Conditional estimates for L-functions in the Selberg class II

2024/10/30 by Neea Palojärvi, Palojärvi, Neea, Aleksander Simonič +1 · 1 citation
Computer Science · Mathematics · #11M06 #11M26 #41A30 #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2410.22711

openalex publication_date 2024/10/30 · openalex created_date 2024/11/14 · openalex updated_date 2026/07/28

Abstract

Assuming the Generalized Riemann Hypothesis, we provide uniform upper and lower bounds with explicit main terms for log|\cL(s)| for σ∈ (1/2,1) and for functions in the Selberg class. In particular, we focus on the region 0≤σ-1/2≪ 1/loglog(\sq|t|^\sdeg). 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.

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