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Zeros of $L$-functions and large partial sums of Dirichlet coefficients
- Publication Year :
- 2024
-
Abstract
- Let $L(s,\pi)=\sum_{n=1}^{\infty}\lambda_{\pi}(n)n^{-s}$ be an $L$-function that satisfies a weak form of the generalized Ramanujan conjecture. We prove that large partial sums of $\lambda_{\pi}(n)$ strongly repel the low-lying zeros of $L(s,\pi)$ away from the critical line. Our results extend and quantitatively improve preceding work of Granville and Soundararajan.<br />Comment: 23 pages
- Subjects :
- Mathematics - Number Theory
11L40, 11N56, 11F66
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2408.03938
- Document Type :
- Working Paper