  weight on the value y over x so that means this is also in some sense on the logarithmic  scale and the weight function is defined together with this gs function and the gs function  has three factors the first factor is relatively small if we are let's say at a distance at  least five and at most log y from the zero then if we are at least a distance log y from  the zero then this factor will be very small that means negligible and with this ps we  have a control over its increase along the imaginary axis or along any vertical line  and we have this property that the row minus row one will be just for those zeros for which  the value is naturally very near to one then for those zeros will be zero and so I go back  to this that the polynomial will be zero at this points which are very near no the polynomial  will be zero for those zeros which are not very near to the distinguished row one in  this gamma less than gamma one less than five and the polynomial will be at the places row  minus row one nearly equal to each other so that means the residues will be nearly equal  to each other in this small group for the small group of zeros.  Now one can show relatively easily that this weight function which is defined by an integral  what we can do that if a is large then we shift the line of integration onto the left  if a is small we shift the line of integration on into the right and if a is so to say in  between that means not too small not too large then we integrate on the line on the imaginary  line itself and then due to this factor which is sigma square minus t square in absolute  value this will be very small if we are far at least the distance log y from the zero  p s takes care to eliminate the residues at the zeros which are near to row one but not  very near to row one and this factor will take care for others but now considering just  the weight it is still this property is not needed just the way how quickly the polynomial  p might increase and what we get will be that we will get either a very small quantity lambda  is just log y so this would be the same as y to minus a for example and this will be  very small if a is large this will be very small if a is small and in between we have  again the same e to the l which is up to some extent lost in the procedure and this is the  way how we show this and so that means it is a technical question now if we know this  weight function then we can already estimate the weight of mean value of our original u  y which was defined earlier in this way so that means delta x divided by x to one plus  row one multiplied by these weight functions and then this weight function has the property  that it localizes the whole integral just for the range y times e to the minus l to  y times e to the 2l apart from a loss of factor e to the l and so that means that we can really  forget about the tails of this integration and we get a localized average of the delta  and this is exactly this d y what we will in this way we can connect the localized average  with the full average of u y now if we go further to v y then this v y can be which is  a weight of mean value of the remainder term can be expressed by integral and this integral  is can be proved easily by partial summation we get the logarithmic derivative of zeta  there and this means that the original mean value as we see here if we change the integration  in this way then we get an integration for the function h s plus row one so we get an  integral here in this way and we see that the residues occur at the places then s plus  row one is equal to row that means if s is equal to row minus row one and this g s is  small if row one is more far than five already from the original point row one this p is  equal to zero that means g is also equal to zero if row is not in R but I forgot to write  here it's here if so it's a distance is at least at most five in this case this polynomial  p takes care for eliminating the residues at those zeros which are not in R so these are  the zeros which are at the distance at most five maybe quite near to row one but not very  near in the sense that not in this small group are so in this place is the function is analytic  and this means that we get actually type of we can call it a power sum but I'm not quite  precise here because I should write here that row is not equal in R plus oh no no that's  okay that for all the zeros which are not equal to not in this size very near to R and but  fortunately let me see oh not in R yes so that means okay yeah please continue Janosch  Christian should ask the question after your talk yes that's thank you thank you so that  means in this way we have the residues for all the zeros of the zeta function which are  not near not very near to the original one and in this way we will get so we can call  it a power sum but the value of this to evaluate this power sum is very easy excuse me for a  moment yeah yeah so UV is yeah so UV is contains zeros which are so this is not  not correct at all not equal to R yeah this is not correct we have zeros which are very  near to R plus we have zeros which are at least the distance from R so that means these are  not equal to R not in R these are aranials maybe the question refer to this so we are some zeros  which are so if we consider the zeros which are so to say at the distance at least log y  then we get y to minus two as an upper bound here if we consider zeros which are at  at least five from it then again due to another part of the g function we get again something  relatively small not so small like here but relatively small now what appears still  still what remains still is that for those zeros which are in R we get asymptotically one  for for each zeros and so we can evaluate easily the power sum for R and as I mentioned the zeros  which are not in R but at a distance at most five they don't appear at all and in this way  we get for this power sum we get exactly the lower estimate and we get by this lower estimate we  can prove the original CRM as well and thank you for your attention
