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Real zeros of Eisenstein series and Rankin-Selberg L-functions

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Abstract

We prove that the Eisenstein series E(z, s) have no real zeroes for s ∈ (0, 1) when the value of the imaginary part of z is in the range \(\tfrac{1}{5}\) < Im z < 4.94. For very large and very small values of the imaginary part of z, E(z, s) have real zeros in (½, 1), i.e. GRH does not hold for the Eisenstein series. Using these properties, we prove that the Rankin-Selberg L-function attached with the Ramanujan τ-function has no real zeros in the critical strip, except at the central point s = ½.

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Authors and Affiliations

  1. Dolby Laboratories, San Francisco, CA, USA

    C. Bauer

  2. Department of Mathematics, Capital Normal University, Beijing, 100037, P. R. China

    Y. Wang

Authors
  1. C. Bauer
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  2. Y. Wang
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Additional information

The second author is supported by China NSF Grant #10G01034, China 973 Project #2007CB807900-2007CB807903 and Morning Side center.

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Bauer, C., Wang, Y. Real zeros of Eisenstein series and Rankin-Selberg L-functions. Acta Math Hung 115, 13–27 (2007). https://doi.org/10.1007/s10474-007-5102-1

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  • DOI: https://doi.org/10.1007/s10474-007-5102-1

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