  We study the meromorphic continuation and the spectral expansion of the
oppposite sign Kloosterman sum zeta function, $$(2\pi
\sqrt{mn})^{2s-1}\sum_{\ell=1}^\infty \frac{S(m,-n,\ell)}{\ell^{2s}}$$ for
$m,n$ positive integers, to all $s \in \mathbb{C}$. There are poles of the
function corresponding to zeros of the Riemann zeta function and the spectral
parameters of Maass forms. The analytic properties of this function are rather
delicate. It turns out that the spectral expansion of the zeta function
converges only in a left half-plane, disjoint from the region of absolute
convergence of the Dirichlet series, even though they both are analytic
expressions of the same meromorphic function on the entire complex plane.

Number Theory