Mathematische Zeitschrift

, Volume 291, Issue 1–2, pp 661–709 | Cite as

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

  • Patrick Kühn
  • Nicolas RoblesEmail author
  • Dirk Zeindler


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.


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 



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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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut für MathematikUniversität ZürichZurichSwitzerland
  2. 2.Department of MathematicsUniversity of IllinoisUrbanaUSA
  3. 3.Wolfram Research IncChampaignUSA
  4. 4.Department of Mathematics and Statistics, Fylde CollegeLancaster UniversityLancasterUK

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