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. In Japan, perhaps earlier, independently discovered 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.

In note G of Ada Byron's notes on the analytical engine from 1842, there is an algorithm for computer-generated Bernoulli numbers. This distinguishes the Bernoulli numbers as being the subject of the first ever published computer program.

1 Introduction
2 Values of the Bernoulli numbers
3 Asymptotic approximation
4 Inequalities
5 Integral representation and continuation.
6 The relation to the Euler numbers and π
7 An algorithmic view: the Seidel triangle
8 A combinatorial view: alternating permutations
9 Generalizations by polynomials
10 Assorted identities
11 Arithmetical properties of the Bernoulli numbers
- 11.1 p-adic continuity
12 Geometrical properties of the Bernoulli numbers
13 Efficient computation of Bernoulli numbers mod p
14 See also
15 External links
16 References

[edit] Introduction

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

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

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

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/(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).

These definitions can be shown to be equivalent using mathematical induction. The initial condition $B 0 = 1$ is immediate from L'Hôpital's rule. To obtain the recurrence, multiply both sides of the equation by $e x - 1$ . 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 b_n is used in order to distinguish these from the Bell numbers.

[edit] Values of the Bernoulli numbers

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

n	B_n
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	B_n
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 B_n = 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) = −B_n/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.

[edit] Asymptotic approximation

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.

[edit] Inequalities

The following two inequalities (Peter Luschny, 2007) hold for n > 8 and the arithmetic mean of the two bounds is an approximation of order n⁻³ to the Bernoulli numbers B_2n.

$4 \sqrt{ \pi n} \left(\frac{n}{\pi e} \right)^{2n} \left[1 + \frac{1}{24n}\right] < |B_{2 n}| < 4 \sqrt{ \pi n} \left( \frac{n}{ \pi e} \right)^{2n} \left[1+\frac{1}{24n}\left(1+\frac{1}{24n}\right)\right]$

Deleting the squared brackets on both sides and replacing on the right hand side the factor 4 by 5 gives simple inequalities valid for n > 1. These inequalities can be compared to related inequalities for the Euler numbers.

For example the low bound for 2n = 1000 is 5.31870445... × 10¹⁷⁶⁹, the high bound is 5.31870448... × 10¹⁷⁶⁹ and the mean is 5.31870446942... × 10¹⁷⁶⁹.

[edit] Integral representation and continuation.

The integral

$b(s) = 2e^{s i \pi/2}\int_{0}^{\infty} \frac{st^{s}}{e^{2\pi t}-1} \frac{dt}{t}$

has as special values b(2n) = B_2n for n > 0. The integral might be considered as a continuation of the Bernoulli numbers to the complex plane and this was indeed suggested by Peter Luschny in 2004.

For example b(3) = (3/2)ζ(3)Π^-3Ι and b(5) = -(15/2)ζ(5)Π^-5Ι. Here ζ(n) denotes the Riemann zeta function and Ι the imaginary unit. It is remarkable that already Leonhard Euler (Opera Omnia, Ser. 1, Vol. 10, p. 351) considered these numbers and calculated

$p = \frac{3}{2\pi^3}\left(1+\frac{1}{2^3}+\frac{1}{3^3}+etc.\ \right) = 0.0581522\ldots \ ,$

$q = \frac{15}{2\pi^{5}}\left(1+\frac{1}{2^5}+\frac{1}{3^5}+etc.\ \right) = 0.0254132\ldots \ .$

Euler's values are unsigned and real, but obviously his aim was to find a meaningful way to define the Bernoulli numbers at the odd integers n > 1.

[edit] The relation to the Euler numbers and π

The Euler numbers are a sequence of integers intimately connected with the Bernoulli numbers. Comparing the asymptotic expansions of the Bernoulli and the Euler numbers shows that the Euler numbers E_2n are in magnitude approximately (2/π)(4²ⁿ − 2²ⁿ) times larger than the Bernoulli numbers B_2n. In consequence

$\pi \ \sim \ 2 \left(2^{2n} - 4^{2n} \right) \frac{B_{2n}}{E_{2n}} \ .$

This asymptotic equation reveals that π lies in the common root of both the Bernoulli and the Euler numbers. In fact π could be computed from these rational approximations.

Bernoulli numbers can be expressed through the Euler numbers and vice versa. Since for n odd B_n = E_n = 0 (with the exception B₁), it suffices to regard the case when n is even.

