On Ramanujan's formula for $\zeta(1/2)$ and $\zeta(2m+1)$

TL;DR

This paper explores transformation formulas for infinite series related to the Riemann zeta function, deriving Ramanujan's formulas for specific values, including (1/2), (1/k), and (2m+1), and introduces a new identity for (-1/2).

Contribution

It extends Ramanujan's formulas for zeta values by deriving new transformation formulas and identities for (1/2), (1/k), (2m+1), and (-1/2).

Findings

Derived Ramanujan's formula for (1/2)

Obtained Ramanujan's formula for (2m+1)

Discovered a new identity for (-1/2)

Abstract

Page 332 of Ramanujan's Lost Notebook contains a compelling identity for $\zeta(1/2)$, which has been studied by many mathematicians over the years. On the same page, Ramanujan also recorded the series, \begin{align*} \frac{1^r}{\exp(1^s x) - 1} + \frac{2^r}{\exp(2^s x) - 1} + \frac{3^r}{\exp(3^s x) - 1} + \cdots, \end{align*} where $s$ is a positive integer and $r-s$ is any even integer. Unfortunately, Ramanujan doesn't give any formula for it. This series was rediscovered by Kanemitsu, Tanigawa, and Yoshimoto, although they studied it only when $r-s$ is a negative even integer. Recently, Dixit and the second author generalized the work of Kanemitsu et al. and obtained a transformation formula for the aforementioned series with $r-s$ is any even integer. While extending the work of Kanemitsu et al., Dixit and the second author obtained a beautiful generalization of Ramanujan's formula for odd zeta values. In the current paper, we investigate transformation formulas for an infinite series, and interestingly, we derive Ramanujan's formula for $\zeta(1/2)$, Wigert's formula for $\zeta(1/k)$ as well as Ramanujan's formula for $\zeta(2m+1)$. Furthermore, we obtain a new identity for $\zeta(-1/2)$ in the spirit of Ramanujan.