Negative moments of the Riemann zeta-function

Abstract

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 ζ ⁢ ( s ) {\zeta(s)} . For example, integrating | ζ ⁢ ( 1 2 + α + i ⁢ t ) | - 2 ⁢ k {|\zeta(\frac{1}{2}+\alpha+it)|^{-2k}} with respect to t from T to 2 ⁢ T {2T} , we obtain an asymptotic formula when the shift α is roughly bigger than 1 log ⁡ T {\frac{1}{\log T}} and k < 1 2 {k<\frac{1}{2}} . We also obtain non-trivial upper bounds for much smaller shifts, as long as log ⁡ 1 α ≪ log ⁡ log ⁡ T {\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.