Explicit Upper Bounds for L-functions on the critical line

TL;DR

This paper derives explicit upper bounds for general L-functions on the critical line under the Generalized Riemann Hypothesis, with applications to number representation problems and improvements over previous bounds.

Contribution

It provides a new explicit upper bound for L-functions on the critical line assuming GRH, with applications to number theory and simpler proofs than prior work.

Findings

Explicit upper bounds for L-functions on the critical line under GRH

Applications to bounds for integers represented by ternary quadratic forms

Improved bounds over previous results by Ono, Soundararajan, and Reinke

Abstract

We find an explicit upper bound for general $L$-functions on the critical line, assuming the Generalized Riemann Hypothesis, and give as illustrative examples its application to some families of $L$-functions and Dedekind zeta functions. Further, this upper bound is used to obtain lower bounds beyond which all eligible integers are represented by Ramanujan's ternary form and Kaplansky's ternary forms. This improves on previous work of Ono and Soundararajan on Ramanujan's form and Reinke on Kaplansky's form with a substantially easier proof.