Conditional estimates for $L$-functions in the Selberg class II

TL;DR

This paper derives explicit bounds for the logarithm of functions in the Selberg class near the critical line, assuming the Generalized Riemann Hypothesis, and improves known estimates for the Riemann zeta function.

Contribution

It provides uniform upper and lower bounds with explicit main terms for Selberg class functions under RH, including near the critical line and for specific functions like the Riemann zeta.

Findings

Explicit bounds for $ ext{log}| ext{L}(s)|$ in the critical strip

Improved lower bounds for the Riemann zeta function near the critical line

Estimates under assumptions on Dirichlet coefficients and polynomial Euler products

Abstract

Assuming the Generalized Riemann Hypothesis, we provide uniform upper and lower bounds with explicit main terms for $\log{\left|\cL(s)\right|}$ for $\sigma \in (1/2,1)$ and for functions in the Selberg class. In particular, we focus on the region $0\leq\sigma-1/2\ll 1/\log{\log{\left(\sq|t|^{\sdeg}\right)}}$. We also provide estimates under additional assumptions on the distribution of Dirichlet coefficients of $\cL(s)$ on prime numbers. Moreover, by assuming a polynomial Euler product representation for $\cL(s)$, we establish both uniform bounds and completely explicit estimates by also assuming the strong $\lambda$-conjecture. In addition to providing estimates for a large set of functions, our results improve the best known estimates for specific functions in the Selberg class including the lower bounds for the Riemann zeta function close to the critical line.