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Upper bounds on a two-term exponential sum*

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Abstract

We obtain upper bounds for mixed exponential sums of the type\(S(\chi ,f,p^m ) = \sum\nolimits_{x = 1}^{p^n } {\chi (x)e} _{p^m } (ax^n + bx)\) where pm is a prime power with m⩾ 2 and X is a multiplicative character (mod pm). If X is primitive or p⫮(a, b) then we obtain |S(χ,f,p m)| ⩽2np 2/3 m. If X is of conductor p and p⫮( a, b) then we get the stronger bound |S(χ,f,p m)|⩽np m/2.

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Authors and Affiliations

  1. Department of Mathematics, Kansas State University, 66506, Manhattan, KS, USA

    Todd Cochrane

  2. Department of Mathematics, Tsinghua University, 100084, Beijing, China

    Zheng Zhiyong

Authors
  1. Todd Cochrane
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  2. Zheng Zhiyong
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Corresponding authors

Correspondence to Todd Cochrane or Zheng Zhiyong.

Additional information

This paper is dedicated to Prof. Wang Yuan on the occasion of his 70th birthday.

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Cochrane, T., Zheng, Z. Upper bounds on a two-term exponential sum*. Sci. China Ser. A-Math. 44, 1003–1015 (2001). https://doi.org/10.1007/BF02878976

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  • DOI: https://doi.org/10.1007/BF02878976

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