quantum field theory and number theory

*"I was immediately led to the idea that somehow passing from the
integers to the primes is very similar to passing from quantum field
theory, as we observe it, to the elementary particles, whatever they
are."*

A. Connes quoted in K. Sabbagh, *Dr.
Riemann's Zeros* (Atlantic, 2002), p.204

any thoughts on what Connes is
implying here?

P. Woit, "Towards a grand unified theory of mathematics and physics" (preprint 06/2015)

[abstract:] "Wigner's "unreasonable effectiveness of mathematics" in physics can be understood as a reflection of a deep and unexpected unity between the fundamental structures of mathematics and of physics. Some of the history of evidence for this is reviewed, emphasizing developments since Wigner's time and still poorly understood analogies between number theory and quantum field theory."

D.J. Broadhurst, "Euler sums in quantum
field theory"

The above is an appendix to the following:

J.M. Borwein and R. Girgensohn, "Evaluation of
triple Euler sums", *Electronic J. Combinatorics* **3** (1996), R23

D.J. Broadhurst, "On the enumeration of irreducible *k*-fold Euler
sums and their roles in knot theory and field theory", *Journal of
Mathematical Physics* (to appear?)

[abstract:] "A generating function is given for the number, *E*(*l*,*k*), of irreducible *k*-fold
Euler sums, with all possible alternations of sign, and exponents summing to *l*. Its form is remarkably simple: $\sum_n E(k+2n,k) x^n =
\sum_{d|k}\mu(d) (1-x^d)^{-k/d}/k$, where $\mu$ is the Möbius function. Equivalently, the size of the search
space in which *k*-fold Euler sums of level *l* are reducible to rational linear combinations of irreducible basis terms is
$S(l,k) = \sum_{nk-fold sums with *l* __<__ 7; combined methods
yield bases for all remaining search spaces with $S(l,k)\leq34$. These findings confirm expectations based on Dirk
Kreimer's connection of knot theory with quantum field theory. The occurrence in perturbative quantum electrodynamics
of all 12 irreducible Euler sums with *l* __<__ 7 is demonstrated. It is suggested that no further transcendental
occurs in the four-loop contributions to the electron's magnetic moment. Irreducible Euler sums are found to occur in
explicit analytical results, for counterterms with up to 13 loops, yielding transcendental knot-numbers, up to 23 crossings."

D.J. Broadhurst, "Conjectured
enumeration of irreducible multiple zeta values, from knots and Feynman diagrams" (preprint,
12/96)

[abstract:] "Multiple zeta values (MZVs) are under intense investigation in three arenas - knot theory, number
theory, and quantum field theory - which unite in Kreimer's proposal that field theory assigns MZVs to positive knots, via
Feynman diagrams whose momentum flow is encoded by link diagrams. Two challenging problems are posed by this nexus
of knot/number/field theory: enumeration of positive knots, and enumeration of irreducible MZVs. Both were recently
tackled by Broadhurst and Kreimer (BK). Here we report large-scale analytical and numerical computations that test,
with considerable severity, the BK conjecture that the number, *D*_{{n,k}},
of irreducible MZVs of weight *n* and
depth *k*, is generated by $\prod_{n\ge3}\prod_{k\ge1}(1-x^n y^k) ^{D_{n,k}}=1-\frac{x^3y}{1-x^2}+
\frac{x^{12}y^2(1-y^2)}{(1-x^4)(1-x^6)}$, which is here shown to be consistent with all shuffle identities for the
corresponding iterated integrals, up to weights *n* = 44, 37, 42, 27, at depths *k* = 2, 3, 4, 5, respectively, entailing
computation at the petashuffle level. We recount the field-theoretic discoveries of MZVs, in counterterms, and of Euler
sums, from more general Feynman diagrams, that led to this success."

D.J. Broadhurst, "Where do
the tedious products of zetas come from?", *Nucl. Phys. Proc.
Suppl.* **116** (2003) 432-436

[abstract:] "Lamentably, the full analytical content of the epsilon-expansion of the master
two-loop two-point function, with arbitrary self-energy insertions in 4-2epsilon dimensions,
is still unknown. Here we show that multiple zeta values (MZVs) of weights up to 12 suffice
through O(epsilon^9). Products of primitive MZVs are generated by a processes of
"pseudo-exponentiation"" whose combinatorics faithfully accord with expectations based on
Kreimer's modified shuffle product and on the Drinfeld-Deligne conjecture. The existence of
such a mechanism, relating thousands of complicated rational numbers, enables us to identify
precise and simple combinations of MZVs specific to quantum field theories in even numbers
of spacetime dimensions."

