Bernoulli numbers

The sequence of rational numbers discovered by Jacob Bernoulli [1] in connection with the calculation of the sum of equal powers of natural numbers:

The values of the first Bernoulli numbers are:

All odd-indexed Bernoulli numbers except for are zero, and the signs of alternate. Bernoulli numbers are the values of the Bernoulli polynomials at : ; they also often serve as the coefficients of the expansions of certain elementary functions into power series. Thus, for example,

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

L. Euler in 1740 pointed out the connection between Bernoulli numbers and the values of the Riemann zeta-function for even :

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

Certain relationships involving Bernoulli numbers are:

(the recurrence formula);

The estimates:

hold. Extensive tables of Bernoulli numbers are available; for instance, [2] contains accurate values of for and approximate values for .

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 de nombres de Bernoulli" , Paris (1923)
[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 is an odd prime number that does not divide the numerators of , then has no solutions in . (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)