On the analytic extension of Random Riemann Zeta Functions for some probabilistic models of the primes

TL;DR

This paper investigates the analytic continuation of two probabilistic models of the Riemann Zeta Function, revealing domain limitations and extensions, with implications for understanding prime distributions and the Riemann Hypothesis.

Contribution

It introduces and analyzes two random models of the RZF using pseudo-primes, demonstrating their analytic continuation properties and extending the domain of continuation in one case.

Findings

Analytic continuation is possible almost surely for the Cramér model when Re s > 1/2.

In the second model with local symmetries, the continuation extends to a larger domain.

The study connects probabilistic models of primes with additive number theory problems.

Abstract

The first step in the formulation and study of the Riemann Hypothesis is the analytic continuation of the Riemann Zeta Function (RZF) in the full Complex Plane with a pole at $s=1$. In the current work, we study the analytic continuation of two random versions of RZF using, for $Re s>1$, the Euler representation of ZF in terms of the product of functions over primes. In the first case, we substitute in the Euler product pseudo-prime numbers from the famous Cram\'er Model. In the second case, we use pseudo-primes with local symmetries. We show that in the Cram\'er case analytic continuation is possible $\mathbb{P}$-a.s. for $Res>1/2$, but not through the critical line $Re s=1/2.$ In the second case, we show that the analytic continuation is possible in a larger domain. We also study for the Cram\'er pseudo-primes several problems from Additive Number Theory.