NationMaster - Encyclopedia: Bernoulli number

In mathematics, the Bernoulli numbers are a sequence of rational numbers with deep connections in number theory. Although easy to calculate, the values of the Bernoulli numbers have no elementary description; they are closely related to the values of the Riemann zeta function at negative integers. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ... In mathematics, a rational number (commonly called a fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... In mathematics, the Riemann zeta-function, named after Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ...

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 Kowa - 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. To meet Wikipedias quality standards, this article or section may require cleanup. ... Abraham de Moivre. ... Kowa Seki (Seki Takakazu, 関 孝和) (1642? – October 24, 1708) was a Japanese mathematician. ... As the degree of the Taylor series rises, it approaches the correct function. ... In mathematics, the word tangent has two distinct but etymologically-related meanings: one in geometry and one in trigonometry. ... In mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. ... In mathematics, the Euler-Maclaurin formula provides a powerful connection between integrals (see calculus) and sums. ...

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 one of the first computer programs ever. In 1840 Charles Babbage was invited to give a seminar at the University of Turin about his analytical engine. ... 1842 was a common year starting on Saturday (see link for calendar). ... In mathematics, computing, linguistics, and related disciplines, an algorithm is a procedure (a finite set of well-defined instructions) for accomplishing some task which, given an initial state, will terminate in a defined end-state. ...

1 Introduction
2 Values of the Bernoulli numbers
3 Asymptotic approximation
4 Assorted identities
5 Arithmetical properties of the Bernoulli numbers
- 5.1 p-adic continuity
6 Geometrical properties of the Bernoulli numbers
7 Efficient computation of Bernoulli numbers mod p
8 See also
9 External links
10 References

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): In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ... In mathematics, a coefficient is a multiplicative factor of a certain object such as a variable (for example, the coefficients of a polynomial), a basis vector, a basis function and so on. ... In mathematics, Faulhabers formula, named after Johann Faulhaber, expresses the sum as a (p + 1)th-degree polynomial function of x, the coefficients involving Bernoulli numbers. ...

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

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. In mathematics, Faulhabers formula, named after Johann Faulhaber, expresses the sum as a (p + 1)th-degree polynomial function of x, the coefficients involving Bernoulli numbers. ... In mathematics, before the 1970s, the term umbral calculus was understood to mean the surprising similarities between otherwise unrelated polynomial equations, and certain shadowy techniques that can be used to prove them. ...

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. In mathematics, the Bernoulli numbers Bn were first discovered in connection with the closed forms of the sums for various fixed values of n. ...

Bernoulli numbers may be calculated by using the following recursive formula: A Sierpinski triangle â€”a confined recursion of triangles to form a geometric lattice. ...

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

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: In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural numbers. ... In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural numbers. ...

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

for all values of x of absolute value less than 2π (the radius of convergence of this power series). In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ... In mathematics, the radius of convergence of a power series where the center a and the coefficients cn are complex numbers (which may, in particular, be real numbers) is the nonnegative quantity r (which may be a real number or âˆž) such that the series converges if and diverges if In... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ...

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, Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ... In calculus, lHÃ´pitals rule (alternatively lHospitals rule) uses derivatives to help compute limits with indeterminate forms. ... As the degree of the Taylor series rises, it approaches the correct function. ... The exponential function is one of the most important functions in mathematics. ...

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

By expanding this as a Cauchy product and rearranging slightly, one obtains In mathematics, the Cauchy product, named in honor of Augustin Louis Cauchy, of two strictly formal (not necessarily convergent) series usually, of real or complex numbers, is defined by a discrete convolution as follows. ...

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

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. The Bell numbers, named in honor of Eric Temple Bell, are a sequence of integers arising in combinatorics that begins thus (sequence A000110 in OEIS): In general, Bn is the number of partitions of a set of size n. ...

Values of the Bernoulli numbers

The first few non-zero Bernoulli numbers (sequences A027641 and A027642 in OEIS) are listed below. The On-Line Encyclopedia of Integer Sequences (OEIS) is a web-based searchable database of integer sequences. ...

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 appearance of the peculiar value B₁₂ = −691/2730 suggests that the values of the Bernoulli numbers have no elementary description. 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 at s = 0), and are as a consequence connected to deep number-theoretic properties, and could not be expected to have a trivial formulation. In mathematics, the Riemann zeta-function, named after Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ...

The first few Bernoulli numbers might lead one to assume that they are all small. Later values belie this assumption, however. In fact, it can be shown that

$|B_{2k}| > frac{2 (2k)!}{(2 pi)^{2 k}}$

so that the sequence of Bernoulli numbers diverges quite rapidly for large indices.