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So, thank you for the invitation and I will then switch to full screen mode, which means I will not see myself, but this is not a big problem. And the topic, what I am talking about is large values of the remainder term of the prime number CRM and I will immediately see what I mean exactly by these large values. First, we have to remember the explicit formula for primes and then delta x is the remainder term of the prime number CRM and we see that this depends on zeros of the zeta function and which I will always denote by beta plus i gamma and we can restrict our attention to zeros with height less than x and then we see that the oscillation, what I can say caused alone by a zero row is x to the beta divided by the modulus of row, which means that asymptotically it is x to the beta divided by gamma. Now little would raise the problem, which perhaps is not clear why he raised it, prove an explicit oscillation result for the remainder term, if we suppose the existence of a hypothetical zero with real part at least one half and for simplicity with imaginary part positive. Now the reason why he raised this problem was that earlier the best result was a consequence of the fragment Lindelof CRM and it had two drawbacks. One was that the bound was just x to the beta minus epsilon, so smaller than the oscillation caused by the zero and secondly it were even this weaker bound was ineffective. So that means if we even knew the existence of a concrete zero or a hypothetical zero, we could not see, could not say from what point on and what size is this oscillation. And this problem was then, this is due to the possible interference of the set of zeros, which means that if I consider some of complex numbers, then I cannot be sure that the sum of these complex numbers can be estimated from below by any term of this sum. So that means if we have positive real numbers, then actually we can estimate from below the sum by any term of it, but if we have complex numbers, then this is naturally not the case. Now in order to settle this case, this was, I don't know whether for which particular problem, but actually in order to settle the problem of interference of complex zeros, Turan developed his method Turan's power sum method and this was the most frequent tool used in these investigations as well. Now just to remark that if we consider zeros on the critical line, which we know that there are a lot of zeros there, then we get an even larger oscillation than the one caused by any single zero itself. Maybe we can gain a factor three times the diluted logarithm of x, which I will denote by log 3x and even this oscillation occurs in most plus and minus direction, in most direction, which actually disproved type of conjecture of Riemann, which appeared in his work, where he showed that at least the second form of the remainder term, which appears here, this pi x minus li x is oscillating in both directions. Riemann showed that pi x is always less than li x and this was just in the same year 1914 and it proved that this oscillates in both directions, which is a size even bigger than the oscillation caused by a single zero. Interestingly, in the same year, Lehmann showed that at least for up to 10 million, pi x is less than li x and now we know that at least until 10 to 20 really pi x is less than li x. But we will not concentrate on sign changes now, so we will concentrate just on the size of the absolute value of the remainder term and in this case, it is not so critical which form of the remainder term we consider. So we will work with this delta x, which contains as you see the prime powers as well and therefore it doesn't appear this negative effect, which makes pi x less than li x usually. Now if we define with dy the average of the modulus of delta x and with sy, so phi, the maximum values of the modulus of the remainder term for x less than y, then to run succeeded in 1950 to solve the problem and the lower bound is really bigger than x y to better not minus epsilon. So there is a concrete explicit function and it is also explicitly given independence of the row from which point this estimate is true. Now some eight years later or nine years later, now we will prove the same for the average of the remainder term. The method in both worlds, the most critical part is to run spores some methods I mentioned or edited complex numbers if we have complex numbers and there are some might be much smaller than the largest term for example. And the two round work the other method which up to some extent helped to face with this situation. So in some sense with a factor what we will see later he succeeded to neglect this inter possible interference of zeros. So the result of Turan which was used in both his proof and Knapov's proof was the so-called second mean theorem of Turan's power some method that if we consider power some of n complex numbers, we use a normalization that the largest of this number has modulus one. And then if we consider we cannot state naturally that for every new this value should be large, but what we can state and this is what his second main theorem states that if we consider at least n values and consecutive values of this exponent, then apart from a very significant factor which is much less than one, we will reach the bound one and we have still a factor depending on these coefficients bj. Now in the present problem we cannot we don't need any coefficients. So that means we have a special case of this Turan second mean theorem which can be written in this case so that the new power some of n complex numbers taken if you take the maximum value for n consecutive values. And then this is a function which depends exponentially on the number of variables and it depends somewhat less drastically on the beginning of this interval. Now one can easily see that if we consider for example nth root of unity, then we have n minus one consecutive values always which are exactly zero and then we have one which is large. So that means in some sense this result is optimal and we don't know whether the lower bound is optimal or not. I used not exactly the same result but a similar argument that means it was more generalization of Dirichlet's approximation theorem to show that if we consider oscillation then this oscillation is really apart from a factor one minus epsilon, the oscillation is x to the better not divided by rho not in modulus. So that means the oscillation caused by a zero occurs really from time to time and this seemed to settle so to say to be precise almost completely the oscillation problem of delta x. And here this function c2 of rho not and epsilon was effective as well. The proof used Turan's method in not Turan's power sums but it was similar to the way Turan and Kanapovsky handled their proof their results but the difference was that I used another approximation theorem and I used still the general framework of Turan's result. Now immediately there is a question that what is the optimal oscillation result because we know that we have two zeros to