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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 (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:
- <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:
- <math>
\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 (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A027641) and A027642 (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A027642) in OEIS) are listed below.
| n | Bn |
|---|
| 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 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 properies, 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.
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 the 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 is 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 Argoh-Giuga conjecture postulates that p is a prime number if and only if pBp−1 is congruent to −1 mod p.
p-adic continutity
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 nonpositve 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.)
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