On an asymptotic formula of Ramanujan for a certain theta-type series (Q2717593)

Let \(\tau\) be a positive real parameter and \(\Gamma(s)\) and \(\zeta(s)\) denote the gamma function and the Riemann zeta-function. In Chapter 15 of his Notebook \textit{S. Ramanujan} [Bombay: Tata Institute of Fundamental Research. Vols. 1 and 2 (1957; Zbl 0138.24201)] suggested a way of computing in exact form the error term \(S(\tau)\) in the formula NEWLINE\[NEWLINE \sum_{n=1}^{\infty}{1\over e^{n^2 \tau}-1}={\pi^2\over 6\tau}+ {1\over 2}\sqrt{{\pi\over\tau}}\zeta\left({1\over 2}\right)+{1\over 4}+S(\tau). \tag{1}NEWLINE\]NEWLINE The computation was carried out by \textit{B. C. Berndt} and \textit{R. J. Evans} [Acta Arith. 47, 123-142 (1986; Zbl 0606.10016)]. Another expression of \(S(\tau)\) was found by \textit{D. Klusch} [Acta Arith. 58, 59-64 (1991; Zbl 0724.11022)], a \(\chi\)-analogue formula of (1) was derived by \textit{S. Egami} [Acta Arith. 69, 189-191 (1995; Zbl 0815.11041)]. NEWLINENEWLINENEWLINEIn the paper under review the author generalizes the formula (1) in two directions. He obtains an expression of type (1) for NEWLINE\[NEWLINE \sum_{n=1}^{\infty}{e^{-\alpha n^2\tau}\over(1-e^{-n^2 \tau})^r}\quad (\tau>0, r\geq 1, 0<\alpha\leq r) \tag{2}NEWLINE\]NEWLINE and for NEWLINE\[NEWLINE \sum_{n=1}^{\infty}{e^{-\alpha n^2\tau}\over 1-e^{2\pi i\lambda-n^2 \tau}}\quad (\tau>0, 0<\alpha,\lambda\leq 1). \tag{3}NEWLINE\]NEWLINE To treat these theta-type series the author uses a Mellin transform technique. In the expression for (2) there appears the generalized Hurwitz zeta-function defined by NEWLINE\[NEWLINE \zeta_r (s,\alpha)=\sum_{n=0}^{\infty}{\Gamma(r+n)\over\Gamma(r)n!}(n+\alpha)^{-s} NEWLINE\]NEWLINE for \(\Re s>r\) and by analytic continuation otherwise. For this function the author obtains the functional equation, which is useful in the consideration of (2). Similarly, in the case of (3) the functional properties of the Lerch zeta-function are used.