$B_{n}=\sum_{k=0}^{n-1}\binom{n-1}{k}\frac{n}{4^n-2^n}E_{k} \quad (n=2,4,6,\ldots)$

$E_{n}=\sum_{k=1}^{n}\binom{n}{k-1}\frac{2^k-4^k}{k} B_{k} \quad (n=2,4,6,\ldots)$

These conversion formulas express an inverse relation between the Bernoulli and the Euler numbers. But more important, there is a deep arithmetic root common to both kinds of numbers, which can be expressed through a more fundamental sequence of numbers, also closely tied to π. These numbers are defined for n > 1 as

$S_{n}=2 \left(\frac{2}{\pi}\right)^{n}\sum_{k=-\infty}^{\infty}\left(4k+1\right)^{-n} \quad (k=0,-1,1,-2,2,\ldots)$

and S₁ = 1 by convention. The magic of these numbers lies in the fact that they turn out to be rational numbers. This was first proved by Leonhard Euler 1734 in a landmark paper `De summis serierum reciprocarum' (On the sums of series of reciprocals) and fascinated mathematicians ever since. The first few of these numbers are

$S_{n} = 1,1,\frac{1}{2},\frac{1}{3},\frac{5}{24}, \frac{2}{15},\frac{61}{720},\frac{17}{315},\frac{277}{8064},\frac{62}{2835},\ldots\quad (n=1,2,\ldots)$

The Bernoulli numbers and Euler numbers are best understood as special views of these numbers, selected from the sequence S_n and scaled for use in special applications.

$B_{n} =(-1)^{\left\lfloor n/2\right\rfloor }\left[ n\ \operatorname{even}\right] \frac{n! }{2^n-4^n}\, S_{n}\ , \quad (n=2,3,\ldots) \ ,$

$E_{n} =(-1)^{\left\lfloor n/2\right\rfloor }\left[ n\ \operatorname{even}\right] n! \, S_{n+1} \quad\qquad (n=0,1,\ldots) \ .$

The expression [n even] has the value 1 if n is even and 0 otherwise (Iverson bracket).

These identities show that the quotient of Bernoulli and Euler numbers at the beginning of this section is just the special case of R_n = 2 S_n / S_n+1 when n is even. The R_n are rational approximations to π and two successive terms always enclose the true value of π. Beginning with n = 1 the sequence starts

$2, 4, 3, \frac{16}{5}, \frac{25}{8}, \frac{192}{61}, \frac{427}{136}, \frac{4352}{1385},\ldots \quad \longrightarrow \pi \ .$

These rational numbers also appear in the last paragraph of Euler's paper cited above. But it was only in September 2007 that this classical sequence found its way into the Encyclopedia of Integer Sequences (A132049).

[edit] An algorithmic view: the Seidel triangle

The sequence S_n has another unexpected yet important property: The denominators of S_n divide the factorial (n − 1)!. In other words: the numbers T_n = S_n(n − 1)! are integers.

$T_{n} = 1,1,1,2,5,16,61,272,1385,7936,50521,353792,\ldots \quad (n=1,2,\ldots)$

Thus the above representations of the Bernoulli and Euler numbers can be rewritten in terms of this sequence as

$B_{n} =(-1)^{\left\lfloor n/2\right\rfloor }\left[ n\ \operatorname{even}\right] \frac{n }{2^n-4^n}\, T_{n}\ , \quad (n=2,3,\ldots) \ ,$

$E_{n} =(-1)^{\left\lfloor n/2\right\rfloor }\left[ n\ \operatorname{even}\right] T_{n+1} \quad\quad\qquad(n=0,1,\ldots) \ .$

These identities make it easy to compute the Bernoulli and Euler numbers: the Euler numbers E_n are given immediately by T_2n + 1 and the Bernoulli numbers B_2n are obtained from T_2n by some easy shifting, avoiding rational arithmetic.

What remains is to find a convenient way to compute the numbers T_n. However, already in 1877 Philipp Ludwig von Seidel published an ingenious algorithm which makes it extremely simple to calculate T_n.

Seidel's algorithm:

		(1)
	→	1	(1)
	(2)	2	1	←
→	2	4	5	(5)
(16)	16	14	10	5	←

[begin] Start by putting 1 in row 0 and let k denote the number of the row currently being filled. If k is odd, then put the number on the left end of the row k − 1 in the first position of the row k, and fill the row from the left to the right, with every entry being the sum of the number to the left and the number to the upper. At the end of the row duplicate the last number. If k is even, proceed similar in the other direction. [end]

Seidel's algorithm is in fact much more general (see the exposition of Dominique Dumont (1981)) and was rediscovered several times thereafter.

Similar to Seidel's approach D. E. Knuth and T. J. Buckholtz (1967) gave a recurrence equation for the numbers T_2n and recommended this method for computing B_2n and E_2n ‘on electronic computers using only simple operations on integers’.