conjugate zeros of the zeta function always and therefore I thought also that one minus epsilon could be replaced by two minus epsilon and this would be really the case if rho not was isolated if we would not have nearby other zeros and then the question was up to some extent or almost completely really settled by Silad Reves in 83 he showed that we don't know what is the optimal constant for the zeta function naturally if the Riemann hypothesis is true then the optimal constant is infinity. We remember that we have then an oscillation of size log log x times the log log x times the term of any zero on the critical line and so then this question is interesting just when Riemann hypothesis is false and so therefore it is very hard to say even how to say mathematically precisely what we mean by the optimal constant. Now he proved that the optimal constant is pi over 2 which meant the pi over 2 the result which I stated was true this pi over 2 minus epsilon instead of one minus epsilon on one hand and on the other hand if we consider a larger by the class of functions which include the zeta function as well then at least for this larger class of functions there are some functions for which pi over 2 is really the optimal constant. We could formulate it also that there could appear such intricate configuration of zeros of the zeta function so that the oscillation would be really not exceed pi half minus epsilon times the expected and proved value x to the rho naught over rho naught. Now I proved also several results on these other methods both for the average and the maximum value thereby improving the earlier results of Turan and Knapovsky also Schrager Puchta showed such an improvement and what our goal is that naturally different forms of these CRMs had different conditions concerning the dependence of y from gamma naught that means from if rho naught is given and then from which point on can we assert that the average or the maximum value is large then a lower bound as a function of y and gamma naught that means how large is really the average or the maximum dependent on gamma naught. A certain problem is that whether this large values occur just somewhere between one and y we can state that the average or the maximum is large or we can say that they occur up to some extent near to y so that means near to y means unfortunately not near on the normal scale but near on the logarithmic scale so that means that the logger if for an interval of the form a y to y and this beginning of the interval a y should be at least on the logarithmic scale near to log y. Now again we can consider effective or ineffective estimate and another method of mine gave a result which produced a lower bound which depended on the derivative on the absolute value of the derivative at the place rho naught which for concrete zeros which we know is really completely okay but it would be already somewhat disturbing so it would make the CRM let's say a little bit ineffective if we speak about a hypothetical zero and this could cause problems in some applications. Now I mentioned already that the really difficult case is Bendy and the really interesting case is Bendy rho naught is not on the critical line that means the Riemann hypothesis is really false. Now I mentioned just some results in this direction I proved that actually we can always reach that not only we have values as large as y to the better not divided by y to the better not divided by rho naught or x to the better not divided by rho naught sometimes but also that the average is at least y to the better not multiplied by some constant depending on the imaginary part of the zero and this has a corollary that on the Riemann hypothesis the average of the remainder term in modulus is at least square root y or the average is square root x. Now Krama showed already exactly 100 years ago that on the other hand the average of the remainder term in absolute value is at most constant times square root y so we know exactly apart from a constant the order of magnitude of the average order of the remainder term supposing the Riemann hypothesis this is this estimate here and on the other hand this is still unclear the maximum value of the remainder term between these two bounds. Now this actually served as an introduction for the CRM for which I will also sketch the proof namely that if we have a concrete zero of the zeta function and we know that y is enough large depending on an explicit way from the imaginary part of the zeta I just mentioned that already I mentioned that but what emphasize that all these constants c will be explicit constants. In this case we can localize we can consider the average of the remainder term in modulus and localize in the average in the logarithm scale from a beginning point near to y until y and we get for it a lower estimate which is just with a factor log log y square smaller x minus log log y square smaller than the expected value and concerning the localization I lose again the same log log y square quantity. Now the strategy is that we consider weighted mean value that means as I mentioned already concentrated to interval a y y where log y is near to log y we already specified that actually this a y would be this quantity so log a y is less than log y this quantity log log y square. Now this weighted mean value can be transferred into a sum of residues containing the zeros of the zeta function and up to this point the strategy is same as by Turan, Knapovsky and in my earlier works. The novelty is that the weight is chosen so that only zeros very near from row not should get non negligible weights and in this way we can avoid both Turan's method and both the loss which is caused by using Turan's second mean theorem. This means this quantity what we have here this is actually a loss compared with the expected value and naturally due to this averaging process and due to this procedure of getting rid of other zeros near to row not we have also a loss in the procedure but this loss is just a function x minus log log y squared various in the works of Turan and Knapovsky for example they had a loss of log y divided by log log y and we have instead log log y squared so we have a much smaller loss than in the procedure of Turan and Knapovsky. Now in the present method we will what I said we will get rid of zeros not very near to the critical given zero row not so that means we will consider the construct of weight function which is asymptotically the same for a small group by group I mean just the set so there is no group operation here so for a small set R