On the functional properties of Bessel zeta-functions

Zeta-functions associated with modified Bessel functions are introduced as ordinary Dirichlet series whose coefficients are J-Bessel and K-Bessel functions. Integral representations, transformation formulas, a power series expansion involving the Riemann zeta-function and a recurrence formula are given. The inverse Laplace transform of Weber's first exponential integral is the basic tool to derive the integral representations. As an application, we give a new proof of the Fourier series expansion of the Poincaré series attached to SL (2, ℤ). Copyright © 2007-2014 by IMPAN. All rights reserved.