Bernoulli numbers

The sequence of rational numbers $B_0,B_1,\ldots,$ discovered by Jacob Bernoulli [1] in connection with the calculation of the sum of equal powers of natural numbers:

$$\sum_{k=0}^{m-1}k^n=\frac{1}{n+1}\sum_{s=0}^n\binom nsB_sm^{n+1-s}$$

$$n=0,1,\ldots,\quad m=1,2,\ldots.$$

The values of the first Bernoulli numbers are:

$$B_0=1,\quad B_1=-\frac12,\quad B_2=\frac16,\quad B_3=0,$$

$$B_4=-\frac{1}{30},\quad B_5=0,\quad B_6=\frac{1}{42},\quad B_7=0,$$

$$B_8=-\frac{1}{30},\quad B_9=0,\quad B_{10}=\frac{5}{66},\quad B_{11}=0.$$

All odd-indexed Bernoulli numbers except for $B_1$ are zero, and the signs of $B_{2n}$ alternate. Bernoulli numbers are the values of the Bernoulli polynomials at $x=0$: $B_n=B_n(0)$; they also often serve as the coefficients of the expansions of certain elementary functions into power series. Thus, for example,

$$\frac{x}{e^x-1}=\sum_{\nu=0}^\infty B_\nu\frac{x^\nu}{\nu!},\quad|x|<2\pi$$

(the so-called generating function of the Bernoulli numbers);

$$x\operatorname{cotg}x=\sum_{\nu=0}^\infty(-1)^\nu B_{2\nu}\frac{2^{2\nu}}{(2\nu)!}x^{2\nu},\quad|x|<\pi,$$

$$\operatorname{tg}x=\sum_{\nu=1}^\infty|B_{2\nu}|\frac{2^{2\nu}(2^{2\nu}-1)}{(2\nu)!}x^{2\nu},\quad|x|<\frac\pi2.$$

L. Euler in 1740 pointed out the connection between Bernoulli numbers and the values of the Riemann zeta-function $\zeta(s)$ for even $s=2m$:

$$\zeta(2m)=\sum_{\nu=1}^\infty\nu^{-2m}=\frac{(2\pi)^{2m}}{2(2n)!}|B_{2m}|.$$

Bernoulli numbers are used to express many improper integrals, such as

$$\int\limits_0^\infty\frac{x^{2n-1}dx}{e^{2\pi x}-1}=\frac{1}{4n}|B_{2n}|,\quad n=1,2,\ldots.$$

Certain relationships involving Bernoulli numbers are:

$$B_0=1,\quad\sum_{k=0}^{n-1}\binom nkB_k=0,\quad n\geq2,$$

(the recurrence formula);

$$B_{2n}=(-1)^{n-1}4n\int\limits_0^\infty\frac{x^{2n-1}}{e^{2\pi x}-1}dx,\quad n\geq1;$$

$$B_{2n}=(-1)^{n-1}\frac{2(2n)!}{(2\pi)^{2n}}\sum_{s=1}^\infty\frac{1}{s^{2n}},\quad n\geq1.$$

The estimates:

$$\frac{2(2n)!}{(2\pi)^{2n}}<(-1)^{n-1}B_{2n}\leq\frac{\pi^2(2n)!}{3(2\pi)^{2n}},\quad n\geq1,$$

hold. Extensive tables of Bernoulli numbers are available; for instance, [2] contains accurate values of $B_{2n}$ for $n\leq90$ and approximate values for $n\leq250$.

Bernoulli numbers have found many applications in mathematical analysis, number theory and approximate calculations.

References

[1]	J. Bernoulli, "Ars conjectandi" , Werke , 3 , Birkhäuser (1975) pp. 107–286 (Original: Basle, 1713)
[2]	H.T. Davis, "Tables of the higher mathematical functions" , 2 , Bloomington (1935)
[3]	L. Saalschuetz, "Vorlesungen über die Bernoullischen Zahlen" , Berlin (1893)
[4]	I.I. Chistyakov, "Bernoulli numbers" , Moscow (1895) (In Russian)
[5]	N. Nielsen, "Traité élémentaire des nombres de Bernoulli" , Paris (1923) Zbl 49.0099.03
[6]	V.A. Kudryavtsev, "Summation of powers of natural numbers and Bernoulli numbers" , Moscow-Leningrad (1936) (In Russian)
[7]	N.E. Nörlund, "Volesungen über Differenzenrechnung" , Springer (1924)
[8]	A.O. [A.O. Gel'fond] Gelfond, "Differenzenrechnung" , Deutsch. Verlag Wissenschaft. (1958) (Translated from Russian)
[9]	G.H. Hardy, "Divergent series" , Clarendon Press (1949)

Comments

Bernoulli numbers play an important role in the theory of cyclotomic fields and Fermat's last theorem, see [a1], pp. 40-41, and [a2]. E.g., if $p$ is an odd prime number that does not divide the numerators of $B_2,B_4,\ldots,B_{p-3}$, then $x^p+y^p=z^p$ has no solutions in $x,y,z\in\mathbf N$. (See also Cyclotomic field; Fermat great theorem.)

References

[a1]	S. Lang, "Cyclotomic fields" , Springer (1978)
[a2]	P. Ribenboim, "Thirteen lectures on Fermat's last theorem" , Springer (1979)