A uniformly convergent Series for $\zeta(s)$ and closed Formulas, that include Catalan Numbers

TL;DR

This paper presents a new uniformly convergent series for the Riemann zeta function involving Catalan numbers and binomial coefficients, along with four novel closed-form formulas linking zeta to combinatorial structures.

Contribution

It introduces a series representation of z(s) using Catalan numbers and derives four new closed formulas for specific values of s, revealing combinatorial connections.

Findings

Series converges uniformly for z(s)

Four new closed-form formulas for z(s) at discrete s values

Links between z(s), Catalan numbers, and geometric content

Abstract

There exists an infinite series of ratios by which one can derive the Riemann zeta function $\zeta(s)$ from Catalan numbers and central binomial coefficients which appear in the terms of the series. While admittedly the derivation is not deep it does indicate some combinatorial aspect to the Riemann zeta function. But we actually do find also four additional new closed formulas, which include a formula by which one can compute $\zeta(s)$ for a countably infinite number of discrete positive values for $s$ where the formula contains Catalan numbers not in infinite series. The Riemann zeta function has applications in physics, such as in computations related to the Casimir effect. Our result indicates a link between the Riemann zeta function, combinatorics, Catalan numbers, the central binomial coefficient and the content of a hypersphere, under certain conditions.