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 $\mathbb{R}$, $\frac{h(1-s)}{h(s)}\ll |s{|}^{1-\epsilon }$ in $\text{Re}(s)>1/2$ and $\frac{h(1-s)}{h(s)}\notin L^2(1/2+i\mathbb{R})$, 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 Verdière. This simplifies ideas of W. Müller, J. Lagarias, M. Suzuki, H. Ki, O. Velásquez Castañón, D. Hejhal, L. de Branges and P.R. Taylor.

13 pages; Thm 2 and converse results removed

References in corpus (2)

• The Riemann Hypothesis for Symmetrised Combinations of Zeta Functions
• Majoration du nombre de zéros d'une fonction méromorphe en dehors d'une droite verticale et applications