A mean-square value of the Riemann zeta function over an arithmetic progression

TL;DR

This paper derives an asymptotic formula for the second discrete moment of the Riemann zeta function over an arithmetic progression, showing it matches the continuous mean value's main term.

Contribution

It provides a new asymptotic formula for the discrete second moment of the zeta function over an arithmetic progression, linking discrete and continuous mean values.

Findings

The main term of the discrete second moment equals that of the continuous mean value.

An asymptotic formula for the second discrete moment is established.

The result bridges discrete and continuous analyses of the zeta function.

Abstract

We obtain an asymptotic formula for the second discrete moment of the Riemann zeta function over the arithmetic progression $\frac{1}{2} + in$. It shows that the first main term is equal to that of the continuous mean value.