# Bernoulli numbers

## Introduction

The Bernoulli numbers B_{n} play an important role in several topics of mathematics. These numbers can be defined by the power series

where all numbers B_{n} are zero with odd index n > 1. The even-indexed rational numbers B_{n} alternate in sign. First values are

The values can be computed iteratively by the recurrence formula

which can be written symbolically as

The sequences of the numerators and denominators of B_{n} are A027641 and A027642, respectively.

## Sum of consecutive integer powers

Jacob Bernoulli Bernoulli1713 (1655-1705) introduced a sequence of rational numbers in his Ars Conjectandi, which was published posthumously in 1713.
He used these numbers, later called Bernoulli numbers, to compute the sum of consecutive integer powers.

This formula is given by

where S_{n}(x) is a polynomial of degree n + 1.

## Explicit formulas

An explicit formula was derived by Worpitzky Worpitzky1883 in 1883:

using the symbol

with S_{2}(n,k) being the Stirling numbers of the second kind.

He also gave another formula for S_{n}(x):

Furthermore, one has by means of iterated forward differences the relation

yielding the double sum

## Special values of the Riemann zeta function

The Bernoulli numbers are connected with the Riemann zeta function

on the positive real axis by Euler's celebrated formula for positive even n, also valid for n = 0:

The functional equation of ζ(s) leads to the following formula for negative integer arguments:

## Regular and irregular primes

In 1850 Kummer Kummer1850 introduced two classes of odd primes, later called regular and irregular (see, e.g., Hilbert Hilbert1897;Chap.~31).

An odd prime p is called *regular* if p does not divide the class number of the cyclotomic field
where is the set of p-th roots of unity; otherwise *irregular*.
Kummer then proved that Fermat's Last Theorem is true, that is

has no solution in positive integers x, y, and z, for the case when the exponent n is a regular prime.

He also provided an equivalent definition concerning Bernoulli numbers:

If p does not divide any of the numerators of the Bernoulli numbers B_{2}, B_{4}, …, B_{p−3}, then p is regular.

The first irregular primes below 100 are 37, 59, and 67; see A000928.

In 1915 Jensen Jensen1915 proved that infinitely many irregular primes p exist with the restriction p ≡ 3 (mod 4).
Carlitz Carlitz1954 later gave a short (and weaker) proof without any restriction on p.

Unfortunately, it is still an open question whether infinitely many regular primes exist. However, several computations (see, e.g., Hart, Harvey, and Ong HartHarveyOng2017) suggest that about 60% of all primes are regular, which agree with an expected distribution proposed by Siegel Siegel1964.

## Irregular pairs

The pair (p, ℓ) is called an *irregular pair*, if p divides the numerator of B_{ℓ} where ℓ is even and 2 ≤ ℓ ≤ p−3.

The *index of irregularity* i(p) is defined to be the number of such pairs belonging to p. If i(p) = 0, then p is regular, otherwise irregular.

The first irregular pairs are (37, 32), (59, 44), and (67, 58).
The irregular prime p = 157 is the least prime with i(p) = 2: (157, 62), (157, 110).

## Structure of the denominator

The denominator of B_{n} for positive even n
is given by the famous von Staudt-Clausen theorem, independently found by von Staudt Staudt1840 and Clausen Clausen1840 in 1840:

As a consequence, the denominator is squarefree and divisible by 6.

Given a Bernoulli number B_{n} with n even, Rado Rado1934 showed that there exist infinitely many even m such that

implying that the numbers B_{m} have the same denominator as B_{n}.

A special case is given for n = 2p, where p is an odd prime p ≡ 1 (mod 3):

See A112772, which is a subsequence of A051222; the sequence of the increasing denominators is A090801.

## Structure of the numerator

The unsigned numerator of the *divided Bernoulli number* B_{n}/n for positive even n equals 1 only for n = 2, 4, 6, 8, 10, 14;
otherwise the numerator consists of a product of powers of irregular primes:

Since B_{n}/n is a p-integer for all primes p with p−1 not dividing n, the structure of the numerator of B_{n} is given by

The additional left product is a *trivial factor* of B_{n} that divides n, see A300711.

For the signed numerators of B_{n} and B_{n}/n for even n see A000367 and A001067, respectively.

## Kummer congruences

The *Kummer congruences* describe the most important arithmetical properties of the Bernoulli numbers, which give a modular relation between these numbers.

Let φ denote Euler's totient function. Let n and m be positive even integers and p be a prime with p−1∤n.

If n ≡ m (mod φ(p^{r})) where r ≥ 1, then

Furthermore,

In 1851 Kummer Kummer1851 originally introduced these congruences without Euler factors and hence with restrictions on r and n.
He showed that the second congruence holds for n > r, whereas the first congruence was derived from the latter only for r = 1
(in these cases the Euler factors vanish). Subsequently, these congruences were widely generalized by several authors (see, e.g., Fresnel Fresnel1967).

## Constructing p-adic zeta functions

The values ζ(1−n) = −B_{n}/n and the Kummer congruences lead to the construction of p-adic zeta and L-functions, as introduced by Kubota and Leopoldt KubotaLeopoldt1964 in 1964.
One kind of their constructions deals with p-adic zeta functions defined in certain residue classes; for a detailed theory see Koblitz Koblitz1996;Chap. II.

For a prime p define

Let p ≥ 5 and ℓ ∈ {2, 4, …, p−3} be fixed. Define the p-adic zeta function on by

for p-adic integers s by taking any sequence (t_{ν})_{ν ≥ 1} of nonnegative integers that p-adically converges to s.

