The exponential-type generating function of the Riemann zeta-function revisited

Abstract

Dirichlet series associated with the Poincaré series attached to $$\mathrm{SL}(2,{{\mathbb {Z}}})$$ SL ( 2 , Z ) are introduced. Integral representations and transformation formulas are given, which describe the Voronoï-type summation formula for the exponential-type generating function of the Riemann zeta-function. As an application, a new proof of the Fourier series expansion of holomorphic Poincaré series is given.