specific zeta values
in physics

I.O. Goriachuk and A.L. Kataev, "Riemann $\zeta(4)$ function contributions to $O(\alpha_s^5)$ terms of Adler D-function and Bjorken polarized sum rule in $SU(N_c)$ QCD: Results and consequences" (preprint 11/2020)

[abstract:] "Two renormalization group invariant quantities in quantum chromodinamics (QCD), defined in Euclidean space, namely, Adler D-function of electron-positron annihilation to hadrons and Bjorken polarized deep-inelastic scattering sum rule, are considered. It is shown, that the 5th order corrections to them in $\overline{MS}$-like renormalization prescriptions, proportional to Riemann $\zeta$-function $\zeta(4)$, can be restored by the transition to the C-scheme, with the $\beta$-function, analogous to Novikov, Shifman, Vainshtein and Zakharov exact $\beta$-function in $\mathcal{N} = 1$ supersymmetric gauge theories. The general analytical expression for these corrections in $SU(N_c)$ QCD is deduced and their scale invariance is shown. The $\beta$-expansion procedure for these contributions is performed and mutual cancellation of them in the 5th order of the generalized Crewther identity are discussed."

S. Ouvry and A. Polychronakos, "Lattice walk area combinatorics, some remarkable trigonometric sums and Apéry-like numbers"

[abstract:] "Explicit algebraic area enumeration formulae are derived for various lattice walks generalizing the canonical square lattice walk, and in particular for the triangular lattice chiral walk recently introduced by the authors. A key element in the enumeration is the derivation of some remarkable identities involving trigonometric sums – which are also important building blocks of non trivial quantum models such as the Hofstadter model – and their explicit rewriting in terms of multiple binomial sums. An intriguing connection is also made with number theory and some classes of Apéry-like numbers, the cousins of the Apéry numbers which play a central role in irrationality considerations for $\zeta(2)$ and $\zeta(3)$."

D. Broadhurst, "Feynman's sunshine numbers" (preprint 04/2010)

[abstract:] "This is an expansion of a talk for mathematics and physics students of the Manchester Grammar and Manchester High Schools. It deals with numbers such as the Riemann zeta value zeta(3)=sum_{n>0}1/n^3. Zeta values appear in the description of sunshine and of relics from the Big Bang. They also result from Feynman diagrams, which occur in the quantum field theory of fundamental particles such as photons, electrons and positrons. My talk included 7 reasonably simple problems, for which I here add solutions, with further details of their context."

L. Albert and M. K.-H. Kiessling, "Order and Chaos in some deterministic infinite trigonometric products" (preprint, 09/2016)

[abstract:] "In this paper it is proved that $\prod_{n=1}^\infty
\left[\frac23+\frac13\cos\left(\frac{x}{n^{2}}\right)\right] = e^{- C
\,\sqrt{|x|} +\varepsilon(|x|)},$ with $|\varepsilon(|x|)| \leq K |x|^{1/3}$
for some $K>0$, and with $ C= \int\frac{\sin\xi^2}{2+\cos\xi^2}{\rm{d}}\xi;$
numerically, $C = 0.319905585... \sqrt{\pi}$. As a corollary this confirms a
surmise of Benoit Cloitre. The $O\big(|x|^{1/3}\big)$ error bound is
empirically found to be accurate for moderately sized $|x|$ but not for larger
$|x|$. This difference $\varepsilon(|x|)$ between Cloitre's $\log
\prod_{n\geq1}\left[\frac23 +\frac13\cos\left(\frac{x}{n^{2}}\right)\right]$
and its regular trend $-C\sqrt{|x|}$, although deterministic, appears to be an
"empirically unpredictable" function. A probabilistic investigation of this
phenomenon is carried out, proving that Cloitre's trigonometric product is the
characteristic function of a simple random walk on the real interval
$(-\zeta(2),\zeta(2))$, where $\zeta$ is Riemann's zeta function, in fact, this
random walk is a 'random Riemann-$\zeta$ function with argument 2.' A few
related random walks are studied empirically and compared with their
theoretical distributions and trend distributions. The paper closes with a
paradoxical random-walk scenario and a remark on the Riemann hypothesis."

