Phase operator on $L^2(\mathbb{Q}_p)$ and the zeroes of Fisher and Riemann

TL;DR

This paper explores a novel operator framework on p-adic spaces to potentially relate the distribution of Riemann zeta zeroes to phase operators in quantum mechanics, extending to Dirichlet L-functions.

Contribution

It constructs a phase operator conjugate to the Vladimirov derivative on $L^2(Q_p)$ and discusses its potential connection to the zeroes of the Riemann zeta function and Dirichlet L-functions.

Findings

Constructed a phase operator on $L^2(p^{-1}Z_p)$.

Proposed a framework linking p-adic operators to zeroes of zeta functions.

Extended the approach to Dirichlet L-functions.

Abstract

The distribution of the non-trivial zeroes of the Riemann zeta function, according to the Riemann hypothesis, is tantalisingly similar to the zeroes of the partition functions (Fisher and Yang-Lee zeroes) of statistical mechanical models studied by physicists. The resolvent function of an operator akin to the phase operator, conjugate to the number operator in quantum mechanics, turns out to be important in this approach. The generalised Vladimirov derivative acting on the space $L^2(\mathbb{Q}_p)$ of complex valued locally constant functions on the $p$-adic field is rather similar to the number operator. We show that a `phase operator' conjugate to it can be constructed on a subspace $L^2(p^{-1}\mathbb{Z}_p)$ of $L^2(\mathbb{Q}_p)$. We discuss (at physicists' level of rigour) how to combine this for all primes to possibly relate to the zeroes of the Riemann zeta function. Finally, we extend these results to the family of Dirichlet $L$-functions, using our recent construction of Vladimirov derivative like pseudodifferential operators associated with the Dirichlet characters.