nLab Bernoulli number

Motivation

In Lie theory

In Lie theory, Bernoulli numbers appear as coefficients in the linear part of the Hausdorff series (and the recursive relation for the Dynkin Lie polynomials appearing in the Hausdorff series); this has consequences in deformation theory. The (determinant of the square root of the) inverse of its generating function appears (for variables in an adjoint representation) in an expression for the Duflo isomorphism in Lie theory and in its generalizations in knot theory etc.

In algebraic topology

In algebraic topology/cohomology Bernoulli numbers appear as the coefficients of the characteristic series of the A-hat genus (see there), and they (or equivalently, their generating functions) also appear in the expression for the Todd class.

The Bernoulli numbers are also proportional to the constant terms of the Eisenstein series and as such appear in the exponential form of the characteristic series of the Witten genus.

Finally they appear as the order of some groups in the image of the J-homomorphism (cf. Adams 65, section 2).

Of course, all of these cases are related to formal group laws. Formal groups bear also some other connections to Bernoulli numbers and generalizations like Bernoulli polynomials.

Elsewhere

The Riemann zeta-function $\zeta$ at negative integral values is proportional to the Bernoulli numbers as

$$\zeta(-n) = - \frac{B_{n+1}}{n+1} \,.$$

Bernoulli numbers appear also in umbral calculus. There are generalizations, for example, Bernoulli polynomials.

They also have applications in analysis (Euler-MacLaurin formula, with applications in numerical analysis).

Definition

The Bernoulli numbers $B_k$ are rational numbers given by their generating function, i.e. by the equation of functions/power series $x \mapsto f(x)$

$$\frac{x}{ e^x -1 } = \sum_{k = 0}^{\infty} \frac{\beta_k}{k !}x^k \,.$$

The $k$th Bernoulli number $B_k \in \mathbb{Q}$ is, depending on convention, either equal to $\beta_k$ (or sporadically, in older literature, to $(-1)^{k-1} \beta_{2k}$). If we take generating function $x+\frac{x}{e^x-1}=\frac{x e^x}{e^x -1}=\frac{-x}{e^{-x}-1}$ this only changes the sign of $B_1$ as all other odd Bernoulli numbers (in standard convention) vanish.

Properties

The von Staudt-Clausen theorem states that

$$B_{2n} + \sum_{p-1|2n} \frac{1}{p} \in\mathbb{Z}.$$

Related concepts

• modified Bernoulli number

• Bernoulli polynomial

• umbral calculus

References

General

Textbook accounts;

• Tom Apostol, Section 12.12 of: Introduction to Analytic Number Theory, Springer 1976 (springer:book/9780387901633)

Exposition:

• Pierre Cartier, chapter 3 in: Mathemagics, Séminaire Lotharingien de Combinatoire 44 (2000), Article B44d (pdf, pdf)

• John Baez, The Bernoulli numbers, 2003 expository notes (pdf)

See also:

• Wikipedia, Bernoulli number

• Wolfram MathWorld: Bernoulli number

• Bernoulli numbers page bernoulli.org

Lie theory, deformation theory, knot theory, geometry

• MathOverflow Todd class and Baker-Campbell-Hausdorff, or the curious number 12

• N. Durov, S. Meljanac, A. Samsarov, Z. Škoda, A universal formula for representing Lie algebra generators as formal power series with coefficients in the Weyl algebra, Journal of Algebra 309:1, pp.318-359 (2007) math.RT/0604096, MPIM2006-62

• Vinay Kathotia, Kontsevich’s universal formula for deformation quantization and the Campbell-Baker-Hausdorff formula, I, math.QA/9811174

• Emanuela Petracci, Functional equations and Lie algebras, PhD thesis, pdf

• E. Meinrenken, Clifford algebras and Lie theory, Springer

• Anton Alekseev, Bernoulli numbers, Drinfeld associators, and the Kashiwara–Vergne problem, slides, pdf

• Dror Bar-Natan, Stavros Garoufalidis, Lev Rozansky, Dylan Thurston, Wheels, wheeling, and the Kontsevich integral of the unknot (q-alg/9703025)

  (modified Bernoulli numbers in the universal Vassiliev invariant of the unknot)

In QFT and string theory

Appearance of Bernoulli numbers in perturbative quantum field theory and string theory:

• Gerald Dunne, Christian Schubert, Bernoulli Number Identities from Quantum Field Theory and Topological String Theory, Communications in Number Theory and Physics, Volume 7 (2013) Number 2, 225 - 249 (arXiv:math/0406610)

In algebraic topology

In algebraic topology:

• John Adams, On the groups $J(X)$ II, Topology 3 (2) (1965) (pdf)

In the context of the A-hat genus the Bernoulli numbers are discussed in

• Matthew Ando, Mike Hopkins, Charles Rezk, Section 10.2 of: Multiplicative orientations of KO-theory and the spectrum of topological modular forms, 2010 (pdf)

Formal groups and Bernoulli polynomials

• Piergiulio Tempesta, Formal groups, Bernoulli-type polynomials and L-series, Comptes Rendus de l Académie des Sciences - Series I - Mathematics 07/2007; 345 doi

• Stefano Marmi, Piergiulio Tempesta, Hyperfunctions, formal groups and generalized Lipschitz summation formulas, Nonlinear Analysis 03/2012; 75:1768-1777 doi

Analysis

• Terence Tao‘s blog: The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation