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The number of zeros of linear combinations of L-functions near the critical line

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Abstract

In this paper, we investigate the zeros near the critical line of linear combinations of L-functions belonging to a large class, which conjecturally contains all L-functions arising from automorphic representations on GL(n). More precisely, if L1, …, LJ are distinct primitive L-functions with J ≥ 2, and bj are any non-zero real numbers, we prove that the number of zeros of \(F(s) = \sum\nolimits_{j = 1}^J {{b_j}{L_j}(s)} \) in the region Re(s) ≥ 1/2+ 1/G(T) and Im(s) ∈ [T, 2T] is asymptotic to \({K_0}TG(T)/\sqrt {\log G(T)} \) uniformly in the range log log T ≤ G(T) ≤ (log T)ν, where K0 is a certain positive constant that depends on J and the Lj. This establishes a generalization of a conjecture of Hejhal in this range. Moreover, the exponent ν verifies \(\nu \asymp 1/J\) as J grows.

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Acknowledgment

We thank the anonymous referee for carefully reading the paper, and for their numerous comments and suggestions.

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Authors and Affiliations

  1. Institut Élie Cartan de Lorraine, Université de Lorraine, BP 70239, 54506, Vandoeuvre-lès-Nancy Cedex, France

    Youness Lamzouri

  2. Department of Mathematics, Research Institute of Basic Sciences, Incheon National University, 119 Academy-ro, Yeonsu-gu, Incheon, 22012, Korea

    Yoonbok Lee

Authors
  1. Youness Lamzouri
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  2. Yoonbok Lee
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Correspondence to Yoonbok Lee.

Additional information

This work was supported by Incheon National University (International Cooperative) Research Grant in 2019.

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Lamzouri, Y., Lee, Y. The number of zeros of linear combinations of L-functions near the critical line. JAMA (2023). https://doi.org/10.1007/s11854-023-0307-y

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  • DOI: https://doi.org/10.1007/s11854-023-0307-y

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