Bounds of divided universal Bernoulli numbers and universal Kummer congruences

Let $p$ be a prime. We obtain good bounds for the $p$-adic sizes of the coefficients of the divided universal Bernoulli number $\tfrac{\hat{B}_n}{n}$ when $n$ is divisible by $p-1$. As an application, we give a simple proof of Clarke's 1989 universal von Staudt theorem. We also establish the universal Kummer congruences modulo $p$ for the divided universal Bernoulli numbers for the case $(p-1)\vert n$, which is a new result.