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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 $$a_n\sim \mu(n)(1-\log(n)/\log(N))$$ for Riemann zeta function.But the numerical computation shows that $$a_n\sim \mu(n)(1-c(n)\log(n)/\log(N)), $$where those coefficients c(n) behave quite irregularly.I draw histograms of c(n) and compute 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 experiment for primitive Dirichlet-L functions with small modulo q(q=3,4,5),$$a_n\sim \chi(n)\mu(n)(1-c(n)\log(n)/\log(N)), $$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 choose N=5000,for Dirichlet L-functions,I choose N=1000.It is beyond the capability of my laptop for larger N(etc,N=10000).

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1 Answer 1

up vote 9 down vote accepted

In this very nice recent paper by Bettin, Conrey and Farmer

http://arxiv.org/abs/1211.5191

it is proven that assuming some very reasonable hypothesis concerning $\zeta(s)$ that the optimal minimizing polynomial is in fact $$ \sum_{n \leq x} \frac{\mu(n)}{n^s} \cdot \bigg ( 1 - \frac{\log n }{\log x} \bigg ) $$ Now it is not certain that the minimizing polynomial is unique. I think you can perturb this polynomial slightly and it will still be a minimizing polynomial.

Concerning the assumption of this paper: Any investigation concerning the optimal Dirichlet polynomial in the Nyman-Beurling criterion will have to assume the Riemann Hypothesis and the simplicity of the zeros (in light of the lower bound obtained by Burnol). In the paper referenced above the simplicitly of the zeros is assumed in a stronger quantitative form, $$ \sum_{T \leq \gamma \leq 2T} \frac{1}{|\zeta'(\rho)|^2} \ll T^{3/2 - \delta} $$ This is still a very reasonable conjecture since we in fact expect that $$ \sum_{T \leq \gamma \leq 2T} \frac{1}{|\zeta'(\rho)|^2} \sim \frac{6}{\pi^3} T $$ This is a conjecture of Gonek. Here, lower bounds which are just a constant factor of $1/2$ away from the conjectured asymptotic have been obtained by Millinovich and Ng.

Possible explanation of the discrepancy with your numerical experiments: Even in the classical case, when you are mollifying the Riemann zeta-function, the optimal Dirichlet polynomial (the one that solves the optimization problem on the nose) has rather complicated arithmetic coefficients. However when you take the length of the Dirichlet polynomial to go to infinity, each coefficient tends to $(1 - \log n / \log x)$. One can then note that the polynomial with those coefficients is also a minimizing polynomial (it doesn't have to be, but luckily it is). I have once tried computing numerics with the two different polynomial. For small numbers they seem to behave quite differently when mollifying. I think what you are seeing here is basically a small numbers issue. This is something that happens a lot when dealing with the Riemann zeta-function or $L$-functions. I recommend the first few pages in the introduction to

http://arxiv.org/abs/0710.5176

for a very clear explanation of another situation where the numerics are misleading.

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Thanks for your informative answer. –  zy_ May 28 '13 at 4:18

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