Negative moments of the Riemann zeta-function

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 Möbius function.

36 pages