A Problem About Bernoulli Numbers

This section was written by Ralph Greenberg.

Definition 4.4.1 (Irregular Prime)   A prime $p$ is said to be irregular if $p$ divides the numerator of a Bernoulli number $B_j$ , where $2 \le j < p-1$ and $j$ is even. (For odd $j$ , one has $B_j = 0$ .)

The index of irregularity for a prime $p$ is the number of such $j$ .'s There is considerable numerical data concerning the statistics of irregular primes - the proportion of $p < x$ which are irregular or which have a certain index of irregularity. (See Irregular primes and cyclotomic invariants to four million, Buhler et al., in Math. of Comp., vol. 61, (1993), 151-153.)

Let

$$C_j = \frac{1}{p}\left(\frac{B_j}{j} - \frac{B_{j+p-1}}{j+ p-1}\right)$$

for each $j$ as above. According to the Kummer congruences, $C_j$ is a $p$ -integer, i.e., its denominator is not divisible by $p$ . But its numerator could be divisible by $p$ . This happens for $p = 13$ and $j = 4$ .

Problem 4.4.2   Obtain numerical data for the divisibility of the numerator of $C_j$ by a prime $p$ analogous to that for the $B_j$ 's.

Motivation: It would be interesting to find an example of a prime $p$ and an index $j$ (with $2 \le j < p-1$ , $j$ even) such that $p$ divides the numerator of both $B_j$ and $C_j$ . Then the $p$ -adic $L$ -function for a certain even character of conductor $p$ (namely, the $p$ -adic valued character $\omega^j$ , where $\omega$ is the character characterized by $\omega(n) \equiv n \pmod{p\mathbb{Z}_p}$ for $n \in \mathbb{Z}$ ) would have at least two zeros. No such example exists for $p < 16,000,000$ . The $p$ -adic $L$ -functions for those primes have at most one zero. If the statistics for the $C_j$ 's are similar to those for the $B_j$ 's, then a probabilistic argument would suggest that examples should exist.

Problem 4.4.3   Computation of $B_j$ for a specific $j$ is very efficient in PARI, hence in SAGE via the command bernoulli. Methods for computation of $B_j\pmod{n}$ for a large range of $j$ are described in Irregular primes and cyclotomic invariants to four million, Buhler et al. Implement the method of Buhler et al. in SAGE.