A weighted one-level density of the non-trivial zeros of the Riemann zeta-function

2022/08/17 by Bettin, Sandro, Fazzari, Alessandro · 2 citations
#FOS: Mathematics #Number Theory (math.NT) #Primary 11M06 #Secondary 11M26

paper · doi:10.48550/arxiv.2208.08421

Abstract

We compute the one-level density of the non-trivial zeros of the Riemann zeta-function weighted by |ζ(\frac12+it)|^2k for k=1 and, for test functions with Fourier support in (-\frac12,\frac12), for k=2. As a consequence, for k=1,2, we deduce under the Riemann hypothesis that T(log T)^1-k2+o(1) non-trivial zeros of ζ, of imaginary parts up to T, are such that ζ attains a value of size (log T)^k+o(1) at a point which is within O(1/log T) from the zero.

Cited by

• Low-lying zeros of Hilbert modular L-functions weighted by powers of central L-values
• Selberg's Central Limit Theorem weighted by Linear Statistics of Zeta Zeros

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