2023/02/14 by Hung M. Bui, Alexandra Florea, Bui, Hung M. +1
Mathematics · #Analytic Number Theory Research #Meromorphic and Entire Functions #Analytic and geometric function theory
paper · pdf · doi:10.48550/arxiv.2302.07226
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 2T, 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\frac1α ≪ 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.