H. Chamati and N.S. Tonchev, "Quantum critical scaling and the Gross–Neveu model in 2+1 dimensions" (preprint 12/2011)

[abstract:] "The quantum critical behavior of the 2+1 dimensional Gross--Neveu model in the vicinity of its zero temperature critical point is considered. The model is known to be renormalisable in the large $N$ limit, which offers the possibility to obtain expressions for various thermodynamic functions in closed form. We have used the concept of finite--size scaling to extract information about the leading temperature behavior of the free energy and the mass term, defined by the fermionic condensate and determined the crossover lines in the coupling ($\g$) -- temperature ($T$) plane. These are given by $T\sim|\g-\g_c|$, where $\g_c$ denotes the critical coupling at zero temperature. According to our analysis no spontaneous symmetry breaking survives at finite temperature. We have found that the leading temperature behavior of the fermionic condensate is proportional to the temperature with the critical amplitude $\frac{\sqrt{5}}3\pi$. The scaling function of the singular part of the free energy is found to exhibit a maximum at $\frac{\ln2}{2\pi}$ corresponding to one of the crossover lines. The critical amplitude of the singular part of the free energy is given by the universal number $\frac13[\frac1{2\pi}\zeta(3)-\mathrm{Cl}_2(\frac{\pi}3)]=-0.274543...$, where $\zeta(z)$ and $\mathrm{Cl}_2(z)$ are the Riemann zeta and Clausen's functions, respectively. Interpreted in terms the thermodynamic Casimir effect, this result implies an attractive Casimir "force". This study is expected to be useful in shedding light on a broader class of four fermionic models."

A. Klümper, D. Nawrath and J. Suzuki, "Correlation functions of the integrable isotropic spin-1 chain: algebraic expressions for arbitrary temperature" (preprint 04/2013)

[abstract:] "We derive algebraic formulas for the density matrices of finite segments of the integrable $su(2)$ isotropic spin-1 chain in the thermodynamic limit. We give explicit results for the 2 and 3 site cases for arbitrary temperature $T$ and zero field. In the zero temperature limit the correlation functions are given in elementary form in terms of Riemann's zeta function at even integer arguments."

P.J. Forrester, "Diffusion processes and the asymptotic bulk gap probability for the real Ginibre ensemble" (preprint 06/2013)

[abstract:] "It is known that the bulk scaling limit of the real eigenvalues for the real Ginibre ensemble is equal in distribution to the rescaled $t \to \infty$ limit of the annihilation process $A + A \to \emptyset$. Furthermore, deleting each particle at random in the rescaled $t \to \infty$ limit of the coalescence process $A + A \to A$, a process equal in distribution to the annihilation process results. We use these inter-relationships to deduce from the existing literature the asymptotic small and large distance form of the gap probability for the real Ginibre ensemble. In particular, the leading form of the latter is shown to be equal to $\exp(-(\zeta(3/2)/(2 \sqrt{2 \pi}))s)$, where $s$ denotes the gap size and $\zeta(z)$ denotes the Riemann zeta function. A determinant formula is derived for the gap probability in the finite $N$ case, and this is used to illustrate the asymptotic formulas against numerical computations."

S. Ouvry, "Random Aharonov-Bohm vortices and some funny
families of integrals" (preprint 02/05)

[abstract:] "A review of the random magnetic impurity model, introduced in the context
of the integer Quantum Hall effect, is presented. It models an electron moving in a plane
and coupled to random Aharonov-Bohm vortices carrying a fraction of the quantum of flux.
Recent results on its perturbative expansion are given. In particular, some funny families
of integrals show up to be related to the Riemann $\zeta(3)$ and $\zeta(2)$."

