Bernoulli number

The Bernoulli numbers are a sequence of rational numbers, usually denoted by B_n, that arise frequently in number theory, analysis, and combinatorics. They were first studied by Jakob Bernoulli in connection with formulas for sums of powers of integers. The first few Bernoulli numbers are B₀ = 1, B₁ = ±1/2 (depending on convention), B₂ = 1/6, B₃ = 0, B₄ = –1/30, B₅ = 0, B₆ = 1/42, B₇ = 0, B₈ = –1/30, B₉ = 0, B₁₀ = 5/66, and so on. Except for B₁, all odd-index Bernoulli numbers are zero. They appear in the expansion of the tangent function, the Euler–Maclaurin formula, the values of the Riemann zeta function at even integers, and in the theory of cyclotomic fields.

Definition

There are several equivalent definitions of Bernoulli numbers. The most common one uses an exponential generating function:

\[
\frac{x}{e^x - 1} = \sum_{n=0}^\infty B_n \frac{x^n}{n!}, \qquad |x| < 2\pi.
\]

Here the sum runs over all nonnegative integers n. The coefficients B_n defined in this way are called the Bernoulli numbers. By convention, B₁ = –1/2. Some authors define the generating function as \(\frac{x e^x}{e^x - 1}\), which gives B₁ = +1/2; the other terms remain the same. The choice of sign for B₁ is a matter of convention and does not affect later properties that involve only even-index Bernoulli numbers.

Alternatively, Bernoulli numbers can be defined recursively by the formula

\[
\sum_{k=0}^n \binom{n+1}{k} B_k = 0, \qquad n \ge 1,
\]

with the initial condition B₀ = 1. This recursion follows from the generating function and is often used to compute the Bernoulli numbers sequentially.

Explicit Formulas and Values

Bernoulli numbers can be expressed by closed forms using Stirling numbers of the second kind or by sums over integer partitions. A standard explicit formula is

\[
B_n = \sum_{k=0}^n \frac{1}{k+1} \sum_{j=0}^k (-1)^j \binom{k}{j} j^n.
\]

For even n (n > 0), Bernoulli numbers alternate in sign and grow in absolute value extremely rapidly. For instance:

• B₀ = 1
• B₁ = –1/2
• B₂ = 1/6
• B₄ = –1/30
• B₆ = 1/42
• B₈ = –1/30
• B₁₀ = 5/66
• B₁₂ = –691/2730
• B₁₄ = 7/6
• B₁₆ = –3617/510
• B₁₈ = 43867/798
• B₂₀ = –174611/330

For odd n ≥ 3, B_n = 0.

Properties

Sign and Vanishing

The Bernoulli numbers satisfy B_2k+1 = 0 for all k ≥ 1. This can be deduced from the generating function, which is an even function in x after subtracting the term –x/2. The only nonzero odd Bernoulli number is B₁ = –1/2.

von Staudt–Clausen Theorem

A fundamental result in number theory is the von Staudt–Clausen theorem, which describes the denominator of B_2n in reduced form. It states that

\[
B_{2n} + \sum_{p-1 \mid 2n} \frac{1}{p}
\]

is an integer, where the sum runs over all primes p such that p–1 divides 2n. Consequently, the denominator of B_2n is the product of those primes. For example, for B₁₂ = –691/2730, the primes p with p–1 dividing 12 are 2, 3, 5, 7, 13; their product is 2·3·5·7·13 = 2730, matching the denominator.

Kummer Congruences

The Bernoulli numbers satisfy congruences modulo primes, known as Kummer congruences. They are essential in the study of Fermat’s Last Theorem and Iwasawa theory. If p is a prime and p–1 does not divide n, and if m ≡ n (mod p–1), then

\[
\frac{B_m}{m} \equiv \frac{B_n}{n} \pmod{p}.
\]

These congruences link Bernoulli numbers to p-adic L-functions.

Growth and Asymptotics

The even-index Bernoulli numbers grow super-exponentially. They satisfy the asymptotic formula

\[
|B_{2n}| \sim 4 \sqrt{\pi n} \left( \frac{n}{\pi e} \right)^{2n}.
\]

This is related to the connection with the Riemann zeta function, since for positive even integers

\[
\zeta(2n) = (-1)^{n+1} \frac{(2\pi)^{2n} B_{2n}}{2 (2n)!}.
\]

Applications

Faulhaber’s Formula

The Bernoulli numbers were originally introduced to give closed expressions for sums of powers of consecutive integers. Faulhaber’s formula states

\[
\sum_{k=1}^m k^n = \frac{1}{n+1} \sum_{j=0}^n (-1)^j \binom{n+1}{j} B_j \, m^{n+1-j}.
\]

For example, using B₀, B₁, and B₂ one obtains the familiar formulas for sums of first powers, squares, etc.

Euler–Maclaurin Formula

The Bernoulli numbers appear as coefficients in the Euler–Maclaurin formula, which relates sums to integrals:

\[
\sum_{k=a}^b f(k) = \int_a^b f(x)\,dx + \frac{f(a)+f(b)}{2} + \sum_{r=1}^\infty \frac{B_{2r}}{(2r)!} \left( f^{(2r-1)}(b) - f^{(2r-1)}(a) \right).
\]

This formula is widely used in numerical analysis and analytic number theory.

Riemann Zeta Function

Bernoulli numbers are intimately connected to the Riemann zeta function. For positive even integers,

\[
\zeta(2n) = (-1)^{n+1} \frac{(2\pi)^{2n} B_{2n}}{2 (2n)!}.
\]

For odd integers, no such simple formula exists. At negative integers, the zeta function is given by

\[
\zeta(1-n) = -\frac{B_n}{n}, \qquad n \ge 1.
\]

This relation provides values of the zeta function at non-positive integers.

Bernoulli Polynomials

Bernoulli numbers are constant terms of Bernoulli polynomials B_n(x), defined by the generating function

\[
\frac{t e^{xt}}{e^t - 1} = \sum_{n=0}^\infty B_n(x) \frac{t^n}{n!}.
\]

These polynomials satisfy B_n(0) = B_n and appear in summation formulas, the theory of the Gamma function, and in the construction of p-adic L-functions.

Topology and Algebraic Geometry

Bernoulli numbers also appear in topology, for instance in the calculation of the Todd class and the A-hat genus of manifolds. They are involved in Hirzebruch’s signature theorem and in the formulae for the Chern numbers of certain varieties.

History

The Bernoulli numbers were first studied by Jakob Bernoulli (also known as Jacques or James Bernoulli) in his posthumously published work Ars Conjectandi (1713). Bernoulli computed many of these numbers in connection with the summation of integer powers. Later, Leonhard Euler discovered their connection with the Riemann zeta function and established many of their analytic properties. In the nineteenth century, Ernst Kummer used Bernoulli numbers in his partial proof of Fermat’s Last Theorem, showing that if a prime p divides the numerator of the Bernoulli numbers B₂, B₄, …, B_p–3, then the theorem holds for exponent p (regular primes). The von Staudt–Clausen theorem was independently proved by Carl von Staudt and Thomas Clausen in 1840. In the twentieth century, Kenkichi Iwasawa developed Iwasawa theory, which deeply involves Bernoulli numbers and their p-adic generalizations.