On an integral summable to $2\xi (s)/(s(s-1))$

Abstract:

Let $\psi (x) = \Sigma _{n = 1}^\infty {e^{ - {n^2}\pi x}}$, and $\chi (u) = {e^{u/2}}(1 + 2\psi ({e^{2u}}))$. The divergent integral $2\smallint _0^\infty \chi (u)\cos$ tu du is shown to be summable for certain complex values of t to the function $2\xi (s)/s(s - 1) = {\pi ^{ - s/2}}\Gamma (1/2s)\zeta (s)$, where $s = 1/2 + it$, and $\zeta (s)$ is the zeta-function of Riemann. The values of a resulting approximation to $2\xi (s)/s(s - 1)$ are computed and its zeros located.