M.V. Cougo-Pinto, C. Farina, J.F.M. Mendes and A.C. Tort, "On the
non-relativistic Casimir effect", *Braz. J. Phys.* **31** no. 1 (2001)

[abstract:] "We compute the Casimir energy for a massive scalar field constrained
between two parallel planes (Dirichlet boundary conditions) in order to investigate its non-relativistic
limit. Instead of employing the usual relativistic dispersion relation \omega(**p**) =
(**p**^{2} + *m*^{2})^{1/2}, we use the non-relativistic
one, \omega(**p**) = **p**^{2}/2*m*. It turns out that the Casimir energy
is zero. We include the relativistic corrections perturbatively and show that at all orders the Casimir energy
remains zero, since each term in the power series in 1/*c*^{2} is proportional to the
Riemann zeta function of a negative even integer. This puzzling result shows that, at least for the free
massive scalar field, the Casimir effect is non-perturbative in the relativistic sense."

F. Brown, "Periods and Feynman amplitudes" (preprint 12/2015)

[abstract:] "Feynman amplitudes in perturbation theory form the basis for most predictions in particle collider experiments. The mathematical quantities which occur as amplitudes include values of the Riemann zeta function and relate to fundamental objects in number theory and algebraic geometry. This talk reviews some of the recent developments in this field, and explains how new ideas from algebraic geometry have led to much progress in our understanding of amplitudes. In particular, the idea that certain transcendental numbers, such as $\pi$, can be viewed as a representation of a group, provides a powerful framework to study amplitudes which reveals many hidden structures."

N. Kurokawa and M. Wakayama, "Casimir effects on Riemann surfaces", *Indagationes Mathematicae* **13** (1) (2002) 63–75

[abstract:] "The Casimir effect whose existence was first predicted by Casimir in 1948 is considered as a manifestation of macroscopic quantum field theory. This force is evaluated theoretically by using the value of the Riemann zeta function at -3. The aim of the present paper is to introduce a similar Casimir energy for a Riemann surface, and to express it by a special value of the Mellin transform of a theta series arising from the heat kernel and also by a weighted integral of the logarithm of the Selberg zeta function."

C. Furtlehner and S. Ouvry, "Integrals
involving four Macdonald functions and their relation to 7zeta(3)/2" (preprint 06/03)

[abstract:] "A family of multiple integrals over four variables is rewritten in
terms of a family of simple integrals involving the product of four modified Bessel
(Macdonald functions). The latter are shown to be related to 7zeta(3)/2. A
generalization to 2*n* integration variables is given which yields only zeta at
odd arguments."

The physics behind these results is concerned with disordered 2-d singular magnetic
fields (Aharonov-Bohm vortices). See reference [1] in the paper.

K. Sakai, M. Shiroishi, Y. Nishiyama and M. Takahashi,
"Third Neighbor
Correlators of Spin-1/2 Heisenberg Antiferromagnet", *Phys. Rev.* E **67**
(2003) 65-101

[abstract:] "We exactly evaluate the third neighbor correlator < *S*_{j}^{z} S_{j}+3^{z} >
and all the possible non-zero correlators <*S*^{alpha}_{j} S^{beta}_{j}+1*S*^{gamma}
_{j}+2*S*^{delta}_{j}+3 > of the spin-1/2 Heisenberg XXX antiferromagnet in the ground state without
magnetic field. All the correlators are expressed in terms of certain combinations of logarithm ln2, the Riemann zeta function zeta(3), zeta(5)
with rational coefficients. The results accurately coincide with the numerical ones obtained by the density-matrix renormalization group method
and the numerical diagonalization."

M. Takahashi, "Correlation function and simplified TBA equations for XXZ chain" (preprint 12/2010)

[abstract:] "The calculation of the correlation functions of Bethe ansatz solvable models is very difficult problem. Among these solvable models spin 1/2 XXX chain has been investigated for a long time. Even for this model only the nearest neighbor and the second neighbor correlations were known. In 1990's multiple integral formula for the general correlations is derived. But the integration of this formula is also very difficult problem. Recently these integrals are decomposed to products of one dimensional integrals and at zero temperature, zero magnetic field and isotropic case, correlation functions are expressed by ${\rm log} 2$ and Riemann's zeta functions with odd integer argument $\zeta(3), \zeta(5), \zeta(7),....$. We can calculate density sub-matrix of successive seven sites. Entanglement entropy of seven sites is calculated. These methods can be extended to XXZ chain up to $n=4$. Correlation functions are expressed by the generalized zeta functions.

