On the functional properties of Bessel zeta-functions
Volume 171 / 2015
Acta Arithmetica 171 (2015), 1-13
MSC: Primary 11M41; Secondary 11F11.
DOI: 10.4064/aa171-1-1
Abstract
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 ${\rm SL}(2,\mathbb {Z})$.
