Bernd C. Kellner
Göttingen, Germany
Email: bk (at) bernoulli.org


Über irreguläre Paare höherer Ordnungen.
Diplomarbeit. Mathematisches Institut der Georg-August-Universität zu Göttingen, Germany, 2002.
Online: irrpairord.pdfDownload  (891 KB)


  1. On irregular prime power divisors of the Bernoulli numbers, Math. Comp. 76 (2007), 405–441.
    Zbl: 1183.11012DOI: 10.1090/S0025-5718-06-01887-4
  2. On asymptotic constants related to products of Bernoulli numbers and factorials, Integers 9 (2009), Article A08, 83–106.
    Zbl: 1163.11014DOI: 10.1515/INTEG.2009.009Link: Integers Vol. 9
  3. On stronger conjectures that imply the Erdős–Moser conjecture, J. Number Theory 131 (2011), 1054–1061.
    Zbl: 1267.11031DOI: 10.1016/j.jnt.2011.01.004
  4. On quotients of Riemann zeta values at odd and even integer arguments, J. Number Theory 133 (2013), 2684–2698.
    Zbl: 1290.11118DOI: 10.1016/j.jnt.2013.02.008
  5. Identities between polynomials related to Stirling and harmonic numbers, Integers 14 (2014), Article A54, 1–22.
    Zbl: 1315.11018Link: Integers Vol. 14
  6. The topology of Stein fillable manifolds in high dimensions II (with an appendix by Bernd C. Kellner), Geom. Topol. 19 (2015), 2995–3030.
    Authors: Jonathan Bowden, Diarmuid Crowley, András I. Stipsicz. Appendix by Bernd C. Kellner.
    Zbl: 1380.32016DOI: 10.2140/gt.2015.19.2995
  7. On a product of certain primes, J. Number Theory 179 (2017), 126–141.
    Zbl: 1418.11045DOI: 10.1016/j.jnt.2017.03.020
  8. Power-sum denominators, Amer. Math. Monthly 124 (2017), 695–709.
    Coauthor: Jonathan Sondow
    Zbl: 1391.11052DOI: 10.4169/amer.math.monthly.124.8.695
  9. The denominators of power sums of arithmetic progressions, Integers 18 (2018), Article A95, 1–17.
    Coauthor: Jonathan Sondow
    Zbl: 1423.11029Link: Integers Vol. 18
  10. On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, Integers 21 (2021), Article A52, 1–21.
    Coauthor: Jonathan Sondow
    Zbl: 07354559Link: Integers Vol. 21
  11. On (self-) reciprocal Appell polynomials: Symmetry and Faulhaber-type polynomials, Integers 21 (2021), Article A119, 1–19.
    Link: Integers Vol. 21
  12. Shifted sums of the Bernoulli numbers, reciprocity, and denominators, Rend. Mat. Appl. VII. Ser. 43 (2022), 151–163.
    Link: Rend. Mat. Appl. VII. Ser. Vol. 43
  13. On primary Carmichael numbers , Integers 22 (2022), Article A38, 1–39.
    Link: Integers Vol. 22
  14. Distribution modulo one and denominators of the Bernoulli polynomials, arXiv: 1708.07119.



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 𝒜 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 𝒵 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 n8 [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]:




Riemann zeta function

Quotient of Riemann zeta values [4, Thm. 1.3, p. 2687]: If n2 is even, then




where ℒ⋆ is a linear functional, which is connected with a special Dirichlet series, and 𝔭n is a certain monic polynomial of degree n. Moreover, if n = p + 1 with p an odd prime, then 𝔮n 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,)(s) has a unique simple zero ξ(p,) in case (p,) is an irregular pair, one has for even n2 [1, Thm. 4.9, p. 16]:




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,) 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




If {·} denotes the fractional part and n1 is a fixed integer, then there is the surprising relation [12]:


The denominator of the nth Bernoulli polynomial Bn(x) can be described by a similar formula ([9] with Jonathan Sondow):