A Dirichlet character analogue of Ramanujan's formula for odd zeta values (Q6564063)

Let \(x>0\), \(N, q\in \mathbb{N}\), \(h\in \mathbb{Z}\) and \(\chi\) be a primitive character modulo \(q\). The paper is concerned with the transformation formula for the Lambert series\N\[\NF(2h-N,x,\chi)=\sum_{r=1}^{q}\sum_{n=1}^{\infty}\frac{\chi(r)n^{N-2h}e^{-\frac{r}{q}n^Nx}}{1-e^{-n^Nx}}\N\]\Nand its conjugate. The Hecke gamma transform gives\N\[\N F(2h-N,x,\chi)=\int_{(c_0)}^{}\Gamma(s)L(s,\chi)\zeta(Ns-N+2h){\left(\frac{x}{q}\right)}^{-s}\, \mathrm{d}s,\tag{1.8}\N\]\Nwhere \(c_0>\max\left\{1,\frac{N-2h+1}{N}\right\}\). The familiar procedure is applied using the Cauchy residue theorem. Shifting the line of integration to \(d_0\sim -\left( \left[2\frac{h}{N} \right]+1 \right)\), computing the residues and transforming the resulting integral by careful argument, into a Lambert series, the main theorem is proved. In the main theorem in the case \(N=1\), then \(N-2h\) is odd, whence it leads to Ramanujan type formula. Also the \(N-2h=-1\) case is excluded, which would lead to Ramanujan's plausible proof of the eta-transformation formula. Among numerous papers on the sequence \(\left\{n^N\right\}\) this paper is in the spirit of [\textit{A. Gupta} and \textit{B. Maji}, J. Math. Anal. Appl. 507, No. 1, Article ID 125738, 24 p. (2022; Zbl 1474.11145)] and provides a closed form for the value \(L\left( \frac{1}{3},\chi_5 \right)\), where \(\chi_5\) is the even primitive character modulo \(5\), thus giving a raison-d'être for considering the \(N\)th power sequence.