A new expansion of the Riemann zeta function

2025/12/12 by Candelpergher, B.
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #Mathematical functions and polynomials

paper · doi:10.48550/arxiv.2512.11405

Abstract

This article presents polynomial expansions for the Dirichlet eta function and Riemann zeta function that are convergent in the critical strip. To do this we introduce a family of hypergeometric polynomials, whose roots lie on the line \\Re(s)=1/2\, and that are related to Meixner-Pollaczek polynomials. We also obtain orthonormal expansions for eta and zeta restricted to the line \\Re(s)=1/2\. The coefficients of these expansions are given explicitly as linear combinations with rational coefficients of log(2), Euler's constant γ, and zeta values at positive integers.

Citations

• On the Meixner-Pollaczek polynomials and the Sturm-Liouville problems
• Other representations of the Riemann Zeta function and an additional reformulation of the Riemann Hypothesis
• On Maslanka's representation for the Riemann zeta-function
• Hypergeometric-like Representation of the Zeta-Function of Riemann

Related

• Expansions of the Riemann Zeta function in the critical strip
• Series expansions for the Riemann zeta function
• A geometrical summation method for the Riemann zêta function
• Analysis of the Riemann zeta function
• Other representations of the Riemann Zeta function and an additional reformulation of the Riemann Hypothesis