D.J. Broadhurst and D. Kreiemer, "Knots and numbers in $\phi^4$ theory
to 7 loops and beyond", *Int. J. Mod. Phys.* C6
(1995) 519-524

[abstract:] "We evaluate all the primitive divergences contributing to the 7-loop
$\beta$-function of $\phi^4$ theory, i.e. all 59 diagrams that are free of subdivergences
and hence give scheme--independent contributions. Guided by the association of diagrams
with knots, we obtain analytical results for 56 diagrams. The remaining three diagrams,
associated with the knots 10_{124}, 10_{139}, and 10_{152}, are
evaluated numerically, to 10 sf. Only one satellite knot with 11 crossings is encountered
and the transcendental number associated with it is found. Thus we achieve an analytical
result for the 6-loop contributions, and a numerical result at 7 loops that is accurate to
one part in 10^{11}. The series of 'zig-zag' counterterms, $\{6\zeta_3,\,20\zeta_5,\,
\frac{441}{8}\zeta_7,\,168\zeta_9,\,\ldots\}$, previously known for *n* = 3,4,5,6 loops,
is evaluated to 10 loops, corresponding to 17 crossings, revealing that the *n*-loop
zig-zag term is $4C_{n-1} \sum_{p>0}\frac{(-1)^{p n - n}}{p^{2n-3}}$, where
$C_n=\frac{1}{n+1}{2n \choose n}$ are the Catalan numbers, familiar in knot theory. The
investigations reported here entailed intensive use of REDUCE, to generate O(10^{4})
lines of code for multiple precision FORTRAN computations, enabled by Bailey's MPFUN routines,
running for ${\rm O}(10^3)$ CPUhours on DecAlpha machines."

J.M. Borwein and D.J. Broadhurst, "Determinations of rational Dedekind-zeta
invariants of hyperbolic manifolds and Feynman knots and links" (preprint 11/98)

[abstract:] "We identify 998 closed hyperbolic 3-manifolds whose volumes are rationally
related to Dedekind zeta values, with coprime integers *a* and *b* giving
$a/b vol(M)=(-D)^{3/2}/(2\pi)^{2n-4} (\zeta_K(2))/(2\zeta(2))$ for a manifold M whose
invariant trace field *K* has a single complex place, discriminant *D*, degree *n*, and
Dedekind zeta value $\zeta_K(2)$. The largest numerator of the 998 invariants of
Hodgson-Weeks manifolds is, astoundingly, *a* = 2^{4} x 23 x 37 x 691 = 9,408,656;
the largest denominator is merely *b* = 9. We also study the rational invariant *a*/*b* for
single-complex-place cusped manifolds, complementary to knots and links, both within
and beyond the Hildebrand-Weeks census. Within the censi, we identify 152 distinct
Dedekind zetas rationally related to volumes. Moreover, 91 census manifolds have volumes
reducible to pairs of these zeta values. Motivated by studies of Feynman diagrams, we
find a 10-component 24-crossing link in the case *n* = 2 and *D* = -20. It is one of 5
alternating platonic links, the other 4 being quartic. For 8 of 10 quadratic fields
distinguished by rational relations between Dedekind zeta values and volumes of Feynman
orthoschemes, we find corresponding links. Feynman links with *D* = -39 and *D* = -84 are
missing; we expect them to be as beautiful as the 8 drawn here. Dedekind-zeta invariants are
obtained for knots from Feynman diagrams with up to 11 loops. We identify a sextic 18-crossing
positive Feynman knot whose rational invariant, *a*/*b*=26, is 390 times that of the cubic
16-crossing non-alternating knot with maximal *D*_{9} symmetry. Our results are secure,
numerically, yet appear very hard to prove by analysis."

