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

Thesis

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

Publications

  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.2009.009
  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, Geom. Topol. 19 (2015), 2995–3030.
    Appendix by Bernd C. Kellner.   Authors: Jonathan Bowden, Diarmuid Crowley, András I. Stipsicz.
    Zbl: 06503559DOI: 10.2140/gt.2015.19.2995
  7. On a product of certain primes, J. Number Theory 179 (2017), 126–141.
    Zbl: 06739165DOI: 10.1016/j.jnt.2017.03.020
  8. Power-Sum Denominators, Amer. Math. Monthly 124 (2017), 695–709.
    Coauthor: Jonathan Sondow
    DOI: 10.4169/amer.math.monthly.124.8.695
  9. The denominators of power sums of arithmetic progressions, arXiv: 1705.05331, submitted.
    Coauthor: Jonathan Sondow
  10. Distribution modulo one and denominators of the Bernoulli polynomials, arXiv: 1708.07119, submitted.

Preprints

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]:

gamma1

with the constants

gamma2

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):

factorial1

with the constant

factorial2

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]:

bnprod1

with the constants

bnprod2

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]:

divisor1

with the convolution function

divisor2

The first case n = 8 is known as Hurwitz's identity (1881)

divisor3

Identities with Stirling and harmonic numbers

Define the polynomials

identsh1

composed of Stirling numbers of the second kind S2(n,k) and harmonic numbers Hk by

identsh2

The Genocchi numbers Gn [OEIS A036968] are related to the Bernoulli numbers by

identsh3

The second identity of Bn is due to Worpitzky (1883):

identsh4

The central value is known to be

identsh5

Involving the harmonic numbers leads to the identities [5, Thm. 1.3, p. 3]:

identsh6

and

identsh7

Riemann zeta function

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

zetaquot1

with

zetaquot2

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]:

bnstruct1

where

bnstruct2

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:

bnstruct3

Moreover, the formulas are valid for all irregular pairs (p,) with

bnstruct4

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])

bpdenom1

where sp(n) denotes the sum of the base-p digits of n. The finite product can be written with explicit sharp bounds as

bpdenom2

where

bpdenom3

If {·} denotes the fractional part and n1 is a fixed integer, then there is the surprising relation [10]:

bpdenom4

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

bpdenom5