The Kummer congruences then states for r ≥ 1 that

when s ≡ s' (mod p^{r−1}) for nonnegative integers s and s'.
Since is dense in , the p-adic zeta function ζ_{(p,ℓ)}(s) uniquely extends, by means of the Kummer congruences, to a continuous function on .

## Zeros of p-adic zeta functions

The p-adic zeta function ζ_{(p,ℓ)}(s) can be written as a special Mahler expansion (Kellner Kellner2007):

with integral coefficients

One has the relation

Existence of a unique simple zero (Kellner Kellner2007):

If (p, ℓ) is an irregular pair and a_{1} ∈ , that is

then the p-adic zeta function ζ_{(p,ℓ)}(s) has a unique simple zero ξ_{(p,ℓ)} ∈ .
So far, no counterexample has been found violating the above congruence.

Example: In the case (p, ℓ) = (37, 32) one computes that

For more p-adic digits see Kellner Kellner2007.

## Irregular pairs of higher order

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 < φ(p^{r}) = (p−1)p^{r−1}.
For r = 1 this gives the usual definition of irregular pairs, since the condition p divides B_{n}/n is then equal to p divides B_{n}.

A zero of the p-adic zeta function ζ_{(p,ℓ)}(s) describes the irregular pairs (p, n) of higher order with n ≡ ℓ (mod p−1),
and vice versa (Kellner Kellner2007). For example

and

Irregular pairs of higher order can be effectively and easily computed using Bernoulli numbers with small indices. By this means one can even predict the extremely huge index of the first occurrence of the power 37^{37} as listed above; see A251782.

## Conjecture on the structure of the Bernoulli numbers

Under the assumption that every p-adic zeta function ζ_{(p,ℓ)}(s) has a unique simple zero ξ_{(p,ℓ)}
in case (p, ℓ) is an irregular pair, one has for even n ≥ 2 (Kellner Kellner2007):

where

The denominator can be described by poles (always lying at 0) and the numerator by zeros of p-adic zeta functions. Equivalently, the formula reads for the Bernoulli numbers:

The first product gives the trivial factor, the second product describes the product over irregular prime powers, and the third product yields the denominator of B_{n}.

Moreover, the formulas are valid for all irregular pairs (p, ℓ) with

This follows by computations of irregular pairs and cyclotomic invariants in that range by Hart, Harvey, and Ong HartHarveyOng2017. So far, no counterexample is known.

## Class numbers of imaginary quadratic fields

Let h(d) denote the class number of the imaginary quadratic field
of discriminant d < −4.
There is the following connection with the Bernoulli numbers due to Carlitz Carlitz1953.

If p > 3 is a prime with p ≡ 3 (mod 4), then

since h(−p) < p. This implies that p cannot divide the above Bernoulli number.
Therefore, an irregular pair (p, (p+1)/2) cannot exist when p ≡ 3 (mod 4).

## Asymptotic formulas

The Minkowski-Siegel mass formula states for positive integers n = 2k with 8 ∣ n that

where the sum runs over all even unimodular lattices Λ in dimension n and Aut(Λ) is the automorphism group of Λ.

The products of (divided) Bernoulli numbers with explicit asymptotic constants (Kellner Kellner2009) are given by

with

where is the Glaisher-Kinkelin constant A074962
and is the product over all Riemann zeta values at even positive integer arguments A080729.

# Euler numbers

## Introduction

The Euler numbers E_{n} may be defined by the power series of the hyperbolic secant function

which is an even function implying that all E_{n} = 0 with odd index n.
The even-indexed numbers E_{n} are integers and alternate in sign. The first values (A028296) are

The values can be computed iteratively for even n ≥ 2 by the recurrence formula

which can be written symbolically as

## E-irregular primes and pairs

A prime p is called *E-irregular*, if p divides at least one of the Euler numbers E_{2}, E_{4}, …, E_{p−3};
otherwise p is *E-regular*.

The pair (p, ℓ) is called an *E-irregular pair*, if p divides E_{ℓ} where ℓ is even and 2 ≤ ℓ ≤ p−3.
The *index of E-irregularity* i_{E}(p) is defined to be the number of such pairs belonging to p.

The first E-irregular pairs are (19, 10), (31, 22), and (43, 12); see A120337.
The E-irregular prime p = 241 is the least prime with i_{E}(p) = 2: (241, 210), (241, 238).

In 1954 Carlitz Carlitz1954 proved that infinitely many E-irregular primes exist. Later Ernvall Ernvall1975 showed the more specialized result that infinitely many E-irregular primes p ≢ ±1 (mod 8) exist.

As in the case of the Bernoulli numbers, it is still an open question whether infinitely many E-regular primes exist.

## Conjecture on the structure of the Euler numbers

For the Euler numbers one can state a similar conjectural formula as in the case of the Bernoulli numbers, though it is a bit more complicated.

One may conjecturally state for even n ≥ 2 that

where ξ_{(p,ℓ)} is the unique simple zero of a certain p-adic L-function
associated with an E-irregular pair (p, ℓ) ∈ when ℓ ≠ 0,
respectively, with a rare *exceptional* prime p with (p, 0) ∈ in case ℓ = 0.

## Class numbers of imaginary quadratic fields

Let h(d) denote the class number of the imaginary quadratic field of discriminant d < −4.
Due to Carlitz Carlitz1953 one has the following connection with the Euler numbers.

If p is a prime with p ≡ 1 (mod 4), then

since h(−4p) < p. Therefore p cannot divide the above Euler number. Consequently, an E-irregular pair (p, (p−1)/2) cannot exist when p ≡ 1 (mod 4).