Conditional estimates for $L$-functions in the Selberg class II
Neea Palojärvi, Aleksander Simonič
TL;DR
Under GRH, this work derives uniform, explicit bounds for $\log|\mathcal{L}(s)|$ for $\mathcal{L}$ in the Selberg class across $\sigma\in(1/2,1)$, with refined control when $\sigma-1/2$ is tiny. The authors reduce the problem to a zero-sum via the functional equation, and then bound the zero-sum with extremal bandlimited majorants/minorants and the Guinand-Weil explicit formula, while carefully estimating sums over primes and gamma-factors. They provide both asymptotic and completely explicit bounds, including near the critical line and in the presence of a polynomial Euler product together with the strong $\lambda$-conjecture, yielding improved constants and new explicit lower bounds for $\zeta(s)$. The results advance the precision and generality of conditional bounds in the Selberg class, with potential applications to zero-distribution questions and explicit verifications of GRH up to large heights.
Abstract
Assuming the Generalized Riemann Hypothesis, we provide uniform upper and lower bounds with explicit main terms for $\log{\left|\cL(s)\right|}$ for $σ\in (1/2,1)$ and for functions in the Selberg class. In particular, we focus on the region $0\leqσ-1/2\ll 1/\log{\log{\left(\sq|t|^{\sdeg}\right)}}$. 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 $λ$-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.