Pair Correlation of zeros of Dirichlet $L$-Functions: A possible path towards the conjectures of Chowla, Elliott-Halberstam and Montgomery
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915797272625152 |
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| author | Kandhil, Neelam Languasco, Alessandro Moree, Pieter |
| author_facet | Kandhil, Neelam Languasco, Alessandro Moree, Pieter |
| contents | 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 $χ\pmod{q}$ for which $L(\frac{1}{2},χ)=0$ is of order less than $q^{1/2+ε}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_19762 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Pair Correlation of zeros of Dirichlet $L$-Functions: A possible path towards the conjectures of Chowla, Elliott-Halberstam and Montgomery Kandhil, Neelam Languasco, Alessandro Moree, Pieter Number Theory 11M26, 11N05 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 $χ\pmod{q}$ for which $L(\frac{1}{2},χ)=0$ is of order less than $q^{1/2+ε}$. |
| title | Pair Correlation of zeros of Dirichlet $L$-Functions: A possible path towards the conjectures of Chowla, Elliott-Halberstam and Montgomery |
| topic | Number Theory 11M26, 11N05 |
| url | https://arxiv.org/abs/2411.19762 |