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On Linnik and Selberg’s Conjecture About Sums of Kloosterman Sums

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Algebra, Arithmetic, and Geometry

Part of the book series: Progress in Mathematics ((PM,volume 270))

Summary

We examine the Linnik and Selberg Conjectures concerning sums of Kloosterman sums, in all its aspects (x, m and n). We correct the precise form of the Conjecture and establish an analogue of Kuznetzov’s 1/6 exponent in the mn aspect. This, perhaps somewhat surprisingly, is connected with the transional ranges associated with asymptotics of Bessel Functions of large order.

2000 Mathematics Subject Classifications: 11Fxx, 11Lxx

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Author information

Authors and Affiliations

  1. Department of Mathematics, Princeton University, Princeton, NJ, USA

    Peter Sarnak & Jacob Tsimerman

  2. Institute for Advanced Study, Princeton, NJ, USA

    Peter Sarnak

Authors
  1. Peter Sarnak
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  2. Jacob Tsimerman
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Corresponding author

Correspondence to Peter Sarnak .

Editor information

Editors and Affiliations

  1. Courant Inst. Mathematical, New York University, Mercer St. 251, New York, 10012, USA

    Yuri Tschinkel

  2. Eberly College of Science, Pennsylvania State University, University Park, 16802, USA

    Yuri Zarhin

Additional information

Dedicated to Y. Manin on the Occasion of his 70th Birthday

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Sarnak, P., Tsimerman, J. (2009). On Linnik and Selberg’s Conjecture About Sums of Kloosterman Sums. In: Tschinkel, Y., Zarhin, Y. (eds) Algebra, Arithmetic, and Geometry. Progress in Mathematics, vol 270. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4747-6_20

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