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Bagchi’s Theorem for Families of Automorphic Forms

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Exploring the Riemann Zeta Function

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Abstract

We prove a version of Bagchi’s Theorem and of Voronin’s Universality Theorem for the family of primitive cusp forms of weight 2 and prime level, and discuss under which conditions the argument will apply to a general reasonable family of automorphic L-functions.

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Notes

  1. 1.

    We assume \( q\geqslant 17 \) to ensure this property; it also holds for q = 11.

  2. 2.

    Here and below, it is important that the “almost surely” property holds for all s, which is the case because we work with random holomorphic functions, and not with particular evaluations of these random functions at specific points s.

  3. 3.

    These assumptions could be easily weakened, as has been done for Voronin’s Theorem.

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Acknowledgements

This work was partially supported by a DFG-SNF lead agency program grant (grant 200021L_153647).

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Authors and Affiliations

  1. ETH Zürich – D-MATH, Rämistrasse 101, 8092, Zürich, Switzerland

    E. Kowalski

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  1. E. Kowalski
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Correspondence to E. Kowalski .

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Editors and Affiliations

  1. Department of Mathematics, University of Michigan, Ann Arbor, Michigan, USA

    Hugh Montgomery

  2. Institut für Mathematik, Universität Zürich, Zürich, Switzerland

    Ashkan Nikeghbali

  3. Institut für Mathematik, Universität Zürich, Zürich, Switzerland

    Michael Th. Rassias

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Kowalski, E. (2017). Bagchi’s Theorem for Families of Automorphic Forms. In: Montgomery, H., Nikeghbali, A., Rassias, M. (eds) Exploring the Riemann Zeta Function. Springer, Cham. https://doi.org/10.1007/978-3-319-59969-4_8

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