The Bernoulli Number Page

More than 7.4 million digits were omitted in the middle of the numerator!

The irregular pairs of higher order describe the first appearance of higher powers of irregular prime factors of B_n/n. An irregular pair (p,n) of order r has the property that p^r divides B_n/n with n < p^r-1(p-1). Note that n is always an even positive integer. There exists a criterion to check whether the sequence of irregular pairs of higher order is unique. It has been proven for all irregular primes below 12 million that there are only unique sequences. Writing sequences p-adically these pairs of higher order provide an approximation of a uniquely existing zero of the p-adic zeta function associated with an irregular pair. For definition and properties see link [6] below.

Example:

Irregular pairs of higher order can be effectively and easily calculated using Bernoulli numbers with small indices. By this means one can even predict the very large index of the first occurence of the power 37³⁷ as listed above.

Table of irregular pairs of order 10 for irregular primes below 1000: irrpairord.txt

A conjectural structural formula for the Bernoulli numbers

Assuming that all sequences of irregular pairs of higher order are unique resp. each p-adic zeta function associated with an irregular pair has a unique zero, one can describe the structure of divided Bernoulli numbers resp. the value of the Riemann zeta function at negative odd integer arguments as follows:

Under the proposed assumption, one can give some interpretation of the formula above. The denominator can be described by poles (always lying at 0) and the numerator by zeros of p-adic zeta functions measuring the distance to them using the p-adic metric induced by the standard ultrametric absolute value | |_p. Note that this formula is valid for all irregular primes below 12 million. For detailed statements see link [6] below. The formula conjecturally states for the Bernoulli numbers that

A conjectural structural formula for the Euler numbers

Similarly, for the Euler numbers E_n, n > 0 and even, one can state also a conjectural formula:

Here (p,l) are irregular pairs associated with the Euler numbers and the ξ _(p,l)  are certain zeros of  p-adic L-functions.

Asymptotic formula

The product of Bernoulli numbers is described by the following asymptotic formula

with an asymptotic constant C₂ = 4.855096646522...

where C₁ = 1.8210174514992... is the product over all values of the Riemann zeta function at even positive integers and A = 1.2824271291... is the Glaisher-Kinkelin constant