Several years ago I derived new thermodynamic Bethe ansatz equation for XXZ chain. This is quite different with Yang-Yang type TBA equations and contains only one unknown function. This equation is very useful to get the high temperature expansion. In this paper we get the analytic solution of this equation at $\Delta=0$."

J. Lepowsky, "Vertex operator algebras
and the zeta function" (preprint 09/99)

[abstract:] "We announce a new type of "Jacobi identity" for vertex operator algebras, incorporating values of the Riemann zeta function
at negative integers. Using this we "explain" and generalize some recent work of S. Bloch's relating values of the zeta function with the
commutators of certain operators and Lie algebras of differential operators."

V. Ravindran, J. Smith and W.L. van Neerven, "Two-loop
corrections to Higgs boson production", *Nucl. Phys.* B **704** (2005) 332-348

O. M. Ogreid and P. Osland,
"Some
infinite series related to Feynman diagrams", *Journal of
Computational and Applied Mathematics* **140** (2002) 659-671

M. D. Pollock,
"The
non-invariance of the heterotic superstring four-action under the
modular transformation", *Physics Letters B* **411**
(1997) 68-72

R.J. Meyer, "Nearest-neighbor
approximation for the dipole moment of conducting-particle chains",
*Journal of Electrostatics* **33** (1994) 133-146

A. A. Gogolin, "Electron
localization and hopping conductivity in one-dimensional disordered systems", *Physics
Reports* **86** (1982) 1-53

B.W. Ninham and M. Bostrom, "Screened Casimir forces at finite temperatures:
A possible role in nuclear interactions", *Phys. Rev. A* **67**
(2003)

B.W. Ninham and M. Bostr m, "The effect of intervening plasma on
Casimir free forces at finite temperatures"

excerpts from E. Schrödinger's *Statistical Thermodynamics*
wherein appear values of zeta at 3/2, 5/2 and various even integers

Notes on the Stefan-Boltzmann Law
involving \zeta(4)

S. Walston,
"The Planck spectrum, Riemann zeta functions, and
modern cryptography"

H. van Erkelens' notes on relativistic gases and the Riemann zeta function

M. Kaneko, N. Kurokawa, and M. Wakayama,
"A variation of Euler's
approach to values of the Riemann zeta function", *Kyushu J. Math.* **57** no. 1
(2003) 175-192

[abstract:] "An elementary method of computing the values at negative integers of the
Riemann zeta function is presented. The principal ingredient is a new *q*-analogue of the
Riemann zeta function. We show that for any argument other than 1 the classical limit of
this *q*-analogue exists and equals the value of the Riemann zeta."

E. Bach, "The
complexity of number-theoretic constants", *Information Processing Letters*
**62** (1997) 145-152

[abstract:] "We show that Artin's constant and two related
number-theoretic quantities can be computed to *t* bits of
precision using *t*^{3 + o(1)} bit operations. The
factor implied by the symbol o(1) depends on the cost of the
underlying arithmetic, but for practical purposes can be taken as log
*t*. As a by-product of this work, we estimate the complexity of
computing Bernoulli numbers **and evaluating the Riemann zeta function
at positive integers**. We also give examples of constants that seem
hard to compute, such as Brun's twin prime constant and the exact
density of primes for which a given base is a primitive root. This
last cannot be computed quickly unless factorization of certain RSA
moduli is easy."

J. Borwein, D. Bradley and R. Crandall, "Computational strategies
for the Riemann zeta function", *J. Comp. App. Math.* **121** (2000) 247-296

[Abstract:] "We provide a compendium of evaluation methods for the Riemann zeta
function, presenting formulae ranging from historical attempts to recently found convergent
series to curious oddities old and new. We concentrate primarily on practical computational
issues, such issues depending on the domain of the argument, the desired speed of computation,
and the incidence of what we call 'value recycling'."

Apéry's constant (zeta(3))
A. van der Poorten, "A proof that Euler missed...Apéry's proof
of the irrationality of zeta(3)" - an informal report

M. Hoffman's notes
on multiple zeta values (with applications to quantum field theory)

M. Marcolli's survey article "Number Theory in Physics" contains notes on the role of multiple zeta values
in quantum field theory, *etc.*

polylogarithms

Clausen Functions