A spectral interpretation of zeros of certain functions

Abstract

We prove that all the zeros of certain meromorphic functions are on the critical line $\text{Re}(s)=1/2$, and are simple (except possibly when $s=1/2$). We prove this by relating the zeros to the discrete spectrum of an unbounded self-adjoint operator. Specifically, we show for $h(s)$ a meromorphic function with no zeros in $\text{Re}(s)>1/2$ and no poles in $\text{Re}(s)<1/2$, real-valued on $\R$, $\frac{h(1-s)}{h(s)}\ll |s|^{1-\epsilon}$ in $\text{Re}(s)>1/2$ and $\frac{h(1-s)}{h(s)}\notin L^2(1/2+i\R)$, the only zeros of $h(s)\pm h(1-s)$ are on the critical line. One instance of such a function $h$ is $h(s)=\xi(2s)$, the completed zeta-function. We use spectral theory suggested by results of Lax-Phillips and Colin de Verdi\`{e}re. This simplifies ideas of W. M\"{u}ller, J. Lagarias, M. Suzuki, H. Ki, O. Vel\'{a}squez Casta\~{n}\'{o}n, D. Hejhal, L. de Branges and P.R. Taylor.