Polygamma theory, the Li/Keiper constants, and validity of the Riemann Hypothesis

2005/07/17 by Mark W. Coffey, Coffey, Mark W.
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.MP

paper · pdf · doi:10.48550/arxiv.math-ph/0507042

28 pages, 4 figures

arxiv created 2005/07/17 · openalex publication_date 2005/07/17 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Riemann hypothesis is equivalent to the Li criterion governing a sequence of real constants, that are certain logarithmic derivatives of the Riemann xi function evaluated at unity. We investigate a related set of constants c_n, n = 1,2,..., showing in detail that the leading behaviour (1/2) ln n of lambda_n/n is absent in c_n. Additional results are presented, including a novel explicit representation of c_n in terms of the Stieltjes constants gamma_j. We conjecture as to the large-n behaviour of c_n. Should this conjecture hold, validity of the Riemann hypothesis would follow.

Citations

• Quantum Computing and Zeroes of Zeta Functions
• A proof of the Bieberbach conjecture
• The Positivity of a Sequence of Numbers and the Riemann Hypothesis

Related

• Some possible approaches to the Riemann Hypothesis via the Li/Keiper constants
• Simplifications of the Keiper/Li approach to the Riemann Hypothesis
• A sharpening of Li's criterion for the Riemann Hypothesis
• Densities, Laplace Transforms and Analytic Number Theory
• Effective method of computing Li's coefficients and their properties