The second shifted moment of the Riemann zeta function

TL;DR

This paper derives an asymptotic formula for the second shifted moment of the Riemann zeta function, considering shifts with small real parts and large imaginary parts up to nearly $T^2$, advancing understanding of zeta function behavior.

Contribution

It provides a new asymptotic formula for the second shifted moment of the Riemann zeta function with large imaginary shifts and small real parts, extending previous results.

Findings

Asymptotic formula for the second shifted moment established.

Handles shifts with imaginary parts up to $T^{2- ext{epsilon}}$.

Enhances understanding of the zeta function's moments in critical regions.

Abstract

We prove an asymptotic formula for the second moment (up to height $T$) of the Riemann zeta function with two shifts. The case we deal with is where the real parts of the shifts are very close to zero and the imaginary parts can grow up to $T^{2-\epsilon}$, for any $\epsilon>0.$