On Ramanujan's formula for ζ(1/2) and ζ(2m + 1)

Abstract

Page 332 of Ramanujan's Lost Notebook contains a compelling identity for ζ(1/2), which has been studied by many mathematicians over the years. On the same page, Ramanujan also recorded the series,1rexp⁡(1sx)−1+2rexp⁡(2sx)−1+3rexp⁡(3sx)−1+⋯, 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 ζ(1/2), Wigert's formula for ζ(1/k) as well as Ramanujan's formula for ζ(2m+1). Furthermore, we obtain a new identity for ζ(−1/2) in the spirit of Ramanujan.