V. I. Arnold rediscovered Seidel's algorithm in 1991 and later Millar, Sloane and Young popularized Seidel's algorithm under the name boustrophedon transform.

[edit] A combinatorial view: alternating permutations

Around 1880, three years after the publication of Seidel's algorithm, Désiré André proved a now classic result of combinatorial analysis. Looking at the first terms of the Taylor expansion of the trigonometric functions tan x and sec x André made a startling discovery.

$\tan x = 1\frac{x}{1!} + 2\frac{x^3}{3!} + 16\frac{x^5}{5!} + 272\frac{x^7}{7!} + 7936\frac{x^9}{9!} + \cdots$

$\sec x = 1 + 1\frac{x^2}{2!} + 5\frac{x^4}{4!} + 61\frac{x^6}{6!} + 1385\frac{x^8}{8!} + 50521\frac{x^{10}}{10!} + \cdots$

The coefficients are the Euler numbers of odd and even index, respectively. In consequence the ordinary expansion of tan x + sec x has as coefficients the rational numbers S_n.

$\tan x + \sec x = 1 + 1x + \frac{1}{2}x^2 + \frac{1}{3}x^3 + \frac{5}{24}x^4 + \frac{2}{15}x^5 + \frac{61}{720}x^6 + \cdots$

André then succeeded by means of a recurrence argument to show that the alternating permutations of odd size are enumerated by the Euler numbers of odd index (also called tangent numbers) and the alternating permutations of even size by the Euler numbers of even index (also called secant numbers).

[edit] Generalizations by polynomials

The Bernoulli polynomials can be regarded as generalizations of the Bernoulli numbers the same as the Euler polynomials are generalizations of the Euler numbers. However, the most beautiful generalization of this kind is the sequence of the Euler–Worpitzky–Chen polynomials W_n(x), which have only integer coefficients, in contrast to the rational coefficients of the Bernoulli and Euler polynomials. These polynomials are closely related to whole family of numbers under consideration here.

The sequence W_n(0) gives the signed tangent numbers and the sequence W_n(1) the signed secant numbers, the coefficients of the hyperbolic functions tanh(−x) and sech(−x) in exponential expansion, respectively. And the sequence W_n − 1(0) n / (2ⁿ − 4ⁿ) gives the Bernoulli numbers for n > 1.

[edit] Assorted identities

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

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

[edit] 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) 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 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 us 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.

[edit] 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^−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 $ζ p (s)$ for all p-adic integers $\Bbb{Z}_p,\,$ the p-adic Zeta function.

[edit] 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_(n) is the numerator of B_4n/n, then

2 2 n - 2 (1 - 2 2 n - 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.)

[edit] Efficient computation of Bernoulli numbers mod p

In some applications it is useful to be able to compute the Bernoulli numbers B₀ through B_p − 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).

[edit] See also

Bernoulli conjecture
poly-Bernoulli numbers

[edit] External links

[edit] References

L. Euler, "De summis serierum reciprocarum", Opera Omnia I.14, E 41, 73-86.
L. Seidel, "Über eine einfache Entstehungsweise der Bernoullischen Zahlen und einiger verwandten Reihen", Sitzungsber. Münch. Akad., 4 (1877), 157--187.
D. André, "Développements de sec x et tan x.", Comptes Rendus Acad. Sci., 88 (1879), 965--967.
D. André, "Mémoire sur les permutations alternées", J. Math., 7 (1881), 167--184.
Entringer R. C., "A combinatorial interpretation of the Euler and Bernoulli numbers", Nieuw. Arch. V. Wiskunde, 14 (1966), 241--6.
D. E. Knuth and T. J. Buckholtz, "Computation of Tangent, Euler, and Bernoulli Numbers", Mathematics of Computation, 21 (1967), 663--688.
D. Dumont and G. Viennot, "A combinatorial interpretation of Seidel generation of Genocchi numbers", Ann. Discrete Math., 6 (1980), 77--87.
D. Dumont, "Matrices d'Euler-Seidel", Séminaire Lotharingien de Combinatoire, 1981. http://emis.u-strasbg.fr/journals/SLC/opapers/s05dumont.html
A. Ayoub, "Euler and the Zeta Function", Amer. Math. Monthly, 74 (1981), 1067--1086.
V. I. Arnold, "Bernoulli-Euler updown numbers associated with function singularities, their combinatorics and arithmetics", Duke Math. J., 63 (1991), 537--555.
Buhler, J., Crandall, R., Ernvall, R., Metsankyla, T., and Shokrollahi, M. "Irregular Primes and Cyclotomic Invariants to 12 Million." Journal of Symbolic Computation, Volume 31, Issues 1–2, January 2001, pages 89–96.

Categories: Number theory | Topology | Integer sequences