A New Representation of the Riemann Zeta Function
Mahipal Gurram
TL;DR
The paper develops a novel discrete representation of the Riemann zeta function by expressing it as the limiting difference of two structured double sums, $A_n(s)$ and $B_n(s)$, with $A_n(s)=\sum_{j,k=1}^{n} \frac{1}{\max(j,k)^{s+1}}$ and $B_n(s)=\sum_{j,k=1}^{n} \frac{1}{(j+k)^{s+1}}$. By relating $A_n(s)$ to generalized harmonic numbers $H_n^{(s)}$ and employing a polygamma-based framework, the authors derive a main identity for $\zeta(s)$ and, in the limit $n\to\infty$, obtain a double-series representation $\zeta(s)=\sum_{j,k=1}^{\infty}\left(\frac{1}{\max(j^{s+1},k^{s+1})}-\frac{1}{(j+k)^{s+1}}\right)$. They further show a relation for odd arguments, $\zeta(2s+1)=2\zeta(2s)-\sum_{j,k=1}^{\infty}\frac{1}{(\max(j,k))^{s+1}}$, and give a concrete $\zeta(3)$ identity, together with polygamma-based bounds. Overall, the work provides an alternative analytic lens that connects discrete harmonic sums, polygamma asymptotics, and classical zeta representations, with potential computational and theoretical implications.
Abstract
In this paper,we develop a novel representation of the zeta function expressed as the limiting difference between two structured double sums. This approach leads to a new and elegant identity involving maximum functions and additive terms, providing theoretical insights. The derivation relies on generalized harmonic series and polygamma functions, linking classical analysis with contemporary summation techniques.