a bibliography of
Bernoulli numbers

S.C. Woon, "Analytic
continuation of Bernoulli numbers, a new formula for the Riemann zeta
function, and the phenomenon of scattering of zeros"

"The new concept of dynamics of the zeros of analytic continued
polynomials is introduced, and an interesting phenomenon of 'scatterings'
of the zeros of [the Bernoulli polynomials] B_{s}(z) is
observed"

S.C. Woon, "A new
representation of the Riemann zeta function \zeta(*s*)"

[abstract:] "A generalization of a well-known relation between the Riemann zeta function
representation of the Riemann zeta function in terms of a nested series of Bernoulli numbers."

C. Musès, "Applied
hypernumbers: computational concepts", *Applied Mathematics and
Computation* **3** (1977) 211-226

[abstract:] "The key importance of hypernumbers in enlarging and
fruitfully generalizing (as distinct from abstraction of a sterile
sort) algebra, function theory and computation is discussed, with
specific examples and theorems. The rich serendipity of hypernumber
research is shown in the author's recent findings; for example, those
generalizing the Bernoulli numbers for any real, complex, or
countercomplex index *s*, as
$B_s = -s!2\cos(\pi s/2) \zeta(s)/(2\pi)^{s}$, where $\zeta$ is
Riemann's Zeta function; whence, *e.g.*, *B*_{0} =1,
*B*_{2} = 1/6, *B*_{1/2} = 1/2 zeta(1/2), and
*B*_{3}/*B*_{5}
= - (80^{2})^{-1}zeta(3)/zeta(5), results like the
last two being unknown and unobtainable before. As in APL computer
language, the symbol "!" is used to denote Gauss' function: the
factorial of unrestricted argument."

C. Musès, "Some new considerations on the Bernoulli numbers, the
factorial function, and Riemann's zeta function", *Applied Mathematics
and Computation* **113** (2000) 1-21.

Curiously, although published,
this includes a supposed proof of the RH on page 21. Musès died around the time of
publication. You should be able to view a PDF version of this article at
this site
(just log in as a guest).

P.R. Subramanian, "Generating
functions for angular momentum traces", *Journal of Physics* A
**19** (1986) 2667-2670.

[Abstract:] "Generating functions for Tr(J_{lambda}^{2p}) are
obtained, one of them being the character of a representation of the three-dimensional
pure rotation group. Recurrence relations for the Bernoulli numbers and
the Riemann zeta functions are deduced."

P.R. Subramanian, "Evaluation
of Tr(J_{lambda}^{2p}) using the Brillouin function",
*Journal of Physics* A **19** (1986) 1179-1187.

[Abstract:] "Obtains expressions for Tr(J_{lambda}^{2p}) in terms of the Brillouin
function. Standard properties of Tr(J_{lambda}^{2p}) are derived from them. Sum rules for the
Bernoulli numbers and the Riemann zeta functions are deduced as corollaries."

A. Kuznetsov, "Nontrivial zeros of
the Riemann zeta function as the limit of eigenvalues of nonsymmetric matrices" (preprint 2006)

[abstract:] "We construct a family of $2n \times 2n$ matrices $B_{2n}$, such that the spectrum of
$\sigma(B_{2n})$ converges to the set of nontrivial zeros of the Riemann zeta function $\zeta(s)$.
The coefficients of these matrices are given explicitly as finite sums of Bernoulli numbers."

M. de Gosson, B. Dragovich and A. Khrennikov, "Some
*p*-adic differential equations"

"We investigate various properties of p-adic differential equations which have as a solution
an analytic function of the form $F_k (x) = \sum_{n\geq 0} n! P_k (n) x^n$, where $P_k (n) = n^k
+ C_{k-1} n^{k-1} + ...+ C_0$ is a polynomial in n with $C_i\in Z$ (in a more general case
$C_i\in Q$ or $C_i\in C_p$). For some special classes of $P_k (n)$, as well as for the general
case, the existence of the corresponding linear differential equations of the first- and
second-order for $F_k (x)$, is shown. In some cases such equations are constructed. For the
second-order differential equations there is no other analytic solution of the form
$\sum a_n x^n$. Due to the fact that the corresponding inhomogeneous first-order differential
equation exists one can construct infinitely many inhomogeneous second-order equations with the
same analytic solution. Relation to some rational sums with the Bernoulli numbers and to
$F_k (x)$ for some $x\in Z$ is considered. Some of these differential equations can be related
to p-adic dynamics and p-adic information theory."

G. Everest, Y. Puri and T. Ward,
"Integer sequences counting periodic points"

[Abstract:] "An existing dialogue between number theory and dynamical systems is advanced.
A combinatorial device gives necessary and sufficient conditions for a sequence of
non-negative integers to count the periodic points in a dynamical system. This is applied to
study linear recurrence sequences which count periodic points. Instances where the
*p*-parts of an integer sequence themselves count periodic points are studied. The
Mersenne sequence provides one example, and the denominators of the Bernoulli numbers provide
another. The methods give a dynamical interpretation of many classical congruences such as
Euler-Fermat for matrices, and suggest the same for the classical Kummer congruences satisfied
by the Bernoulli numbers."

A. Veselov and J. Ward,
"On the real roots of the Bernoulli polynomials and the Hurwitz zeta-function"
(1999)

U. Muller and C. Schubert,
"A quantum field
theoretical representation of Euler-Zagier sums"

[abstract:] "We establish a novel representation of arbitrary
Euler-Zagier sums in terms of weighted vacuum graphs. This
representation uses a toy quantum field theory with infinitely many
propagators and interaction vertices. The propagators involve
Bernoulli polynomials and Clausen functions to arbitrary orders. The
Feynman integrals of this model can be decomposed in terms of an
algebra of elementary vertex integrals whose structure we
investigate. We derive a large class of relations between multiple
zeta values, of arbitrary lengths and weights, using only a certain
set of graphical manipulations on Feynman diagrams. Further uses and
possible generalizations of the model are pointed out."

M.W. Coffey, "Series representations of the Riemann and Hurwitz zeta functions and series and integral representations of the first Stieltjes constant" (preprint 06/2011)

"We develop series representations for the Hurwitz and Riemann zeta functions in terms of generalized Bernoulli numbers (N\"{o}rlund polynomials), that give the analytic continuation of these functions to the entire complex plane. Special cases yield series representations of a wide variety of special functions and numbers, including log Gamma, the digamma, and polygamma functions. A further byproduct is that $\zeta(n)$ values emerge as nonlinear Euler sums in terms of generalized harmonic numbers. We additionally obtain series and integral representations of the first Stieltjes constant $\gamma_1(a)$. The presentation unifies some earlier results."

G. Ottarsson,
"The Ladder Hypothesis"

[Abstract:] "In a paper from 18 August 2001 available at
www.islandia.is/gko/010818.pdf,
a thermoelectric generator was constructed from a large number of series
connected parallelepipeds. The hot and/or cold reservoir was made of some
electrically conductive metal, and the fluid was to some extent conductive to the
electrical ground. This topology generated a number of small
capacitors, each formed by two parallelepiped crystal faces and the grounded
thermal reservoir. When analysing the frequency behaviour of
such a device, rational polynomials manifested themselves and proved
to be a rich source of advanced mathematical relations."

These relations involve the Riemann zeta function, Bernoulli
numbers, the Gamma function, Euler's constant and Stirling's Formula.

zeta
functions related to poly-Bernoulli numbers