Spectral interpretation of Riemann zeta zeros

Abstract

It is a well-known problem to identify the nontrivial zeros of the Riemann zeta function in terms of an eigenvalue problem. We here find such an eigenvalue problem for second order differential operators on the half-line. In a sense, our analysis pushesthe analysis of the zeta function over to the study of the Jacobi theta function, which may be thought of as the fundamental solution of the heat (or Schr\"odinger) equation on the unit circle (or the semi-infinite cylinder, if time is added). The eigenvalue problem takes the form $LD u+\alpha Lu=0$, where $L$ and $D$ are first-order differential operators, of which only $L$ involves the theta function. In a formal sense, then, $\alpha$ is an eigenvalue of the twisted operator $-LDL^{-1}$. Based on this formal thinking, we develop the notion of self-adjointness of the pair $(LD,L)$, to adapt the Hilbert-P\'olya idea to the spectral problem at hand.