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Bounds on short character sums and L-functions with characters to a powerful modulus

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Abstract

We combine a classical idea of Postnikov (1956) with the method of Korobov (1974) for estimating double Weyl sums, deriving new bounds on short character sums when the modulus q has a small core Πp|qp. Using this estimate, we improve certain bounds of Gallagher (1972) and Iwaniec (1974) for the corresponding L-functions. In turn, this allows us to improve the error term in the asymptotic formula for primes in short arithmetic progressions modulo a power of a fixed prime. As yet another application of our bounds, we substantially extend the classical zero-free region (which might include Siegel zeros). Finally, we improve the previous best value \(L=\frac{12}{5}=2.2\) of the Linnik constant for primes in arithmetic progressions modulo powers of a fixed prime to L < 2.1115.

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Acknowledgements

The authors are very grateful to Glyn Harman for various discussions concerning [8, Theorem 1.2]. We also thank the anonymous referee for numerous important comments and suggestions.

The first author was supported in part by a grant from the University ofMissouri Research Board. The second author was supported by ARC Grant DP170100786.

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Authors and Affiliations

  1. Department of Mathematics, University of Missouri, Columbia, MO, 65211, USA

    William D. Banks

  2. Department of Pure Mathematics, University of New South Wales, Sydney, NSW, 2052, Australia

    Igor E. Shparlinski

Authors
  1. William D. Banks
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  2. Igor E. Shparlinski
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Correspondence to Igor E. Shparlinski.

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Banks, W.D., Shparlinski, I.E. Bounds on short character sums and L-functions with characters to a powerful modulus. JAMA 139, 239–263 (2019). https://doi.org/10.1007/s11854-019-0060-4

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  • DOI: https://doi.org/10.1007/s11854-019-0060-4

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