2023/08/17 by Anushree Gupta, Gupta, Anushree, Md Kashif Jamal +5
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Primary 11M06 #Secondary 11J81
paper · pdf · doi:10.48550/arxiv.2308.08988
openalex publication_date 2023/08/17 · openalex created_date 2023/08/22 · openalex updated_date 2026/07/28
In 2001, Kanemitsu, Tanigawa, and Yoshimoto studied the following generalized Lambert series, ∑n=1∞ \fracnN-2h exp(nN x)-1, for N ∈ ℕ and h∈ ℤ with some restriction on h. Recently, Dixit and the last author pointed out that this series has already been present in the Lost Notebook of Ramanujan with a more general form. Although, Ramanujan did not provide any transformation identity for it. In the same paper, Dixit and the last author found an elegant generalization of Ramanujan's celebrated identity for ζ(2m+1) while extending the results of Kanemitsu et al. In a subsequent work, Kanemitsu et al. explored another extended version of the aforementioned series, namely, ∑r=1q∑n=1∞ \fracχ(r)nN-2hexp(-(r)/(q)nN x)1-exp(-nN x), where χ denotes a Dirichlet character modulo q, N∈ 2ℕ and with some restriction on the variable h. In the current paper, we investigate the above series for \it any N ∈ ℕ and h ∈ ℤ. We obtain a Dirichlet character analogue of Dixit and the last author's identity and there by derive a two variable generalization of Ramanujan's identity for ζ(2m+1). Moreover, we establish a new identity for L(1/3, χ) analogous to Ramanujan's famous identity for ζ(1/2).