Estimates for $L$-functions in the critical strip under GRH with effective applications

TL;DR

This paper derives explicit bounds for L-functions near the 1-line under GRH, extending Littlewood's results to the Selberg class, and applies these bounds to estimate the Mertens function effectively.

Contribution

It generalizes Littlewood's conditional bounds to the Selberg class and provides effective estimates for the Mertens function assuming GRH.

Findings

Explicit bounds for log L(s) and L'(s)/L(s) near the 1-line under GRH

Extension of Littlewood's result to functions in the Selberg class

Conditional effective estimates for the Mertens function

Abstract

Assuming the Generalized Riemann Hypothesis, we provide explicit upper bounds for moduli of $\log{\mathcal{L}(s)}$ and $\mathcal{L}'(s)/\mathcal{L}(s)$ in the neighbourhood of the 1-line when $\mathcal{L}(s)$ are the Riemann, Dirichlet and Dedekind zeta-functions. To do this, we generalize Littlewood's well known conditional result to functions in the Selberg class with a polynomial Euler product, for which we also establish a suitable convexity estimate. As an application we provide conditional and effective estimate for the Mertens function.