nLab modified Bernoulli number

Idea

A variant of the Bernoulli numbers introduced in Zagier 98.

Properties

Periodicity

The modified Bernoulli numbers $B_n$ are 12-periodic for $n$ odd Zagier, (A.4) (A.18).

References

General

Introduced in

• Don Zagier, A modified Bernoulli number, Nieuw Archief voor Wiskunde, 16:63–72, 1998 (pdf)

• Don Zagier, Hecke operators and periods of modular forms, Israel Math. Conf. Proc., 3 (1990), 321–336 (hdl:21.11116/0000-0004-3936-0)

Further discussion in

• Don Zagier, Curious and Exotic Identities for Bernoulli Numbers (pdf)

• Atul Dixit, Victor H. Moll, Christophe Vignat, The Zagier modification of Bernoulli numbers and a polynomial extension. Part I (arXiv:1209.4110)

In Chern-Simons theory

The “Wheels theorem”, saying that the perturbative Chern-Simons Wilson loop observable of the unknot is, as a universal Vassiliev invariant, a series of wheel-shaped Jacobi diagrams with coefficients the modified Bernoulli numbers, is due to

• Dror Bar-Natan, Thang T Q Le, Dylan Thurston, Two applications of elementary knot theory to Lie algebras and Vassiliev invariants, Geom. Topol. Volume 7, Number 1 (2003), 1-31 (euclid.gt/1513883092)

following

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