Asymptotic approximation

For even n the Bernoulli numbers can be approximated by

$|B_{n}| sim 2 sqrt{2pi n} left(frac{n}{2 pi e}right)^n left(frac{120 n^2 + 9}{120 n^2 -1}right)^n$

This formula (Peter Luschny, 2007) is based on the well known 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

$|B(1000)| approx 0.5318704469415522033ldotstimes 10^{1770} ,$

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

Assorted identities

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

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

The nth cumulant of the uniform probability distribution on the interval [−1, 0] is B_n/n. // Cumulants of probability distributions In probability theory and statistics, the cumulants Îºn of the probability distribution of a random variable X are given by In other words, Îºn/n! is the nth coefficient in the power series representation of the logarithm of the moment-generating function. ... In mathematics, the continuous uniform distributions are probability distributions such that all intervals of the same length are equally probable. ... In mathematics and statistics, a probability distribution, more properly called a probability density, assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ...

The following relations, due to Ramanujan, provide a more efficient method for calculating Bernoulli numbers: Ramanujan Srinivasa Aiyangar Ramanujan (Tamil: ஸ்ரீனிவாஸ ஐயங்கார் ராமானுஜன்) (December 22, 1887 – April 26, 1920) was a groundbreaking Indian mathematician. ...

$mequiv 0,bmod,6qquad {{m+3}choose{m}}B_m={{m+3}over3}-sum_{j=1}^{m/6}{m+3choose{m-6j}}B_{m-6j}$

$mequiv 2,bmod,6qquad {{m+3}choose{m}}B_m={{m+3}over3}-sum_{j=1}^{(m-2)/6}{m+3choose{m-6j}}B_{m-6j}$

$mequiv 4,bmod, 6qquad{{m+3}choose{m}}B_m=-{{m+3}over6}-sum_{j=1}^{(m-4)/6}{m+3choose{m-6j}}B_{m-6j}$

An identity of Carlitz:

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

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 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. Ernst Eduard Kummer (29 January 1810 in Sorau, Brandenburg, Prussia - 14 May 1893 in Berlin, Germany) was a German mathematician. ... Pierre de Fermat Problem II.8 in the Arithmetica of Diophantus, annotated with Fermats comment which became Fermats Last Theorem (edition of 1670). ...

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. In mathematics the theory of algebraic number fields gives rise to a finite abelian group constructed from each such field, its ideal class group. ... In mathematics, the n-th roots of unity or de Moivre numbers, named after Abraham de Moivre (1667 - 1754), are complex numbers located on the unit circle. ... The Herbrand-Ribet theorem is a strengthening of Kummers theorem to the effect that the prime p divides the class number of the cyclotomic field of pth roots of unity if and only if p divides the denominator of the nth Bernoulli number Bn for some n, 0 < n... In number theory, the Ankeny-Artin-Chowla congruence is a result published in 1951 by N.C. Ankeny, Emil Artin and S. Chowla. ... In mathematics, algebraic K-theory is an advanced part of homological algebra concerned with defining and applying a sequence Kn(R) of functors from rings to abelian groups, for n = 0,1,2, ... . Here for traditional reasons the cases of K0 and K1 are thought of in somewhat different terms...

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. In number theory, the von Staudt-Clausen theorem is a result on the fractional part of Bernoulli numbers. ...

The Agoh-Giuga conjecture postulates that p is a prime number if and only if pB_p−1 is congruent to −1 mod p. In number theory the Agoh-Giuga conjecture on the Bernoulli numbers Bk postulates that p is a prime number if and only if pBp−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^−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 $a notequiv 1, bmod, p-1$ , 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. The p-adic number systems were first described by Kurt Hensel in 1897. ...

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 In mathematics, an exotic sphere is a differential manifold M, such that from a topological point of view M is a sphere, but not from the point of view of its differential structure. ... In mathematics, a parallelizable manifold M is a smooth manifold of dimension n having vector fields V1, ..., Vn, such that at any point P of M the tangent vectors Vi, P provide a basis of the tangent space at P. Equivalently, the tangent bundle is a trivial bundle, so that...

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.)

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). Vandivers conjecture concerns a property of algebraic number fields. ... In mathematics, regular primes are a certain kind of prime numbers. ... The Big O notation is a mathematical notation used to describe the asymptotic behavior of functions. ...

External links

The Bernoulli Number Page
The first 498 Bernoulli Numbers from Project Gutenberg
The first 10,000 Bernoulli numbers
Bernoulli number programs at LiteratePrograms

Project Gutenberg (often abbreviated as PG) is a volunteer effort to digitize, archive, and distribute cultural works. ...

References

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.

Categories: Number theory | Topology | Integer sequences

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PlanetMath: Bernoulli number (164 words)
	and, in fact, the Bernoulli numbers are usually defined as the coefficients that appear in such expansion.
	These combinatorial numbers occur in a number of contexts; the most elementary is perhaps that they occur in the formulas for the sum of the
	This is version 8 of Bernoulli number, born on 2001-10-15, modified 2006-11-29.

Bernoulli number - Wikipedia, the free encyclopedia (1035 words)
	In mathematics, the Bernoulli numbers are a sequence of rational numbers with deep connections in number theory.
	The Bernoulli numbers may also be defined using the technique of generating functions.
	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.

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