Simple zeros of primitive Dirichlet L-functions and the asymptotic large sieve

Creators: Chandee, Vorrapan; Lee, Yoonbok; Liu, Sheng-Chi; Radziwiłł, Maksym

Abstract

Assuming the generalized Riemann hypothesis, we show using the asymptotic large sieve that 91% of the zeros of primitive Dirichlet L-functions are simple. This improves on earlier work of Özlük which gives a proportion of at most 86%. We further compute the q-analogue of the Pair Correlation Function F(α) averaged over all primitive Dirichlet L-functions in the range |α| < 2. Previously such a result was available only when the average included all the characters χ. As a corollary of our results, we obtain an asymptotic formula for a sum over characters similar to the one encountered in the Barban–Davenport–Halberstam Theorem.

Additional Information

© 2013 Published by Oxford University Press. Received: 10 December 2012; Published: 21 March 2013. The first author is supported by a CRM-ISM fellowship. The fourth author is partially supported by a NSERC PGS-D award. This work was initiated during the Arithmetic Statistics MRC program at Snowbird. We would like to thank Brian Conrey for his guidance throughout the project and for providing us with many unpublished materials on the asymptotic large sieve. We would also like to thank Kannan Soundararajan for pointing out the relation of our work to the Barban-Davenport-Halberstam Theorem. Finally, we would like to thank the referee for many valuable comments.

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