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arXiv · 2411.19762

Pair Correlation of zeros of Dirichlet $L$-Functions: A possible path towards the conjectures of Chowla, Elliott-Halberstam and Montgomery

Neelam Kandhil·Alessandro Languasco·Pieter Moree
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Abstract

Assuming the Generalized Riemann Hypothesis and a pair correlation conjecture for the zeros of Dirichlet $L$-functions, we establish the truth of a conjecture of Montgomery (in its corrected form stated by Friedlander and Granville) on the magnitude of the error term in the prime number theorem in arithmetic progressions. As a consequence, we obtain that, under the same assumptions, the Elliott-Halberstam conjecture holds true. As another consequence, under the same assumptions, we will show that the number of Dirichlet characters $\chi \pmod{q}$ for which $L(\frac{1}{2},\chi)=0$ is of order less than $q^{1/2+\epsilon}$.

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Neelam Kandhil, Alessandro Languasco, Pieter Moree. 2024-11-29. Pair Correlation of zeros of Dirichlet $L$-Functions: A possible path towards the conjectures of Chowla, Elliott-Halberstam and Montgomery. https://doi.org/10.1007/s00208-026-03383-y

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