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)\in\Psi^{\rm irr}_1 \atop n \equiv l \mods{p-1}} |p
(\chi_{(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
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.