Explicit Subconvexity Estimates for Dirichlet $L$-functions

Abstract

Given a Dirichlet character χ modulo a prime q and its associated L-function, L(s, χ), we provide an explicit version of Burgess' estimate for |L(s, χ)|. We use partial summation to provide bounds along the vertical lines ℜs = 1 − r −1 , where r is a parameter associated with Burgess' character sum estimate. These bounds are then connected across the critical strip using the Phragmén-Lindelöf principle. In particular, for σ ∈ 1 2 , 9 10 , we establish|L (σ + it, χ)| ≤ (1.105)(0.692) σ q

Short Character Sums and Their Applications

“…We also include applications to norm-Euclidean cyclic fields and least kth power nonresidues. Some of this research has been published in [1][2][3].…”