Now, I am new to functional analysis. So please dont be harsh.


I was going through some papers by Balazard & Saias, Baez-Duarte, etc. that discussed and delved deep into details of approximating the characteristic function $\chi(0,1]$.


Now, what appears to me is, if we can construct a function, using functions of the form $f(x) = \sum c\_k \rho(\frac{\theta\_k}{x})$ where $\rho(x)$ is the fractional part of $x$ that stay constant in the interval (0,1] and satisfying $\sum c\_k \theta\_k = 0$, then we have a proof of RH.


But then I saw that they have also discussed the function $g(x) = - \sum \mu(n) \rho(1/nx) = \chi(t)$ which they say holds for every positive $t$ but in the sense of pointwise convergence, and then stated that in the words of Balazard and Saias's paper,
*"Unfortunately, this feeling is, at least to a certain extent, a mirage"*


My question is:



> 
> Why is g(x) not the fulfilling the criteria of being that "function" which mathematicians are trying to find to serve as the approximation for $\chi(0,1]$? I would really appreciate any sort of clarification for this characteristic function that is being searched.
> 
> 
> 


Thanks,


The papers that I am reading includes:



> 
> The Nyman-Beurling equivalent form for the Riemann hypothesis -- Michel Balazard and Eric Saias
> 
> 
> Convergence and the Riemann hypothesis -- Jungseob Lee
> 
> 
> A Note on Nyman-Beurling's equivalent formulation of the Riemann hypothesis -- Jean Francois Burnol
> 
> 
> A Closure Problem related to the Riemann zeta funtion -- Arne Beurling
> 
> 
> New versions of the Nyman-Beurling criterion for the Riemann hypothesis
> -- Luis Baez Duarte
> 
> 
> 



On the Nyman-Beurling equivalent form for RH
The Nyman-Beurling Theorem is: $\zeta(s)$ has no zeroes for $\sigma>1/p$ if and only if the subspace of functions of the form $\sum\_kc\_k\rho(\alpha\_k/x)$, for $0\le\alpha\_k\le 1$ with $\sum\_k c\_k\alpha\_k$=0, is dense in $L^p(0,1)$ (which is equivalent to the closure containing the characteristic function $\chi\_{(0,1)}$).


The series $\sum\_k\mu(k)\rho\big(1/(kt)\big)$ converges to $\chi\_{(0,1)}(t)$ pointwise and in $L^1$, but not in $L^p$ for any $p>1$. Note that, in this formulation, $L^1$ convergence doesn't even imply nonvanishing on the edge of the critical strip. Similar sums have been tried. It's known that sums of this form converge very slowly (if at all, of course).


By the way, I gathered this from page 2 of [this paper of Baez-Duarte](http://arxiv.org/pdf/math/0202141v2) (plus a little extra googling).


