The Riemann Hypothesis for Symmetrised Combinations of Zeta Functions

Abstract

This paper studies combinations of the Riemann zeta function, based on one defined by P.R. Taylor, which was shown by him to have all its zeros on the critical line. With a rescaled complex argument, this is denoted here by ${\cal T}_-(s)$, and is considered together with a counterpart function ${\cal T}_+(s)$, symmetric rather than antisymmetric about the critical line. We prove that ${\cal T}_+(s)$ has all its zeros on the critical line, and that the zeros of both functions are all of first order. We establish a link between the zeros of ${\cal T}_-(s)$ and of ${\cal T}_+(s)$ with those of the zeros of the Riemann zeta function $\zeta(2 s-1)$, which enables us to prove that, if the Riemann hypothesis holds, then the distribution function of the zeros of $\zeta (2 s-1)$ agrees with those for ${\cal T}_-(s)$ and of ${\cal T}_+(s)$ in all terms which do not remain finite as $t\rightarrow \infty$.