Bernoulli number

In mathematics, the Bernoulli numbers Bn were first discovered in connection with the closed forms of the sums

<math>\sum_{k=0}^{m-1} k^n = 0^n + 1^n + 2^n + \cdots + {(m-1)}^n <math>

for various fixed values of n. The closed forms are always polynomials in m of degree n+1 and are called Bernoulli polynomials. The coefficients of the Bernoulli polynomials are closely related to the Bernoulli numbers, as follows:

<math>\sum_{k=0}^{m-1} k^n = {1\over{n+1}}\sum_{k=0}^n{n+1\choose{k}} B_k m^{n+1-k}.<math>

For example, taking n to be 1, we have 0 + 1 + 2 + ... + (m−1) = 1/2 (B0 m2 + 2 B1 m1) = 1/2 (m2m).

The Bernoulli numbers were first studied by Jakob Bernoulli, after whom they were named by Abraham de Moivre.

Bernoulli numbers may be calculated by using the following recursive formula:

<math>\sum_{j=0}^m{m+1\choose{j}}B_j = 0<math>

plus the initial condition that B0 = 1.

The Bernoulli numbers may also be defined using the technique of generating functions. Their exponential generating function is x/(ex − 1), so that:


\frac{x}{e^x-1} = \sum_{n=0}^{\infin} B_n \frac{x^n}{n!} <math> for all values of x of absolute value less than 2π (the radius of convergence of this power series).

Sometimes the lower-case bn is used in order to distinguish these from the Bell numbers.

The first few Bernoulli numbers (sequences A027641 ( and A027642 ( in OEIS) are listed below.


It can be shown that Bn = 0 for all odd n other than 1. The appearance of the peculiar value B12 = −691/2730 signals that the values of the Bernoulli numbers have no elementary description; in fact they are essentially values of the Riemann zeta function at negative integers, and are associated to deep number-theoretic properties, and so cannot be expected to have a trivial formulation.

The Bernoulli numbers also appear in the Taylor series expansion of the tangent and hyperbolic tangent functions, in the Euler-Maclaurin formula, and in expressions of certain values of the Riemann zeta function.

In note G of Ada Byron's notes on the analytical engine from 1842 an algorithm for computer-generated Bernoulli numbers was described for the first time.


Assorted identities

Leonhard Euler expressed the Bernoulli numbers in terms of the Riemann zeta as

<math>B_{2k}=2(-1)^{k+1}\frac {\zeta(2k)\; (2k)!} {(2\pi)^{2k}}. <math>

The nth cumulant of the uniform probability distribution on the interval [−1, 0] is Bn/n.

Arithmetical properties of the Bernoulli numbers

The Bernoulli numbers can be expressed in terms of the Riemann zeta function as Bn = − nζ(1 − n), which means in essence they are the values of the zeta function at negative integers. As such, they could be expected to have and do have deep arithmetical properties, a fact discovered by Kummer in his work on Fermat's last theorem.

Divisibility properties of the Bernoulli numbers are related to the ideal class groups of cyclotomic fields by a theorem of Kummer and its strengthening in the Herbrand-Ribet theorem, and to class numbers of real quadratic fields by Ankeny-Artin-Chowla. We also have a relationship to algebraic K-theory; if cn is the numerator of Bn/2n, then the order of <math>K_{4n-2}(\Bbb{Z})<math> is −c2n if n is even, and 2c2n if n is odd.

Also related to divisibility is the von Staudt-Clausen theorem which tells us if we add 1/p to Bn for every prime p such that p − 1 divides n, we obtain an integer. This fact immediately allows us to characterize the denominators of the non-zero Bernoulli numbers Bn as the product of all primes p such that p − 1 divides n; consequently the denominators are square-free and divisible by 6.

The Agoh-Giuga conjecture postulates that p is a prime number if and only if pBp−1 is congruent to −1 mod p.

p-adic continuity

An especially important congruence property of the Bernoulli numbers can be characterized as a p-adic continuity property. If b, m and n are positive integers such that m and n are not divisible by p − 1 and <math>m \equiv n\, \bmod\,p^{b-1}(p-1)<math>, then

<math>(1-p^{m-1}){B_m \over m} \equiv (1-p^{n-1}){B_n \over n} \,\bmod\, p^b.<math>

Since <math>B_n = -n\zeta(1-n)<math>, this can also be written

<math>(1-p^{-u})\zeta(u) \equiv (1-p^{-v})\zeta(v)\, \bmod \,p^b\,,<math>

where u = 1 − m and v = 1 − n, so that u and v are nonpositive and not congruent to 1 mod p − 1. This tells us that the Riemann zeta function, with <math>1-p^z<math> taken out of the Euler product formula, is continuous in the p-adic numbers on odd negative integers congruent mod p − 1 to a particular <math>a \not\equiv 1\, \bmod\, p-1<math>, and so can be extended to a continuous function <math>\zeta_p(z)<math> for all p-adic integers <math>\Bbb{Z}_p,\,<math> the p-adic Zeta function.

Geometrical properties of the Bernoulli numbers

The Kervaire-Milnor formula for the order of the cyclic group of diffeomorphism classes of exotic (4n−1)-spheres which bound parallelizable manifolds for <math>n \ge 2<math> involves Bernoulli numbers; if B is the numerator of B4n/n, then <math>2^{2n-2}(1-2^{2n-1})B<math> is the number of such exotic spheres. (The formula in the topological literature differs because topologists use a different convention for naming Bernoulli numbers; this article uses the number theorists' convention.)

See also

External links

fr:Nombre de Bernoulli it:Numeri di Bernoulli ja:ベルヌーイ数 nl:Bernoulli-getal zh:伯努利数


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