The positivity technique and low-lying zeros of Dirichlet L-functions

2025/03/20 by Tianyu Zhao, Zhao, Tianyu · 2 citations
#math.NT

paper · pdf · doi:10.48550/arxiv.2503.15832

Abstract

Assuming the generalized Riemann hypothesis, we sharpen the error terms in a number of estimates related to the distribution of low-lying zeros of Dirichlet L-functions. The refinements, building upon earlier work of Omar, come from the use of the positivity technique, which involves choosing certain test functions in the explicit formula. We also improve a result of Hughes and Rudnick on the proportion of Dirichlet L-functions modulo a large prime q having small first zeros. In addition, some of our results are generalized to a larger class of L-functions, and explicit conditional estimates are provided for the lowest zeros of Dirichlet L-functions.

Citations

• Hilbert spaces and low-lying zeros of L-functions
• Counting zeros of Dirichlet L-functions
• Nonvanishing of Dirichlet L-functions, II
• A note on the zeros of zeta and L-functions
• Nonvanishing of Quadratic Dirichlet L-Functions at S = &#189
• On character sums and class numbers

Cited by

• Unconditional estimates on the argument of Dirichlet L-functions with applications to low-lying zeros
• Conditional estimates on the argument of Dirichlet L-functions with applications to low-lying zeros

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