Bernoulli Number

The Bernoulli numbers  are a sequence of signed rational numbers that can be defined by the exponential generating function

(1)

These numbers arise in the series expansions of trigonometric functions, and are extremely important in number theory and analysis.

There are actually two definitions for the Bernoulli numbers. To distinguish them, the Bernoulli numbers as defined in modern usage (National Institute of Standards and Technology convention) are written , while the Bernoulli numbers encountered in older literature are written  (Gradshteyn and Ryzhik 2000). In each case, the Bernoulli numbers are a special case of the Bernoulli polynomials  or  with  and .

The Bernoulli number and polynomial should not be confused with the Bell numbers and Bell polynomial, which are also commonly denoted  and , respectively.

Bernoulli numbers defined by the modern definition are denoted  and sometimes called "even-index" Bernoulli numbers. These are the Bernoulli numbers returned, by example, by the Wolfram Language function BernoulliB[n].

The Bernoulli number  can be defined by the contour integral

(2)

where the contour encloses the origin, has radius less than  (to avoid the poles at ), and is traversed in a counterclockwise direction (Arfken 1985, p. 413).

The first few Bernoulli numbers  are

			(3)
			(4)
			(5)
			(6)
			(7)
			(8)
			(9)
			(10)
			(11)
			(12)
			(13)
			(14)
			(15)

(OEIS A000367 and A002445), with

(16)

for , 2, ....

The numbers of digits in the numerator of  for the , 4, ... are 1, 1, 1, 1, 1, 3, 1, 4, 5, 6, 6, 9, 7, 11, ... (OEIS A068399), while the numbers of digits in the corresponding denominators are 1, 2, 2, 2, 2, 4, 1, 3, 3, 3, 3, 4, 1, 3, 5, 3, ... (OEIS A092904). Both are plotted above.

The denominator of  is given by

(17)

where the product is taken over the primes  (Ex. 6.54 in Graham et al. 1994), a result which is related to the von Staudt-Clausen theorem.

The number of digits in the numerators of  for , 1, ... are 1, 1, 83, 1779, 27691, 376772, 4767554, 57675292, 676752609, 7767525702, ... (OEIS A103233), while the corresponding numbers of digits in the denominator are 1, 2, 5, 9, 13, 16, 24, ... (OEIS A114471). The values of the denominators of  for , 1, ... are 66, 33330, 342999030, 2338224387510, 9355235774427510, ... (OEIS A139822).

 for 1806, but for no other  (Kellner 2005).

The running maxima of denominators are 1, 6, 30, 42, 66, 2730, 14322, 1919190, ... (OEIS A100194), which occur for , 4, 6, 8, 12, 14, 32, 38, ... (OEIS A100195).

The fraction of  with even  that have denominator 6 is strictly positive (Jensen 1915), with similar results for other denominators (Erdős and Wagstaff 1980, Moreno and Wagstaff 2005).

Interestingly, a higher proportion of Bernoulli denominators equal 6 than any other value (Sunseri 1980), and the fraction of even Bernoulli numbers with denominator 6 is close to 1/6 (Erdős and Wagstaff 1980). S. Plouffe (pers. comm., Feb. 12, 2007) computed the fraction of even Bernoulli numbers with denominator 6 up to  and found it to be 0.1526... and still slowly decreasing.

The numbers of Bernoulli numbers less than or equal to 1, 10, , ... having denominator 6 are 0, 1, 10, 87, 834, ... (OEIS A114648), which approaches the decimal expansion of . The above histogram shows the fraction of denominators having given denominators for index up to . Ranked in order of frequency, the first few denominators appear to be 6, 30, 42, 66, 510, ... (OEIS A114649).

The only known Bernoulli numbers  having prime numerators occur for , 12, 14, 16, 18, 36, and 42 (OEIS A092132), corresponding to 5, , 7, , 43867, , and 1520097643918070802691 (OEIS A092133), with no other primes for  (E. W. Weisstein, Feb. 27, 2007). Wagstaff maintains a page of factorizations of Bernoulli number numerators.

