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On mean values of mollifiers and L-functions associated to primitive cusp forms

Abstract

We study the second moment of the L-function associated to a holomorphic primitive cusp form of even weight perturbed by a new family of mollifiers. This family is a natural extension of the mollifers considered by Conrey and by Bui, Conrey and Young. As an application, we improve the current lower bound on critical zeros of holomorphic primitive cusp forms.

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Acknowledgements

The first author wishes to acknowledge partial support from SNF grant PP00P2 138906. The second author wishes to thank Keiju Sono for a cordial correspondence while working on similar results. Sono’s results in [31] for the Riemann zeta-function overlap with our computations and these were produced independently of ours. The authors are extremely grateful to the anonymous referees for their comments and suggestions. Their corrections have removed inaccuracies and greatly increased the clarity of the manuscript.

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Correspondence to Nicolas Robles.

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Kühn, P., Robles, N. & Zeindler, D. On mean values of mollifiers and L-functions associated to primitive cusp forms. Math. Z. 291, 661–709 (2019). https://doi.org/10.1007/s00209-018-2099-9

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Keywords

  • Dirichlet polynomial
  • Mollifier
  • Zeros on the critical line
  • Ratios conjecture technique
  • Autocorrelation
  • Holomorphic cusp form
  • Modular forms
  • Generalized Möbius functions

Mathematics Subject Classification

  • Primary 11M26
  • Secondary 11M06
  • 11N64