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 λ \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 ζ ( s ) \zeta (s) for λ → ∞ \lambda \to \infty . The remaining trajectories at least in the region considered, approach the zero trajectories of the incomplete gamma function.