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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(\frac{1}{2}+\alpha+it)|^{-2k}}with respect totfromTto {2T}, we obtain an asymptotic formula when the shift α is roughly bigger than {\frac{1}{\log T}}and {k<\frac{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.
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Large Deviation Estimates of Selberg’s Central Limit Theorem and Applications
Abstract For $$V\sim \alpha \log \log T$$ with $$0<\alpha <2$$, we prove $$\begin{align*} & \frac{1}{T}\textrm{meas}\{t\in [T,2T]: \log|\zeta(1/2+ \textrm{i} t)|>V\}\ll \frac{1}{\sqrt{\log\log T}} e^{-V^{2}/\log\log T}. \end{align*}$$This improves prior results of Soundararajan and of Harper on the large deviations of Selberg’s Central Limit Theorem in that range, without the use of the Riemann hypothesis. The result implies the sharp upper bound for the fractional moments of the Riemann zeta function proved by Heap, Radziwiłł, and Soundararajan. It also shows a new upper bound for the maximum of the zeta function on short intervals of length $$(\log T)^{\theta }$$, $$0<\theta <3$$, that is expected to be sharp for $$\theta> 0$$. Finally, it yields a sharp upper bound (to order one) for the moments on short intervals, below and above the freezing transition. The proof is an adaptation of the recursive scheme introduced by Bourgade, Radziwiłł, and one of the authors to prove fine asymptotics for the maximum on intervals of length $$1$$.
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- PAR ID:
- 10457115
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- ISSN:
- 1073-7928
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/imrn/rnad176
