Bernoulli number

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In mathematics, the Bernoulli numbers are a sequence of rational numbers with deep connections in number theory. They are closely related to the values of the Riemann zeta function at negative integers.

In Europe, they were first studied by Jakob Bernoulli, after whom they were named by Abraham de Moivre, and independently discovered, perhaps earlier, by Seki Takakazu. They 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.

Curiously, 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. This distinguishes the Bernoulli numbers as being the subject of the first published computer program ever.

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The Bernoulli numbers Bn 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. The coefficients of these polynomials are closely related to the Bernoulli numbers, as follows (this is known, not entirely justly, as Faulhaber's formula):

\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). See Faulhaber's formula for more details on this, including an umbral form.

One may also write

\sum_{k=0}^{m-1} k^n = \frac{B_{n+1}(m)-B_{n+1}(0)}{n+1},

where Bn + 1(m) is the (n + 1)th-degree Bernoulli polynomial.

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

\sum_{j=0}^m{m+1\choose{j}}B_j = 0
Failed to parse (syntax error): \sum_{j=0}^{m-1}{{m+1}/choose{j}}B_j = {m+1)B_m


for m > 0, and 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!}

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

These definitions can be shown to be equivalent using mathematical induction. The initial condition B0 = 1 is immediate from L'Hôpital's rule. To obtain the recurrence, multiply both sides of the equation by ex − 1. Then, using the Taylor series for the exponential function,

x = \left( \sum_{j=1}^{\infty} \frac{x^j}{j!} \right) \left( \sum_{k=0}^{\infty} \frac{B_k x^k}{k!} \right).

By expanding this as a Cauchy product and rearranging slightly, one obtains

x = \sum_{m=0}^{\infty} \left( \sum_{j=0}^{m} {m+1 \choose j} B_j \right) \frac{x^{m+1}}{(m+1)!}.

It is clear from this last equality that the coefficients in this power series satisfy the same recurrence as the Bernoulli numbers.

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

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

n Bn
0 1
1 −1 / 2 = −0.5
2 1 / 6 ≈ 0.1667
4 −1 / 30 ≈ −0.0333
6 1 / 42 ≈ 0.02381
8 −1 / 30 ≈ −0.0333
10 5 / 66 ≈ 0.07576
12 −691 / 2730 ≈ −0.2531
14 7 / 6 ≈ 1.1667
n Bn
16 −3617 / 510 ≈ −7.0922
18 43867 / 798 ≈ 54.9712
20 −174611 / 330 ≈ −529.124
22 854513 / 138 ≈ 6192.12
24 −236364091 / 2730 ≈ −86580.3
26 8553103 / 6 ≈ 1425517
28 −23749461029 / 870 ≈ −27298231
30 8615841276005 / 14322 ≈ 601580874
32 −7709321041217 / 510 ≈ −15116315767

It can be shown that Bn = 0 for all odd n other than 1. The Bernoulli numbers do have an explicit formula involving choice functions which is rather complicated. In fact they may be derived in a simple way from the values of the Riemann zeta function at negative integers (since ζ(1−n) = −Bn/n for all integers n greater than 1, but not at n = 1 since the zeta-function is −1/2 when its argument is 0), and are as a consequence connected to deep number-theoretic properties, and could not be expected to have a trivial formulation.

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

B_{2n} = (-1)^{n+1}\frac {2(2n)!} {(2\pi)^{2n}} \left[1+\frac{1}{2^{2n}}+\frac{1}{3^{2n}}+\frac{1}{4^{2n}}+\cdots\right].

The first few Bernoulli numbers might lead one to assume that they are all small. Later values belie this assumption, however. In fact, since the factor in the squared brackets is greater than 1 from this representation follows

|B_{2n}| > \frac{2 (2n)!}{(2 \pi)^{2 n}}

so that the sequence of Bernoulli numbers diverges quite rapidly for large indices. Substituting an asymptotic approximation for the factorial function in this formula gives an asymptotic approximation for the Bernoulli numbers. For example

|B_{2 n}| \sim 4 \sqrt{\pi n} \left(\frac{n}{ \pi e} \cdot \frac{480 n^2 + 9}{480 n^2 -1}\right)^{2n}.

This formula (Peter Luschny, 2007) is based on the connection of the Bernoulli numbers with the Riemann zeta function and on an approximation of the factorial function given by Gergő Nemes in 2007. For example this approximation gives

|B(1000)| \approx 5.318704469415522033\ldots\times 10^{1769} \,

which is off only by three units in the least significant digit displayed.

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

The following relations, due to Ramanujan, provide a more efficient method for calculating Bernoulli numbers:

m\equiv 0\,\bmod\,6\qquad {{m+3}\choose{m}}B_m={{m+3}\over3}-\sum_{j=1}^{m/6}{m+3\choose{m-6j}}B_{m-6j}
m\equiv 2\,\bmod\,6\qquad {{m+3}\choose{m}}B_m={{m+3}\over3}-\sum_{j=1}^{(m-2)/6}{m+3\choose{m-6j}}B_{m-6j}
m\equiv 4\,\bmod\, 6\qquad{{m+3}\choose{m}}B_m=-{{m+3}\over6}-\sum_{j=1}^{(m-4)/6}{m+3\choose{m-6j}}B_{m-6j}.

An identity of Carlitz:

(-1)^m \sum_{r=0}^m {m \choose r} B_{n+r} = (-1)^n \sum_{s=0}^n {n \choose s} B_{m+s}.

The Bernoulli numbers can be expressed in terms of the Riemann zeta function as Bn = − nζ(1 − n) for integers n > 1 (the formula is off by a sign at n = 1, as ζ(0) = -1/2) which intimately relates them to 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 K_{4n-2}(\Bbb{Z}) 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.

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 Bn = − nζ(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−s 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 ζp(s) for all p-adic integers \Bbb{Z}_p,\, the p-adic Zeta function.

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(n) is the numerator of B4n/n, then

22n − 2(1 − 22n − 1)B(n)

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.)

In some applications it is useful to be able to compute the Bernoulli numbers B0 through Bp − 3 modulo p, where p is a prime; for example to test whether Vandiver's conjecture holds for p, or even just to determine whether p is an irregular prime. It is not feasible to carry out such a computation using the above recursive formulae, since at least (a constant multiple of) p2 arithmetic operations would be required. Fortunately, faster methods have been developed (see Buhler et al) which require only O(p (log p)2) operations (see big-O notation).

  • Buhler, J., Crandall, R., Ernvall, R., Metsankyla, T., and Shokrollahi, M. "Irregular Primes and Cyclotomic Invariants to 12 Million." J. Symb. Comput. 11, 1-8, 2000.
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