D.J. Broadhurst and D. Kreimer, "Association
of multiple zeta values with positive knots via Feynman diagrams up to 9 loops",
*Phys. Lett. B* **393** (1997) 403

D.J. Broadhurst, "Massive
3-loop Feynman diagrams reducible to SC* primitives of algebras of the sixth root of unity",
*Eur. Phys. J.* **C8** (1999) 311-333

D. Broadhurst, "Feynman integrals, $L$-series and Kloosterman moments" (preprint 02/2016)

[abstract:] "This work lies at an intersection of three subjects: quantum field theory, algebraic geometry and number theory, in a situation where dialogue between practitioners has revealed rich structure. It contains a theorem and 7 conjectures, tested deeply by 3 optimized algorithms, on relations between Feynman integrals and $L$-series defined by products, over the primes, of data determined by moments of Kloosterman sums in finite fields. There is an extended introduction, for readers who may not be familiar with all three of these subjects. Notable new results include conjectural evaluations of non-critical $L$-series of modular forms of weights 3, 4 and 6, by determinants of Feynman integrals, an evaluation for the weight 5 problem, at a critical integer, and formulas for determinants of arbitrary size, tested up to 30 loops. It is shown that the functional equation for Kloosterman moments determines much but not all of the structure of the $L$-series. In particular, for problems with odd numbers of Bessel functions, it misses a crucial feature captured in this work by novel and intensively tested conjectures. For the 9-Bessel problem, these lead to an astounding compression of data at the primes."

J.M. Borwein, D.M. Bradley, D.J. Broadhurst, and P. Lisonek,
"Combinatorial
aspects of multiple zeta values", *Electronic
Journal of Combinatorics* **5** (1998), R38

D.J. Broadhurst, J.M. Borwein, and D.M. Bradley, "Evaluation of
*k*-fold Euler/Zagier sums: a compendium of results for arbitrary
*k*", *Electronic Journal of Combinatorics* **4** (2)
(1997), R5

J. Borwein, D. Broadhurst, J. Kamnitzer,
"Central binomial sums,
multiple Clausen values and zeta values" (preprint 04/00)

D.J. Broadhurst and Dirk Kreimer's number/field/knot theory bibliography

"Construing renormalization as a skein operation on link diagrams that
encode momentum flow, new connections between field theory, knot
theory, and number theory, have been forged, and intensively
investigated to 7-loop order. This has also given a better
understanding of which Euler sums are irreducible."

D. Kreimer, *Knots
and Feynman Diagrams* (Cambridge U.P., 2000)

[excerpt from publisher's description:] "Beginning with a summary of key ideas from
perturbative quantum field theory and an introduction to the Hopf algebra structure of
renormalization, early chapters discuss the rationality of
ladder diagrams and simple link diagrams. The necessary basics of knot theory are then presented
and **the number-theoretic relationship between the topology of Feynman diagrams and knot theory
is explored**. Later chapters discuss four-term relations motivated by the discovery of
Vassiliev invariants in knot theory and draw a link to algebraic structures recently observed
in noncommutative geometry."

C. Bergbauer and D. Kreimer, "New algebraic aspects
of perturbative and non-perturbative Quantum Field Theory" (preprint 04/2007)

[abstract:] "In this expository article we review recent advances in our understanding of the combinatorial and algebraic structure
of perturbation theory in terms of Feynman graphs, and Dyson-Schwinger equations. Starting from Lie and Hopf algebras of Feynman
graphs, perturbative renormalization is rephrased algebraically. The Hochschild cohomology of these Hopf algebras leads the way to
Slavnov-Taylor identities and Dyson-Schwinger equations. We discuss recent progress in solving simple Dyson-Schwinger equations in the
high energy sector using the algebraic machinery. Finally there is a short account on **a relation to** algebraic geometry and
**number theory**: understanding Feynman integrals as periods of mixed (Tate) motives."

This is from D. Kreimer's summary
of his research interests:

"**Feynan Diagrams, Knot Theory and Number Theory**

To what extent is a coefficient of ultraviolent divergence in a Feynman integral uniquely
determined by the topology of the underlying graph?

It turned out to be true that the topology of a Feynman graph can be related to braid-positive knots. This establishes a knot-to-number dictionary: if and only if a certain braid-positive knot is obtained from a graph, the evaluation of this graph will produce a corresponding transcendental number as its coefficient of ultraviolet divergence

My results...led to the conclusion that Feynman diagrams obtained from a field theory in even dimensions all evaluate to the same number-class up to the seven-loop level, the limit
of computational ability at this time, although we believe this result to be true in general.
Recently Kontsevich conjectured a related result. The precise determination of this generic
number class at high loop orders is an important open problem for number theorists and
computational physicists alike.

**Combinatorics of perturbative Quantum Field Theory**

The elimination of ultraviolet divergences by local counterterms, commonly known as
Bogoliubov-Parasuik-Hepp-Zimmermann (BPHZ) renormalization, is achieved by a recursion
whose solution is Zimmermann's forest formula. In the summer of 1997 I discovered that
this algebraic structure establishes a Hopf algebra structure on Feynman graphs...The
primitive elements of this Hopf algebra are primitive graphs considered in the previous section, and the determination of all the algebraic relations between them leads back to the
number theory discussed above."

