Negative moments of the Riemann zeta-function

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