# CARMA Colloquium

## 4:00 pm

## Thursday, 14^{th} Jul 2011

**V129, Mathematics Building**

# Prof Ross McPhedran

(The University of Sydney)
*The Riemann Hypothesis for Combinations of Zeta Functions*

This paper studies combinations of the Riemann zeta function, based on one defined by P.R. Taylor, and 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 by a graphical argument 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 also establish a link between the zeros of ${\cal T}_-(s)$ and of ${\cal T}_+s)$ with zeros of the Riemann zeta function $\zeta(2 s-1)$, and between the distribution functions of the zeros of the three functions.