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- Title
Negative moments of the Riemann zeta-function.
- Authors
Bui, Hung M.; Florea, Alexandra
- 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) . For example, integrating | ζ (1 2 + α + i t) | - 2 k 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.
- Subjects
RIEMANN hypothesis; MOBIUS function; ZETA functions
- Publication
Journal für die Reine und Angewandte Mathematik, 2024, Vol 2024, Issue 806, p247
- ISSN
0075-4102
- Publication type
Article
- DOI
10.1515/crelle-2023-0091