The following table summarizes record computations of the th Bernoulli number , including giving the number of digits in the numerator.

	digits in numerator	denominator	date	reference
		14977732474858443510		Fee and Plouffe
		584711591137493802510	2002	Plouffe (2002)
		936123257411127577818510	Dec. 16, 2002	Kellner
		9601480183016524970884020224910	Feb. 10, 2003	Kellner
		936123257411127577818510	Oct. 8, 2005	O. Pavlyk (pers. comm.)
		9601480183016524970884020224910	Feb. 2008	O. Pavlyk (2008)
		394815332706046542049668428841497001870	Oct. 2008	D. Harvey (2008)

The denominator of  (mod 1) is given by the von Staudt-Clausen theorem, which also implies that the denominator of  is squarefree (Hardy and Wright 1979). Another curious property is that the fractional part of  has a decimal expansion period that divides , and there is a single digit before that period (Conway 1996). In particular, the periods of  for , 4, ... are 1, 1, 6, 1, 2, 6, 1, 16, 18, 2, 22, ... (OEIS A112828), and the corresponding values of  are 2, 4, 1, 8, 5, 2, 14, 1, 1, 10, ... (OEIS A112829).

Consider the generating function

(18)

which converges uniformly for  and all  (Castellanos 1988). Taking the partial derivative gives

			(19)
			(20)
			(21)

The solution to this differential equation can be found using separation of variables as

(22)

so integrating gives

			(23)
			(24)

But integrating (24) explicitly gives

			(25)
			(26)
			(27)

so

(28)

Solving for  and plugging back into (◇) then gives

(29)

(Castellanos 1988). Setting  and adding  to both sides then gives

(30)

Letting  then gives

(31)

for .

The Bernoulli numbers may also be calculated from

(32)

The Bernoulli numbers are given by the double sum

(33)

where  is a binomial coefficient. They also satisfy the sum

(34)

which can be solved for  to give a recurrence relation for computing . By adding  to both sides of (34), it can be written simply as

(35)

where the notation  means the quantity in question is raised to the appropriate power  and all terms of the form  are replaced with the corresponding Bernoulli numbers .

as well as the interesting sums

			(36)
			(37)
			(38)

(Lehmer 1935, Carlitz 1968, Štofka 2014), as well as the nice sum identity

(39)

(Gosper).

An asymptotic series for the even Bernoulli numbers is

(40)

Bernoulli numbers appear in expressions of the form , where , 2, .... Bernoulli numbers also appear in the series expansions of functions involving , , , , , , , , and .

An analytic solution exists for even orders,

			(41)
			(42)

for , 2, ..., where  is the Riemann zeta function. Another intimate connection with the Riemann zeta function is provided by the identity

(43)

An integral in terms of the Euler polynomial is given by

(44)

where  is an Euler polynomial (J. Crepps, pers. comm., Apr. 2002).

Bernoulli first used the Bernoulli numbers while computing . He used the property of the figurate number triangle that

(45)

along with a form for  which he derived inductively to compute the sums up to  (Boyer 1968, p. 85). For , the sum is given by

(46)

where again the notation  means the quantity in question is raised to the appropriate power  and all terms of the form  are replaced with the corresponding Bernoulli numbers . Note that it is common (e.g., Carlitz 1965) to simply write  with the understanding that after expansion,  is replaced by .

Written explicitly in terms of a sum of powers,

			(47)
			(48)
			(49)

where

(50)

Taking  gives Bernoulli's observation that the coefficients of the terms  sum to 1,

(51)

Ramanujan gave a number of curious infinite sum identities involving Bernoulli numbers (Berndt 1994).

Plouffe (pers. comm., Jun. 21, 2004) conjectured that the fractional parts of positive Bernoulli numbers of the form  satisfy either  or . However, there are many counterexamples, the first few of which occur for  (found by Plouffe also on Jun. 21, 2004), 6216210, 8128890, 10360350, 13548150, ... (OEIS A155125). Interestingly, all of these are numbers having a large number of factors in their primes factorizations, as summarized in the following table. The indices of these numbers having incrementally smallest value of  are given by 2072070, 6216210, 10360350, 18648630, 31081050, 35225190, 93243150, ... (OEIS A155126), which appear to tend to occur at positions in the original list that are powers of 2 (1, 2, 4, 8, 16, 18, 64, ...).

	factorization of
2072070		0.6664435068
6216210		0.6588649656
8128890		0.6648723198
10360350		0.6564013890

The older definition of the Bernoulli numbers, no longer in widespread use, defines  using the equations

			(52)
			(53)

or

			(54)
			(55)

for  (Whittaker and Watson 1990, p. 125). The  Bernoulli numbers may be calculated from the integral

(56)

and analytically from

(57)

for , 2, ..., where  is the Riemann zeta function.

The Bernoulli numbers  are a superset of the archaic ones  since

(58)

The first few Bernoulli numbers  are

			(59)
			(60)
			(61)
			(62)
			(63)
			(64)
			(65)
			(66)
			(67)
			(68)
			(69)

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