ON AN ASYMPTOTIC FORMULA OF RAMANUJAN

FOR A CERTAIN THETA-TYPE SERIES

Masanori KATSURADA

Mathematics, Hiyoshi Campus, Keio University

Let τ be a positive real parameter, s a complex variable and Γ(s) and ζ(s) denote the gamma and the Riemann zeta-function respectively. In Chapter 15 of his celebrated notebook [Ra], Ramanujan suggested a way of computing in exact form the error term of the asymptotic formula

∑_{n=1}^{∞} \frac{1}{e^{n^{2}τ} − 1} = \frac{π^{2}}{6τ} + \frac{1}{2}\sqrt{\frac{π}{τ}} ζ(\frac{1}{2}) + \frac{1}{4} + o(1)

as τ → +0. The computation was carried out by Berndt and Evans [BE] in terms of Poisson’s summation device. They found that the o(1) term above can be expressed as

\frac{1}{2}\sqrt{\frac{π}{τ}} ∑_{n=1}^{∞} \frac{1}{\sqrt{n}}\{\frac{sinh(2π\sqrt{πn/τ}) − sin(2π\sqrt{πn/τ})}{cosh(2π\sqrt{πn/τ}) − cos(2π\sqrt{πn/τ})} − 1\}.

Let α be a positive real parameter and r a positive integer. In this talk we aim at showing that a similar asymptotic formula exists for a more general series of the type

∑_{n=1}^{∞} \frac{e^{−αn^{2}τ}}{(1 − e^{−n^{2}τ})^{r}} .

The key method for our treatments of the theta-type series is a Mellin transform technique. In the proof we apply the functional properties of the genealized Hurwitz zeta-function ζ_{r}(s, α) = ∑_{n=0}^{∞}{(r)_{n}/n!}(n + α)^{−s} (Re s > r), where (r)_{n} = Γ(r + n)/Γ(r) denotes Pochhammer’s symbol. It should be remarked that ζ_{r}(s, α) is a particular case \underline{ω} = (1, …, 1) of Barnes’ multiple zeta-function ζ_{r}(s, α | \underline{ω}) (cf. [Ba]).

References

[Ba] E. W. Barnes, The theory of the multiple gamma function, Trans. Cambridge Philos. Soc. 19 (1904), 265–387.

[BE] B. C. Berndt and R. J. Evans, Chapter 15 of Ramanujan’s Second Notebook: Part 2, Modular forms, Acta Arith. XLVII (1986), 123–142.

[Ra] S. Ramanujan, Notebooks, 2 volumes, Tata Institute of Fundamental Research, Bombay,