Hi


is there any lower bound for $\Re\zeta(1+it)$. 


I did try with computer until some ordinate and I saw $\Re\zeta(1+it)>0$.


If it is true, is there any reference to prove it.
thanks



lower bound for $\Re\zeta(1+it)$
There is no lower bound for Re$(\zeta(1+it))$. This can be found in Lamzouri's paper <http://www.math.uiuc.edu/~lamzouri/distribzeta.pdf> (see e.g. page 3, formula (5)). (Choose $\tau$ large enough, $\theta\_1=3\pi/4$ and $\theta\_2=5\pi/4$ for and substract). The results of Lamzouri however also implies that on average the argument of $\zeta(1+it)$ is small and that Re$(\zeta(1+it))$ is positive more often than it is negative.


