2021/06/09 by Anushree Gupta, Gupta, Anushree, Bibekananda Maji +1
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Primary 11M06 #Secondary 11J81
paper · pdf · doi:10.48550/arxiv.2106.04797
openalex publication_date 2021/06/09 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
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, (1r)/(exp(1s x) - 1) + (2r)/(exp(2s x) - 1) + (3r)/(exp(3s x) - 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.