**D.J.
Broadhurst's homepage**

popular
article by Ivars Peterson on PSLQ algorithm as a generalisation
of Euclid's algorithm, which refers to Broadhurt's application to QFT

M. Marcolli and A. Connes, "From
physics to number theory via noncommutative geometry. Part II: Renormalization, the Riemann-Hilbert correspondence,
and motivic Galois theory", from *Frontiers in Number Theory,
Physics, and Geometry: On Random Matrices, Zeta Functions, and Dynamical Systems* (Springer, 2006)

Yu. Manin and M. Marcolli, "Holography principle and arithmetic
of algebraic curves", *Adv. Theor. Math. Phys.* **5** (2001), no. 3, 617–650.

[abstract:] "According to the holography principle (due to G. 't Hooft, L. Susskind, J. Maldacena, *et al.*), quantum gravity
and string theory on certain manifolds with boundary can be studied in terms of a conformal field theory on the boundary. Only a few
mathematically exact results corroborating this exciting program are known. In this paper we interpret from this perspective several
constructions which arose initially in the arithmetic geometry of algebraic curves. We show that the relation between hyperbolic geometry
and Arakelov geometry at arithmetic infinity involves exactly the same geometric data as the Euclidean AdS_{3} holography of
black holes. Moreover, in the case of Euclidean AdS_{2} holography, we present some results on bulk/boundary correspondence
where the boundary is a non-commutative space."

M. Marcolli's survey article
"Number Theory in Physics" contains material
on the role of multiple zeta values in QFT, *etc.*

J. G. Dueñas, N. F. Svaiter and G. Menezes, "One-loop effective action and the Riemann zeros" (preprint 05/2014)

[abstract:] "We present a remarkable connection between the asymptotic behavior of the Riemann zeros and one-loop effective action in Euclidean scalar field theory. We show that in a two-dimensional space, the asymptotic behavior of the Fourier transform of two-point correlation functions fits the asymptotic distribution of the non-trivial zeros of the Riemann zeta function. We work out an explicit example, namely the non-linear sigma model in the leading order in $1/N$ expansion."

G. Menezes and N.F. Svaiter, "Quantum field theories and prime numbers spectrum" (preprint 11/2012)

[abstract:] "The Riemann hypothesis states that all nontrivial zeros of the zeta function lie on the critical line $\Re(s)=1/2$. Hilbert and Pólya suggested a possible approach to prove it, based on spectral theory. Within this context, some authors formulated the question: is there a quantum mechanical system related to the sequence of prime numbers? In this Letter we assume that there is a class of hypothetical physical systems described by self-adjoint operators with countable infinite number of degrees of freedom with spectra given by the sequence of primes numbers. We prove a no-go theorem. We show that the generating functional of connected Schwinger functions of such theories cannot be constructed."

T. Ohsaku, "The anomalous Nambu–Goldstone Theorem in relativistic/nonrelativistic quantum field theory" (preprint 12/2013)

[abstract:] "The anomalous Nambu–Goldstone (NG) theorem which is found as a violation of counting law of the number of NG bosons of the normal NG theorem in nonrelativistic and Lorentz-symmetry-violated relativistic theories is studied in detail, with emphasis on its mathematical aspect from Lie algebras, geometry to number theory. The basis of counting law of NG bosons in the anomalous NG theorem is examined by Lie algebras (local) and Lie groups (global). A quasi-Heisenberg algebra is found generically in various symmetry breaking schema of the anomalous NG theorem, and it indicates that it causes a violation/modification of the Heisenberg uncertainty relation in an NG sector which can be experimentally confirmed. The formalism of effective potential is presented for understanding the mechanism of anomalous NG theorem with the aid of our result of Lie algebras. After an investigation on a bosonic kaon condensation model with a finite chemical potential as an explicit Lorentz-symmetry-breaking parameter, a model Lagrangian approach on the anomalous NG theorem is given for our general discussion. Not only the condition of the counting law of true NG bosons, but also the mechanism to generate a mass of massive NG boson is also found by our examination on the kaon condensation model. Furthermore, the generation of a massive mode in the NG sector is understood by the quantum uncertainty relation of the Heisenberg algebra, obtained from a symmetry breaking of a Lie algebra, which realizes in the effective potential of the kaon condensation model. Hence the relation between a symmetry breaking scheme, a Heisenberg algebra, a mode-mode coupling, and the mechanism of mass generation in an NG sector is established. **Finally, some relations between the Riemann hypothesis and the anomalous NG theorem are presented.**"