of zeta zeros near to row not it is asymptotically the same naturally these zeros are very near to row not then at the distance basically this C naught we can say it's five so at the distance at most five for all the other zeros the weight function will be just zero and in this way those residues will not appear at all you know and due to other part of the weight function the weights there will be not zero but negligible the weights for the residues if we are more far than five from this point now the in fact the original CRM was formulated in such a way that the upper bound was by now we consider in on the logarithmic scale an interval symmetric to by and this will be our estimate and here what we should possibly keep in mind is that script L is log log y squared lambda is log y and L is log log y so capital L is log log y lambda is log y and script L is log log y squared and script L is which on in the exponential scale appears both in the loss and the both in the localization and this will be the exact form in which I will prove our result and the beginning is a procedure what was observed by Turan that if we have a zero and we have near to that zero near means just that the on the logger on the the height is not too far from it but the zero is little bit more on to the right then we take this other zero instead of our original zero and so with step by step but the number of steps is at least at the most just log log y or we can log y we can go further to another zero what we call an extreme right hand zero and this extreme right hand zero has the property that its real part is at least as large as the real part of the original zero so in this oscillation CRM which has gained by it the imaginary part is can be somewhat larger but this is not decisive and still we have a control about it the original imaginary part was at most square root of lambda and by this procedure it will be just lambda squared and lambda is just log y so that means it will be still near to log y log y squared as at most but on the other hand more on the right and this is an important point of view that for this zero we have already if we go a little bit far to the right with a quantity one over log y then we will have a large so if we go a little bit more to the right then already in the how to say vicinity but not vicinity but in a rather large height so that means three times log y higher or three times log y lower there is no zero as at all so that means in some direction and at least we solved the isolation of row not but we are very far from saying that the row is an isolated zero so this procedure is just how to say routine used always by Turan and Knobsky and also myself in these works and this is just a nearly trivial starting point for the procedure and if so that means our final zero is this gamma wave K beta wave K row K wave this is what we substitute the original row not by this zero but then comes the idea is that we take a very small epsilon and then we consider zeros which can be reached from one from the other by a chain of zeta zeros so that neighboring zeta zeros are at a distance at most epsilon from each other but this epsilon is not a small absolute constant but this epsilon is something which is very small so first we consider already zeros which could be reached from each other by a chain of zeros at most with a distance at most as you see one over log y and this is even in the denominator multiplied so that means this one over log log y to the third power so these zeros are very near to each other if in this group there would be just one zero then this wave could be right called an isolated zero already and then we could so to say go further with this procedure but what we know just is that it might be so that means definitely the starting set of zeros was contained at most log log y that means L zeros this is by the result of back line and this was later we can so to say nest into each other smaller and smaller groups of zeta zeros so that they would be very near to each other but any zero outside that small set would be much more far from these zeros than those zeros are from each other so that means this whole we repeat the same procedure and we take all these sets which of zeros which could be reached so equivalence classes of the zeros which could be reached from other by a chain with a smaller epsilon to smaller means by a factor log log y third smaller and if we always consider and continue our work from the smallest set then this whole procedure must finish quite soon so the number of step should be just log L that means much much smaller and in this way in this way we will get very soon just really not an isolated but a set of zeros what we will call our by our this the property that they are very near to each other any other zero is already much further from it much further means still it may be a small distance but much further with a power with a factor log log y to the second power and then we consider an arbitrary zero from this set our and we will have so that means basically the same extreme right hand property will be true for this zero maybe with a little bit changed parameters and after which we consider polynomial we construct such a polynomial which is so to say since the residues will occur at the races robin minus row prime if so that means in this way we will get such a polynomial that if we consider any other zero which is not in this small group but at most the distance five from it then we will have a property that will be interesting in our case so that means if we are not in this small set of of equivalent zeros then the polynomial is zero and we have so this is one thing for those zeros which are in this small group this polynomial at the place row minus row one is asymptotically the same exactly one one plus or the one over log log y and we know bound for the value of this polynomial at different places a roughly speaking e to the l or s absolute value to the l so on in an exponential scale we have this l which is log log y squared l squared ten times or forty times i don't know how a hundred times i don't know this is the value of l and this will be equal to 40 times l will be the actual loss in the procedure and so this l will have an important role but that means we have a control over the size of the polynomial and here comes really the weighted mean value of delta we consider this delta x divided by x to one plus row one times weight on the value y over x so that means this is also in some sense on the logarithmic scale and the weight function is defined together