It is well known that the Riemann Hypothesis is true iff $$\lim_{N \rightarrow \infty}d_N=0$$,where

$$d_{N}^{2}=\inf_{A_N}\frac{1}{2 \pi}\int_{-\infty}^{\infty}\vert 1-\zeta A_N(1/2+it)\vert^2\frac{dt}{1/4+t^2}$$

and the inf is over all the Dirichlet Polynomials of length N.I'm very interested in finding a minimizing polynomial $A_N=\sum_{n=1}^N\frac{a_n}{n^s}$, which is equivalent to minimizing a quadratic form.

I made several experiment with the method in this paper (Le critère de Beurling et Nyman pour l’hypothèse de Riemann: aspects numériques), and it is possible to extend this experiment to Dirichlet L-functions associated to primitive characters.

For a square-free number n, I expect that $$a_n\sim \mu(n)\left(1-\frac{\log n}{\log N}\right)$$ for Riemann zeta function. But the numerical computation shows that $$a_n\sim \mu(n)\left(1-c(n)\frac{\log n}{\log N}\right), $$where those coefficients c(n) behave quite irregularly. I drew histograms of c(n) and computed the average of c(n) over square-free numbers. The average of c(n) is far from 1. It is about 0.8. I made several experiments for primitive Dirichlet-L functions with small modulo q(q=3,4,5),$$ a_n\sim \chi(n)\mu(n)\left(1-c(n)\frac{\log n}{\log N}\right), $$ and the average of c(n) is still about 0.8 (the sum is over square-free number n, (n,q)=1).

**Question:** It is amazing that this minimizing polynomial is different from Selberg's mollifier (where c(n) is equal to 1).

Is it possible to give an explanation why the cofficients c(n) is close to 0.8 rather than close to 1?(RMT,etc.)? Is it possible that c(n) has limiting distribution?

p.s.

For Riemann zeta function, I chose N=5000, and for Dirichlet L-functions, I chose N=1000. It is beyond the capability of my laptop for larger N (etc,N=10000).