Hurwitz Zeta Function - Zeros

Zeros

If q=1 the Hurwitz zeta function reduces to the Riemann zeta function itself; if q=1/2 it reduces to the Riemann zeta function multiplied by a simple function of the complex argument s (vide supra), leading in each case to the difficult study of the zeros of Riemann's zeta function. In particular, there will be no zeros with real part greater than or equal to 1. However, if 0<q<1 and q≠1/2, then there are zeros of Hurwitz's zeta function in the strip 1s)<1+ε for any positive real number ε. This was proved by Davenport and Heilbronn for rational and non-algebraic irrational q, and by Cassels for algebraic irrational q.