M.W. Coffey, "On a three-dimensional symmetric Ising tetrahedron, and
contributions to the theory of the dilogarithm and Clausen functions" (preprint 01/2008)

[abstract:] "Perturbative quantum field theory for the Ising model at the three-loop level yields a tetrahedral Feynman diagram $C(a,b)$ with
masses $a$ and $b$ and four other lines with unit mass. The completely symmetric tetrahedron $C^Tet \equiv C(1,1)$ has been of interest from
many points of view, with several representations and conjectures having been given in the literature. We prove a conjectured exponentially fast
convergent sum for $C(1,1)$, as well as provide further analytic support for an empirical relation for $C(1,1)$ as a remarkable difference of Clausen
function values. Our presentation includes Propositions extending the theory of the dilogarithm $Li_2$ and Clausen $Cl_2$ functions, as well as their
relation to other special functions of mathematical physics. The results strengthen connections between Feynman diagram integrals, volumes in
hyperbolic space, number theory, and special functions and numbers, specifically including dilogarithms, Clausen function values, and harmonic
numbers."

M.W. Coffey, "Alternative evaluation of a ln tan integral
arising in quantum field theory" (preprint, 10/2008)

[Abstract:] "A certain dilogarithmic integral $I_7$ turns up in a number of contexts including Feynman diagram
calculations, volumes of tetrahedra in hyperbolic geometry, knot theory, and conjectured relations in analytic number
theory. We provide an alternative explicit evaluation of a parameterized family of integrals containing this
particular case. By invoking the Bloch--Wigner form of the dilogarithm function, we produce an equivalent result,
giving a third evaluation of $I_7$. We also alternatively formulate some conjectures which we pose in terms of values
of the specific Clausen function $Cl_2$."

H. Furusho, "*p*-adic
multiple zeta values I - *p*-adic multiple polylogarithms and the *p*-adic KZ
equation"

[abstract:] "Our main aim in this paper is to give a foundation of the theory of
*p*-adic multiple zeta values. We introduce (one variable) *p*-adic multiple
polylogarithms by Coleman's *p*-adic iterated integration theory. We define *p*-adic
multiple zeta values to be special values of *p*-adic multiple polylogarithms. We
consider the *p*-adic KZ equation and introduce the *p*-adic Drinfel'd associator
by using certain two fundamental solutions of the *p*-adic KZ equation. We show that
our *p*-adic multiple polylogarithms appear on coefficients of a certain fundamental
solution of the *p*-adic KZ equation and our *p*-adic multiple zeta values appear
on coefficients of the *p*-adic Drinfel'd associator. We show various properties of
*p*-adic multiple zeta values, which are sometimes analogous to the complex case and
are sometimes peculiar to the *p*-adic case, via the *p*-adic KZ equation."

L. Guo, S. Paycha and B. Zhang, "Renormalization of conical zeta values and the Euler–Maclaurin formula" (preprint 06/2013)

[abstract:] "We equip the space of convex rational cones with a connected coalgebra structure, which we further generalize to decorated cones by means of a differentiation procedure. Using the convolution product $\ast$ associated with the coproduct on cones we define an interpolator $\mu:= I^{\ast(-1)}\ast S$ as the $\ast$ quotient of an exponential discrete sum $S$ and an exponential integral $I$ on cones. A generalization of the algebraic Birkhoff decomposition to linear maps from a connected coalgebra to a space with a linear decomposition then enables us to carry out a Birkhoff–Hopf factorization $S:= S_-^{\ast (-1)}\ast S_+ $ on the map $S$ whose "holomorphic part" corresponds to $S_+$. By the uniqueness of the Birkhoff–Hopf factorization we obtain $\mu=S_+$ and $I=S_-^{\ast (-1)}$ so that this renormalization procedure à la Connes and Kreimer yields a new interpretation of the local Euler–Maclaurin formula on cones of Berline and Vergne. The Taylor coefficients at zero of the interpolating holomorphic function $\mu=S_+$ correspond to renormalized conical zeta values at non-positive integers. When restricted to Chen cones, this yields yet another way to renormalize multiple zeta values at non-positive integers previously investigated by the authors using other approaches.

In the present approach renormalized conical multiple zeta values lie at the cross road of three a priori distinct fields, the geometry on cones with the Euler–Maclaurin formula, number theory with multiple zeta values and renormalization theory with methods borrowed from quantum field theory."

