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:

${\frac {x}{e^{x}-1}}=\sum _{n=0}^{\infty }B_{n}{\frac {x^{n}}{n!}}$

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	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 B_n = 0 for all odd n other than 1. The appearance of the peculiar value B₁₂ = −691/2730 appears to rule out the possibility of a simple closed form for Bernoulli numbers.

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 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 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 c_n is the numerator of B_n/2n, then the order of $K_{4n-2}({\mathbb {Z} })$ 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.

Geometrical properties of the Bernoulli numbers

The Kervaire-Milnor formula for the number of diffentiable structures on the k-sphere for k, where k = 2n-1 for even n involves Bernoulli numbers:

$(-1)^{n \over 2}2^{n-3}(1-2^{n-1})P_{k}a_{n}B_{n}/n$,

where P_k is the order of the stable homotopy group of the k-stem of m-spheres, for m > k+1, and a_n is 1 if n is congruent to 0 mod 4, and 2 if n is congruent to 2 mod 4. (The formula in the topological literature differs because topologists use a different convention for naming Bernoulli numbers; this article uses the number theorist's convention.)

External link

• The first 498 Bernoulli Numbers from Project Gutenberg