characterization of the primes as the minimizers of a certain J criterion

Question

I'm not an advanced number theorist but I think the following problem might be fun.

Let's consider a sorted set of real numbers $ \lambda = \{\lambda_k\}_{k \in \mathbb{N}}$, $\lambda_k \in [2;\infty[$ being thought as primes for a certain "pseudo-integer" set $G_\lambda$ (the set of real numbers which are product of powers of the $\lambda_k$)

From these "primes", we will be able to define an euler product and a zeta function as its analytic continuation :

$\zeta_\lambda(s) = \prod_k \frac{1}{1 - \lambda_k^{-s}}$

Then, let's define (if it converges) :

$J(\lambda,\sigma) = \frac{1}{2i \pi} \ \int_{\sigma-i \infty}^{\sigma + i\infty} \left| \frac{\zeta_\gamma(s)}{s} - \frac{1}{s - 1} + \frac{1}{2 s} \right|^2 ds$

The problem is to find which set $\lambda$ minimizes $J(.,\sigma)$

I thought that if $\sigma = 1$ then the natural primes could be the minimizers, but now I think it's not.

This is because I am thinking to the totally un-rigorous equality :

$J(\lambda,\sigma) = J_2(\lambda,\sigma) = \int_1^\infty \{x\}_\lambda^2 \frac{dx}{x^{2\sigma + 1}} $

where $\{x\}_\lambda = x - 1/2 - E_\lambda(x)$ and $E_\lambda(x) = \sum_{g \in G_\lambda, g \le x} 1$

Now I think that all we can say is that when $\sigma \to 0^+$ the set minimizing $J(.,\sigma)$ is as close to $\mathcal{P}$ as wanted for any "between sets" distance. Again I say that because I am thinking to :

$J_2(\lambda, \epsilon) = \int_1^\infty \{x\}_\lambda^2 \frac{dx}{x^{1 + 2 \epsilon}} $

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The idea behind all this message is that if we write :

$\zeta_\lambda(a,s) = \prod \frac{1}{1 - (\lambda_k + a)^{-s}}$

where the $\lambda_k$ are minimizing $J$, then we can hope to write something like :

$\frac{\partial J}{\partial a}_{(a = 0)} = 0$

with

$\frac{\partial \zeta_\lambda(a,s)}{\partial a}_{(a = 0)} = s \zeta_\lambda(s) \sum \frac{\lambda_k^{-s-1}}{ 1 - \lambda_k^{-s}}$

Then there is the extended problem :

$\min$ with respect to $\lambda$ : $ \ J_F(\lambda,\sigma) = \frac{1}{2i \pi} \ \int_{\sigma-i \infty}^{\sigma + i\infty} \left| \frac{\zeta_\gamma(s)}{s} - F(s) \right|^2 ds$

This leads to a huge set $H$ of zeta-function, $\zeta(s)$ being one of them for at least $F(s) = \zeta(s) / s$.

For each $\sigma,F$ what is the $\lambda$ pseudo-prime set minimizing $J$ ? Do these pseudo-prime sets share some common properties with the natural primes ? In particular, do some subsets of $H$ (related to some $F(s)$ classes) have their own Riemann hypothesis ?

Answer 1

About the RH seen as a minimization problem, see the Nyman-Beurling-Baez-Duarte criterion. The major idea with this approach is that we don't even need to mention the primes.

We start from

RH is true iff for every $\Re(s) > 1/2$ : $$\lim_{N \to \infty} 1-\zeta(s)\sum_{n=1}^N \mu(n) n^{-s} = 0$$

that we can generalize with

RH is true iff there is a sequence of Dirichlet polynomials $A_N(s) =\sum_{n=1}^N a_{n,N} n^{-s}$ such that for every $\Re(s) > 1/2$ : $$\lim_{N \to \infty} 1-\zeta(s)A_N(s) = 0$$ the convergence being locally uniform,

which leads to

$$\lim_{N \to \infty} \int_{\sigma-i\infty}^{\sigma+i\infty} \frac{|1- \zeta(s)A_N(s)|^2}{|s(s-1)|} ds = 0, \qquad (\sigma > 1/2)$$

And with some work, looking carefuly at $\sum_{n=1}^N \mu(n) n^{-1/2-it}$ under the assumption that RH is true, we obtain that $\sigma = 1/2$ works, ie.

RH is true iff there is a sequence of Dirichlet polynomials $A_N(s)$ such that $$\lim_{N \to \infty} \int_{-\infty}^{\infty} \frac{|1- \zeta(1/2+it)A_N(1/2+it)|^2}{t^2+1/4} dt = 0$$