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Bernoulli number

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

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

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:

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

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

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:

$\sum_{j=0}^m{m+1\choose{j}}B_j = 0$

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/(e^x − 1), so that:

$\frac{x}{e^x-1} = \sum_{n=0}^{\infin} B_n \frac{x^n}{n!}$

for all values of x of absolute value less than 2π (the radius of convergence of this power series).

Sometimes the lower-case b_n 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.

n	B_n
0	1
1	−1/2
2	1/6
3	0
4	−1/30
5	0
6	1/42
7	0
8	−1/30
9	0
10	5/66
11	0
12	−691/2730
13	0
14	7/6

It can be shown that B_n = 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.

Contents

1 Assorted identities

2 Arithmetical properties of the Bernoulli numbers

2.1 p-adic continuity

3 Geometrical properties of the Bernoulli numbers

4 External links

Assorted identities

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

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

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

Arithmetical properties of the Bernoulli numbers

The Bernoulli numbers can be expressed in terms of the Riemann zeta function as B_n = − 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 c_n is the numerator of B_n/2n, then the order of $K_{4n-2}(\Bbb{Z})$ is −c_2n if n is even, and 2c_2n if n is odd.

Also related to divisibility is the von Staudt-Clausen theorem which tells is if we add 1/p to B_n 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 B_n 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 pB_p−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 $m \equiv n\, \bmod\,p^{b-1}(p-1)$ , then

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

Since $B n = - n ζ(1 - n)$ , this can also be written

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

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 $1 - p z$ 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 $a \not\equiv 1\, \bmod\, p-1$ , and so can be extended to a continuous function $ζ p (z)$ for all p-adic integers $\Bbb{Z}_p,\,$ 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 $n \ge 2$ involves Bernoulli numbers; if B is the numerator of B_4n/n, then $2 2 n - 2 (1 - 2 2 n - 1) B$ 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.)

External links

Categories: Number theory | Topology | Number sequences

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09-23-2007 01:00:40
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