New integral representations for the square of the Riemann zeta-function
Volume 82 / 1997
Abstract
Introduction. The recent discovery of an analogue of the Riemann-Siegel integral formula for Dirichlet series associated with cusp forms [2] naturally raises the question whether similar formulas might exist for other types of zeta functions. The proof of these formulas depends on the functional equation for the underlying Dirichlet series. In both cases, for ζ(s) and for the cusp form zeta functions, only a simple gamma factor is involved. The next simplest case arises when two such factors occur in the functional equation. The prototype of these Dirichlet series is ζ²(s), and so any investigation might well begin with this example. In the present study we show that, indeed, a formula of Riemann-Siegel type can be found for ζ²(s). The numerous applications of the ordinary Riemann-Siegel integral formula [3] suggest similar ones for our formula too. For instance, it seems very probable to derive an asymptotic expansion for ζ²(s), giving a generalization of Siegel's result [9]. Originally, this expansion is due to Motohashi [6,7], and it depends on the corresponding formula for ζ(s). Consequently, our approach might lead to an independent proof of Motohashi's expansion. This would be of considerable value, since our method applies as well to other Dirichlet series satisfying similar functional equations, like Hecke L series of quadratic fields. As a first step in this direction we give a simple proof of the approximate functional equation for ζ²(s) at the end of the paper. The author is very much indebted to Professor Aleksandar Ivić (Belgrade) who read a preliminary version of this work. His suggestions and criticism led to numerous improvements. Thanks are also due to the referee for pointing out some misprints and unclear passages.
