Explicit Subconvexity Estimates for Dirichlet L-functions

2022/06/22 by Francis, Forrest J.
#11L40 #11M06 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2206.11112

Abstract

Given a Dirichlet character χ modulo 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 \Res = 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)^σq^{(31)/(80)-(2)/(5)σ}(logq)^{(33)/(16)-(9)/(8)σ} |σ+ it|.

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