On generalized Li criterion for a certain class of L-functions

2014/10/16 by Anne-Maria Ernvall-Hytönen, Almasa Odžak, Ernvall-Hytönen, Anne-Maria +5
Mathematics · #11M26 #11M36 #11M41 #Advanced Algebra and Geometry #Analytic Number Theory Research #Analytic and geometric function theory #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11M26 #msc:11M36 #msc:11M41

paper · pdf · doi:10.48550/arxiv.1410.4384

28 pages, 6 figures

arxiv created 2014/10/16 · openalex publication_date 2014/10/16 · arxiv updated 2014/10/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We define generalized Li coefficients, called τ-Li coefficients for a very broad class S^{\sharp \flat} (σ₀, σ₁) of L-functions that contains the Selberg class, the class of all automorphic L-functions and the Rankin-Selberg L-functions, as well as products of suitable shifts of those functions. We prove the generalized Li criterion for zero-free regions of functions belonging to the class S^{\sharp \flat} (σ₀, σ₁), derive an arithmetic formula for the computation of τ-Li coefficients and conduct numerical investigation of τ-Li coefficients for a certain product of shifts of the Riemann zeta function.

Citations

• The Positivity of a Sequence of Numbers and the Riemann Hypothesis

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• Reformulation of the Li criterion for the Selberg class
• On relations equivalent to the generalized Riemann hypothesis for the Selberg class
• Riesz type criteria for L-functions in the Selberg class
• A note on the standard zero-free region for L-functions
• Fourier coefficients of automorphic L-functions over primes in ray classes