Moments of zeta and correlations of divisor-sums: I

TL;DR

This paper analyzes the calculation of moments of the Riemann zeta-function using divisor correlations, identifying limitations of existing methods and clarifying the structure of second and fourth moments.

Contribution

It provides a detailed analysis of moments of the zeta-function and highlights the specific terms missed by standard divisor correlation methods.

Findings

Calculated second and fourth moments of zeta on the critical line

Identified missing terms in standard divisor correlation approaches

Clarified the structure of moments and their calculation challenges

Abstract

We examine the calculation of the second and fourth moments and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations. Previously this approach has proved unsuccessful in computing moments beyond the eighth, even heuristically. A careful analysis of the second and fourth moments illustrates the nature of the problem and enables us to identify the terms that are missed in the standard application of these methods.