  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.
