Complex zeros of an incomplete Riemann zeta function and of the incomplete gamma function

Abstract:

Complex zeros of an incomplete Riemann zeta function and of the incomplete gamma function are calculated as functions of the upper limit $\lambda$ of the definition integrals. It becomes apparent that not all, but only some, of the zero trajectories of the incomplete Riemann zeta function join a zero of the Riemann zeta function $\zeta (s)$ for $\lambda \to \infty$. The remaining trajectories at least in the region considered, approach the zero trajectories of the incomplete gamma function.