A spectral interpretation of zeros of certain functions

Kim Klinger-Logan

Published 2017-06-26Version 1

We prove that all the zeros of a certain family of meromorphic functions are on the critical line $\text{Re}(s)=1/2$ and are simple (except for possibly $s=1/2$), by relating the zeros to the discrete spectrum of unbounded self-adjoint operators. For example, for $h(s)$ a meromorphic function with no zeros in $\text{Re}(s)>1/2$ with $h(s)$ real-valued on $\mathbb{R}$, and $\frac{h(1-s)}{h(s)}\ll |s|^{1-\epsilon}$ in $\text{Re}(s)>1/2$, the only zeros of $h(s)\pm h(1-s)$ are on the critical line. One such instance of this result is $h(s)=\xi_k(2s)$ the completed zeta-function of a number field $k$ or, more generally, many self-dual automorphic $L$-functions. We use spectral theory suggested by results of Lax-Phillips and ColinDeVerdi\`ere. This simplifies ideas of W. M\"uller, J. Lagarias, M. Suzuki, H. Ki, O. Vel\'asquez Casta\~n\'on, D. Hejhal, L. de Branges and P.R. Taylor.