Explicit formula for the discrete Laplace transform of the M\"obius function, related special functions, and a criterion for the Riemann hypothesis

Abstract

In this paper, we assume that all the zeros of the Riemann zeta function are simple. Under this assumption we give an explicit formula for the function $\Phi(e^{-t})=\sum_{n=1}^{\infty}\mu(n)e^{-nt}$, as a function of the values of $\zeta(s)$ and $\zeta'(s)$ at the odd integers and as a function of the zeros of $\zeta(s)$. A structural feature distinguishes this formula from the classical explicit formula for the Mertens function: the poles of $\Gamma(s)$ collide with the trivial zeros of $\zeta(s)$, producing double poles whose residues contain a logarithmic term. Using this formula, we give a criterion for the Riemann hypothesis: the bound $O(x^{-1/2})$ on the transform implies the Riemann hypothesis unconditionally, while the converse direction requires additional hypotheses on the zeros. We also introduce special entire functions related to $\zeta(s)$ and show that they admit absolutely convergent closed forms as M\"obius-weighted series of Bessel functions of rotated argument.