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

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.