Formulas
Asymptotic products
Product of values of the gamma function
Product of increasing powers of gamma values at rational arguments [2, Cor. 18, p. 93]:
with the constants
where γ is Euler's constant [OEIS A001620]
and 
is the Glaisher-Kinkelin constant [OEIS A074962].
Product of factorials
Special product ([2, Thms. 12, 13, p. 88] for k = 5):
with the constant
where (1+√5)/2 is the golden ratio [OEIS A001622].
Product of the Bernoulli numbers
Product of (divided) Bernoulli numbers [2, Thm. 21, p. 95]:
with the constants
where 
is the product over all Riemann zeta values at even positive integer arguments [OEIS A080729].
Number theory
Divisor function
Recurrence formula for the divisor function for even n ≥ 8 [Thesis, Satz 1.1.5, p. 8]:
with the convolution function
The first case n = 8 is known as Hurwitz's identity (1881)
Identities with Stirling and harmonic numbers
Define the polynomials
composed of Stirling numbers of the second kind S2(n,k) and harmonic numbers Hk by
The Genocchi numbers Gn [OEIS A036968] are related to the Bernoulli numbers by
The second identity of Bn is due to Worpitzky (1883):
The central value
is known to be
Involving the harmonic numbers leads to the identities [5, Thm. 1.3, p. 3]:
and
Riemann zeta function
Quotient of Riemann zeta values [4, Thm. 1.3, p. 2687]: If n ≥ 2 is even, then
with
where 
is a linear functional, which is connected with a special Dirichlet series,
and 
is a certain monic polynomial of degree n.
Moreover, if n = p + 1 with p an odd prime,
then 
is an Eisenstein polynomial and therefore irreducible over ℤ[x].
Conjecture on the structure of the Bernoulli numbers
Under the assumption that every p-adic zeta function ζ(p,ℓ)(s) has a unique simple zero ξ(p,ℓ) in case (p,ℓ) is an irregular pair, one has for even n ≥ 2 [1, Thm. 4.9, p. 16]:
where
The denominator can be described by poles (always lying at 0) and the numerator by zeros of p-adic zeta functions. Equivalently, the formula reads for the Bernoulli numbers:
Moreover, the formulas are valid for all irregular pairs (p,ℓ) with
Denominators of the Bernoulli polynomials
The denominator of Bn(x) − Bn, the nth Bernoulli polynomial without constant term, is given by the remarkable formula ([7], [8] with Jonathan Sondow, [OEIS A195441])
where sp(n) denotes the sum of the base-p digits of n. The finite product can be written with explicit sharp bounds as
where
The denominator of the nth Bernoulli polynomial Bn(x) can be described by a similar formula ([9] with Jonathan Sondow):
If {·} denotes the fractional part and n ≥ 1 is a fixed integer, then there is the surprising relation [C]:
