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arXiv · 2602.05511

Some series representing the Riemann zeta function

Jean-François Burnol
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Abstract

Given an integer $b$ at least equal to $2$, we obtain a representation of the Riemann zeta function in the complex plane as a finite linear combination of geometrically convergent series. The coefficients involve partial factorials and rational functions in $b^{s}$ using the Bernoulli numbers. We obtain, for arbitrary $b$, and for $s$ away from the poles, the asymptotic expansion of these rational functions to all orders in inverse powers of their index $m$. Each term of the development involves a periodic function in the base $b$ logarithm of $m$, depending on $s$.

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Jean-François Burnol. 2026-02-05. Some series representing the Riemann zeta function. https://arxiv.org/abs/2602.05511

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