Upper bounds for fractional joint moments of the Riemann zeta function

TL;DR

This paper derives upper bounds for joint moments of the Riemann zeta function and its derivative, aligning with predictions from random matrix theory, for specific powers and ranges.

Contribution

It provides new upper bounds for joint moments of the zeta function and its derivative, extending understanding in a range where predictions suggest sharpness.

Findings

Established upper bounds for joint moments of zeta and its derivative.

Results are consistent with random matrix theory predictions.

Bounds are valid for specified powers and ranges.

Abstract

We establish upper bounds for the joint moments of the $2k^{\text{th}}$ power of the Riemann zeta function with the $2h^{\text{th}}$ power of its derivative for $0 \leq h \leq 1$ and $1 \leq k \leq 2$. These bounds are expected to be sharp based upon predictions from random matrix theory.