A Hybrid Euler-Hadamard product formula for the Riemann zeta function

TL;DR

This paper introduces a hybrid model combining the zeros and primes of the Riemann zeta function using a smoothed explicit formula and random matrix theory, providing insights into its moments on the critical line.

Contribution

It develops a novel hybrid Euler-Hadamard product formula for the zeta function, linking prime numbers and zeros through a statistical model involving random matrices.

Findings

Provides a heuristic for moments of the zeta function on the critical line.

Connects the zeta function's behavior with random matrix theory.

Offers a new perspective on conjectures related to the zeta function.

Abstract

We use a smoothed version of the explicit formula to find an approximation to the Riemann zeta function as a product over its nontrivial zeros multiplied by a product over the primes. We model the first product by characteristic polynomials of random matrices. This provides a statistical model of the zeta function that involves the primes in a natural way. We then employ the model in a heuristic calculation of the moments of the modulus of the zeta function on the critical line. This calculation illuminates recent conjectures for these moments based on connections with random matrix theory.