Negative moments of the Riemann zeta-function

TL;DR

This paper investigates negative moments of the Riemann zeta-function assuming the Riemann Hypothesis, deriving asymptotic formulas and bounds in specific ranges of the shift, advancing understanding of Gonek's conjecture.

Contribution

It provides new asymptotic formulas and bounds for negative moments of the zeta-function in certain ranges, partially confirming Gonek's conjecture.

Findings

Asymptotic formulas for negative moments when shift > 1/log T

Non-trivial upper bounds for smaller shifts

Progress towards Gonek's conjecture

Abstract

Assuming the Riemann Hypothesis we study negative moments of the Riemann zeta-function and obtain asymptotic formulas in certain ranges of the shift in $\zeta(s)$. For example, integrating $|\zeta(1/2+\alpha+it)|^{-2k}$ with respect to $t$ from $T$ to $2T$, we obtain an asymptotic formula when the shift $\alpha$ is roughly bigger than $\frac{1}{\log T}$ and $k < 1/2$. We also obtain non-trivial upper bounds for much smaller shifts, as long as $\log\frac{1}{\alpha} \ll \log \log T$. This provides partial progress towards a conjecture of Gonek on negative moments of the Riemann zeta-function, and settles the conjecture in certain ranges. As an application, we also obtain an upper bound for the average of the generalized M\"{o}bius function.