Explicit formula for the discrete Laplace transform of the Möbius function, related special functions, and a criterion for the Riemann hypothesis
Year: 2026
Type: preprint
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 $Φ(e^{-t})=\sum_{n=1}^{\infty}μ(n)e^{-nt}$, as a function of the values of $ζ(s)$ and $ζ'(s)$ at the odd integers and as a function of the zeros of $ζ(s)$. A structural feature distinguishes this formula from the classical explicit formula for the Mertens function: the poles of $Γ(s)$ collide with the trivial zeros of $ζ(s)$, 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 $ζ(s)$ and show that they admit absolutely convergent closed forms as Möbius-weighted series of Bessel functions of rotated argument.
Source: arXiv (Cornell University)
Author: Liflandsky, Sergey
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7168298325