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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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