A sharpening of Li's criterion for the Riemann Hypothesis

2004/04/10 by André Voros, Voros, André
Mathematics · #11M26 (primary) 30B40 #41A60 (secondary) #Advanced Mathematical Identities #Advanced Mathematical Theories #Analytic Number Theory Research #Complex Variables (math.CV) #FOS: Mathematics #Number Theory (math.NT) #math.CV #math.NT #msc:11M26 #msc:30B40 #msc:41A60

paper · pdf · doi:10.48550/arxiv.math/0404213

1 Latex file, 5 pages, submitted to C.R. Acad. Sci. (Paris) Sér. I. V2: notation corrected in eq.(7); eq.(9) made more precise

openalex publication_date 2004/04/10 · arxiv created 2004/04/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Exact and asymptotic formulae are displayed for the coefficients λ_n used in Li's criterion for the Riemann Hypothesis. In particular, we argue that if (and only if) the Hypothesis is true, λ_n ∼ n(A log n +B) for n → ∞ (with explicit A>0 and B). The approach also holds for more general zeta or L-functions.

Related

• Effective method of computing Li's coefficients and their properties
• Some possible approaches to the Riemann Hypothesis via the Li/Keiper constants
• First applications of generalized Li's criterion to study the Riemann\n zeta-function zeroes location
• Li coefficients as norms of functions in a model space
• Densities, Laplace Transforms and Analytic Number Theory