The structure of Bernoulli numbers

Abstract

We conjecture that the structure of Bernoulli numbers can be explicitly given in the closed form $$ B_n = (-1)^{\frac{n}{2}-1} \prod_{p-1 \nmid n} |n|_p^{-1} \prod\limits_{(p,l)\inΨ^{\rm irr}_1 \atop n \equiv l \mods{p-1}} |p (χ_{(p,l)} - {\textstyle \frac{n-l}{p-1}})|_p^{-1} \prod\limits_{p-1 \mid n} p^{-1} $$ where the $χ_{(p,l)}$ are zeros of certain $p$-adic zeta functions and $Ψ^{\rm irr}_1$ is the set of irregular pairs. The more complicated but improbable case where the conjecture does not hold is also handled; we obtain an unconditional structural formula for Bernoulli numbers. Finally, applications are given which are related to classical results.