Functions and Analysis

    

A New Closed-Form Expansion Related to the Riemann Zeta Function

Authors: Ayoub Zaroual

We establish a closed-form expansion of the Riemann zeta function ζ(s) at any complex point s = a + ib in terms of an auxiliary real-analytic function L(a) built from theBernoulli numbers and the rising factorial. The cornerstone of the derivation is a clean dif-ferentiation identity for the elementary symmetric polynomial function Kp n(a) on consecutive integers, which we prove by a generating-function argument. Specialising the expansion to the critical line Re(s) = 1/2 recasts the Riemann hypothesis as the simultaneous vanishing of two real power series in the imaginary part b.

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[v1] 2026-05-20 22:14:22

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