Bounds on short character sums and L-functions for characters with a smooth modulus

2016/05/24 by Banks, William, Shparlinski, Igor
#11L26 #11L40 #11M06 #11M20 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1605.07553

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 ∏_{p| q}p. 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 region free of Siegel zeros.

Related

• Short character sums for composite moduli
• Sums with the Mobius function twisted by characters with powerful moduli
• The maximum size of short character sums
• Character sums to prime power moduli evaluated at binary quadratic forms
• Estimates for character sums and Dirichlet L-functions to smooth moduli