Von Staudt-Clausen theorem

An important result on the arithmetic of the Bernoulli numbers $B _ { n }$, first published in 1840 by Th. Clausen [a1] without proof, and independently by K.G.C. von Staudt [a2]:

\begin{equation} \tag{a1} B _ { 2 n } = A _ { 2 n } - \sum _ { p - 1 | 2 n } \frac { 1 } { p }, \end{equation}

where $A _ { 2n }$ is an integer and the summation is over all prime numbers $p$ such that $p - 1$ divides $2 n$ (cf. also Prime number). Since $B _ { 1 } = - 1 / 2$, the identity (a1) holds also for $B _ { 1 }$. An immediate consequence of the von Staudt–Clausen theorem is the complete determination of the denominators of the Bernoulli numbers: If $B _ { 2 n } = N _ { 2 n } / D _ { 2 n }$, with $\operatorname { gcd } ( N _ { 2n } , D _ { 2n } ) = 1$, then

\begin{equation*} D _ { 2 n } = \prod _ { p - 1 | 2 n } p. \end{equation*}

The von Staudt–Clausen theorem has been extended in a variety of ways, among them:

1) K.G.C. von Staudt [a3] showed that the integer $A _ { 2n }$ in (a1) has the same parity as the number of primes $p$ such that $p - 1 \mid 2 n$; M.A. Stern [a4] derived a congruence modulo $4$ between these two quantities. Ch. Hermite [a5] found a recurrence relation among the $A _ { 2n }$, and R. Lipschitz [a6] derived an asymptotic relation for the $A _ { 2n }$.

2) The identity (a1) implies that $p B _ { 2 n } \equiv - 1 ( \operatorname { mod } p )$ if $p - 1 \mid 2 n$. L. Carlitz [a7] showed that $p B _ { 2 n } \equiv p - 1 ( \operatorname { mod } p ^ { h + 1 } )$ if $p$ is a prime number and $( p - 1 ) p ^ { h } | 2 n$. A different extension modulo higher powers of $p$ is given in [a8].

3) H.S. Vandiver [a9] extended (a1) to Bernoulli polynomials evaluated at rational arguments: Let $h$ and $k$ be relatively prime integers. If $n$ is even, then

\begin{equation*} k ^ { n } B _ { n } \left( \frac { h } { k } \right) = G _ { n } - \sum \frac { 1 } { p }, \end{equation*}

where $G_n$ is an integer and the summation is over all prime numbers $p$ such that $p - 1 | n$ but $p \nmid k$. If $n$ is odd, then $k ^ { n } B _ { n } ( h / k )$ is an integer, except for $n = 1$ and $k$ odd, in which case $k B _ { 1 } ( h / k ) = G _ { 1 } + 1 / 2$. It has also been shown [a10] that for all integers $h$, $k$, $n$ with $k \neq 0$ and $n \geq 1$, $k ^ { n } ( B _ { n } ( h / k ) - B _ { n } )$ is an integer.

4) Von Staudt [a3] proved a related result on the numerators of the Bernoulli numbers. Combined with (a1), it can be given in the following form: For any integer $n \geq 1$, the denominator of $B _ { n } / n$ is

\begin{equation*} d _ { n } = \prod _ { p - 1 | n } p ^ { 1 + v _ { p } ( n ) }, \end{equation*}

where the product is over all prime numbers $p$ such that $p - 1 | n$, and $v _ { p } ( n )$ denotes the highest power of $p$ dividing $n$.

5) R. Rado [a11] showed that, given a positive integer $n$, there exist infinitely many Bernoulli numbers $B _ { m }$ such that $B _ { m } - B _ { n }$ is an integer.

Numerous results on Bernoulli and allied numbers rely on the von Staudt–Clausen theorem. An early application was the explicit evaluation of Bernoulli numbers; more recent applications lie, for instance, in the theory of $p$-adic $L$-functions; see [a12], p. 56.

The von Staudt–Clausen theorem has been generalized in various directions. In particular, analogues of the theorem exist for most concepts of generalized Bernoulli numbers, among them the generalized Bernoulli numbers associated with Dirichlet characters (see, e.g., [a13]), degenerate Bernoulli numbers [a14], periodic Bernoulli numbers (or cotangent numbers) [a15], Bernoulli–Carlitz numbers [a16], Bernoulli–Hurwitz numbers [a17], and others. Another vast generalization was given by F. Clarke [a18].

References

[a1]	Th. Clausen, "Lehrsatz aus einer Abhandlung über die Bernoullischen Zahlen" Astr. Nachr. , 17 (1840) pp. 351–352
[a2]	K.G.C. von Staudt, "Beweis eines Lehrsatzes die Bernoulli'schen Zahlen betreffend" J. Reine Angew. Math. , 21 (1840) pp. 372–374
[a3]	K.G.C. von Staudt, "De Numeris Bernoullianis" , Erlangen (1845)
[a4]	M.A. Stern, "Über eine Eigenschaft der Bernoulli'schen Zahlen" J. Reine Angew. Math. , 81 (1876) pp. 290–294
[a5]	Ch. Hermite, "Extrait d'une lettre à M. Borchardt (sur les nombres de Bernoulli)" J. Reine Angew. Math. , 81 (1876) pp. 93–95
[a6]	R. Lipschitz, "Sur la représentation asymptotique de la valeur numérique ou de la partie entière des nombres de Bernoulli" Bull. Sci. Math. (2) , 10 (1886) pp. 135–144
[a7]	L. Carlitz, "A note on the Staudt–Clausen theorem" Amer. Math. Monthly , 64 (1957) pp. 19–21
[a8]	Zhi-Hong Sun, "Congruences for Bernoulli numbers and Bernoulli polynomials" Discrete Math. , 163 (1997) pp. 153–163
[a9]	H.S. Vandiver, "Simple explicit expressions for generalized Bernoulli numbers of the first order" Duke Math. J. , 8 (1941) pp. 575–584
[a10]	F. Clarke, I.Sh. Slavutskii, "The integrality of the values of Bernoulli polynomials and of generalised Bernoulli numbers" Bull. London Math. Soc. , 29 (1997) pp. 22–24
[a11]	R. Rado, "A note on Bernoullian numbers" J. London Math. Soc. , 9 (1934) pp. 88–90
[a12]	L.C. Washington, "Introduction to cyclotomic fields" , Springer (1982) (Second ed.: 1996)
[a13]	L. Carlitz, "Arithmetic properties of generalized Bernoulli numbers" J. Reine Angew. Math. , 202 (1959) pp. 174–182
[a14]	L. Carlitz, "A degenerate Staudt–Clausen theorem" Arch. Math. Phys. , 7 (1956) pp. 28–33
[a15]	K. Girstmair, "Ein v. Staudt–Clausenscher Satz für periodische Bernoulli–Zahlen" Monatsh. Math. , 104 (1987) pp. 109–118
[a16]	D. Goss, "Von Staudt for $F _ { q } ( T )$" Duke Math. J. , 45 (1978) pp. 887–910
[a17]	N. Katz, "The congruences of Clausen–von Staudt and Kummer for Bernoulli–Hurwitz numbers" Math. Ann. , 216 (1975) pp. 1–4
[a18]	F. Clarke, "The universal von Staudt theorems" Trans. Amer. Math. Soc. , 315 (1989) pp. 591–603