with this gs function and the gs function has three factors the first factor is relatively small if we are let's say at a distance at least five and at most log y from the zero then if we are at least a distance log y from the zero then this factor will be very small that means negligible and with this ps we have a control over its increase along the imaginary axis or along any vertical line and we have this property that the row minus row one will be just for those zeros for which the value is naturally very near to one then for those zeros will be zero and so I go back to this that the polynomial will be zero at this points which are very near no the polynomial will be zero for those zeros which are not very near to the distinguished row one in this gamma less than gamma one less than five and the polynomial will be at the places row minus row one nearly equal to each other so that means the residues will be nearly equal to each other in this small group for the small group of zeros. Now one can show relatively easily that this weight function which is defined by an integral what we can do that if a is large then we shift the line of integration onto the left if a is small we shift the line of integration on into the right and if a is so to say in between that means not too small not too large then we integrate on the line on the imaginary line itself and then due to this factor which is sigma square minus t square in absolute value this will be very small if we are far at least the distance log y from the zero p s takes care to eliminate the residues at the zeros which are near to row one but not very near to row one and this factor will take care for others but now considering just the weight it is still this property is not needed just the way how quickly the polynomial p might increase and what we get will be that we will get either a very small quantity lambda is just log y so this would be the same as y to minus a for example and this will be very small if a is large this will be very small if a is small and in between we have again the same e to the l which is up to some extent lost in the procedure and this is the way how we show this and so that means it is a technical question now if we know this weight function then we can already estimate the weight of mean value of our original u y which was defined earlier in this way so that means delta x divided by x to one plus row one multiplied by these weight functions and then this weight function has the property that it localizes the whole integral just for the range y times e to the minus l to y times e to the 2l apart from a loss of factor e to the l and so that means that we can really forget about the tails of this integration and we get a localized average of the delta and this is exactly this d y what we will in this way we can connect the localized average with the full average of u y now if we go further to v y then this v y can be which is a weight of mean value of the remainder term can be expressed by integral and this integral is can be proved easily by partial summation we get the logarithmic derivative of zeta there and this means that the original mean value as we see here if we change the integration in this way then we get an integration for the function h s plus row one so we get an integral here in this way and we see that the residues occur at the places then s plus row one is equal to row that means if s is equal to row minus row one and this g s is small if row one is more far than five already from the original point row one this p is equal to zero that means g is also equal to zero if row is not in R but I forgot to write here it's here if so it's a distance is at least at most five in this case this polynomial p takes care for eliminating the residues at those zeros which are not in R so these are the zeros which are at the distance at most five maybe quite near to row one but not very near in the sense that not in this small group are so in this place is the function is analytic and this means that we get actually type of we can call it a power sum but I'm not quite precise here because I should write here that row is not equal in R plus oh no no that's okay that for all the zeros which are not equal to not in this size very near to R and but fortunately let me see oh not in R yes so that means okay yeah please continue Janosch Christian should ask the question after your talk yes that's thank you thank you so that means in this way we have the residues for all the zeros of the zeta function which are not near not very near to the original one and in this way we will get so we can call it a power sum but the value of this to evaluate this power sum is very easy excuse me for a moment yeah yeah so UV is yeah so UV is contains zeros which are so this is not not correct at all not equal to R yeah this is not correct we have zeros which are very near to R plus we have zeros which are at least the distance from R so that means these are not equal to R not in R these are aranials maybe the question refer to this so we are some zeros which are so if we consider the zeros which are so to say at the distance at least log y then we get y to minus two as an upper bound here if we consider zeros which are at at least five from it then again due to another part of the g function we get again something relatively small not so small like here but relatively small now what appears still still what remains still is that for those zeros which are in R we get asymptotically one for for each zeros and so we can evaluate easily the power sum for R and as I mentioned the zeros which are not in R but at a distance at most five they don't appear at all and in this way we get for this power sum we get exactly the lower estimate and we get by this lower estimate we can prove the original CRM as well and thank you for your attention
values of the remainder term of the prime number developed his method Turan's power sum method and if we consider zeros on the critical line, which single zero. Interestingly, in the same year, size of the absolute value of the remainder term which point this estimate is true. Now some eight prove the same for the average of the remainder this is a function which depends exponentially on that I used another approximation theorem and I zeta function naturally if the Riemann hypothesis some results in this direction I proved that that y is enough large depending on an explicit until y and we get for it a lower estimate which given zero row not so that means we will consider one from the other by a chain of zeta zeros so