G.H.E. Duchamp, V.H.N. Minh, A.I. Solomon and S. Goodenough, "An interface between physics and number theory" (preprint 11/2010)

[abstract:] "We extend the Hopf algebra description of a simple quantum system given previously, to a more elaborate Hopf algebra, which is rich enough to encompass that related to a description of perturbative quantum field theory (pQFT). This provides a {\em mathematical} route from an algebraic description of non-relativistic, non-field theoretic quantum statistical mechanics to one of relativistic quantum field theory. Such a description necessarily involves treating the algebra of polyzeta functions, extensions of the Riemann Zeta function, since these occur naturally in pQFT. This provides a link between physics, algebra and number theory. As a by-product of this approach, we are led to indicate {\it inter alia} a basis for concluding that the Euler gamma constant $\gamma$ may be rational."

M.-A. Sanchis-Lozano, J. Fernando Barbero, J. Navarro-Salas, "Prime numbers, quantum field theory and the Goldbach conjecture" (preprint 01/2012)

[abstract:] "Motivated by the Goldbach and Polignac conjectures in Number Theory, we propose the factorization of a classical non-interacting real scalar field (on a two-cylindrical spacetime) as a product of either two or three (so-called primer) fields whose Fourier expansion exclusively contains prime modes. We undertake the canonical quantization of such primer fields and construct the corresponding Fock space by introducing creation operators $a_p^{\dag}$ (labeled by prime numbers $p$) acting on the vacuum. The analysis of our model, based on the standard rules of quantum field theory, suggests intriguing connections between different topics in Number Theory, notably the Riemann hypothesis and the Goldbach and Polignac conjectures. Our analysis also suggests that the (non) renormalizability properties of the proposed model could be linked to the possible validity or breakdown of the Goldbach conjecture for large integer numbers."

B. Fauser and P.D. Jarvis, "The
Dirichlet Hopf algebra of arithmetics" (preprint, 11/05)

[abstract:] "In this work, we develop systematically the 'Dirichlet Hopf algebra of arithmetics' by dualizing
addition and multiplication maps. We study the additive and multiplicative antipodal convolutions which fail to
give rise to Hopf algebra structures, obeying only a weakened (multiplicative) homomorphism axiom. The consequences
of the weakened structure, called a Hopf gebra, e.g. on cohomology are explored. This features multiplicativity
versus complete multiplicativity of number theoretic arithmetic functions. The deficiency of not being a Hopf
algebra is then cured by introducing an 'unrenormalized' coproduct and an 'unrenormalized' pairing. It is then
argued that exactly the failure of the homomorphism property (complete multiplicativity) for non-coprime integers
is a blueprint for the problems in quantum field theory (QFT) leading to the need for renormalization.
Renormalization turns out to be the morphism from the algebraically sound Hopf algebra to the physical and
number theoretically meaningful Hopf algebra. This can be modelled alternatively by employing Rota–Baxter
operators. We stress the need for a characteristic-free development where possible, to have a sound starting
point for generalizations of the algebraic structures. The last section provides three key applications:
symmetric function theory, quantum (matrix) mechanics, and the combinatorics of renormalization in QFT which
can be discerned as functorially inherited from the development at the number-theoretic level as outlined here.
Hence the occurrence of number theoretic functions in QFT becomes natural."

U. Müller and C. Schubert,
"A quantum field
theoretical representation of Euler-Zagier sums" (preprint 08/99)

[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."

D. Wohl, "Selberg integrals, multiple zeta
values and Feynman diagrams" (preprint 06/02)

[Abstract:] "We prove that there is an isomorphism between the Hopf Algebra of Feynman
diagrams and the Hopf algebra corresponding to the Homogenous Multiple Zeta Value ring *H*
in *C*<<*X,Y*>> . In other words, Feynman diagrams evaluate to Multiple Zeta Values
in all cases. This proves a recent conjecture of Connes-Kreimer, and others including
Broadhurst and Kontsevich.

The key step of our theorem is to present the Selberg integral as discussed in Terasoma [22]
as a Functional from the Rooted Trees Operad to the Hopf algebra of Multiple Zeta Values. This
is a new construction which provides illumination to the relations between zeta values,
associators, Feynman diagrams and moduli spaces.
An immediate implication of our Main Theorem is that by applying Terasoma's result and using
the construction of our Selberg integral-rooted trees functional, we prove that the Hermitian
matrix integral as discussed in Mulase [18] evaluates to a Multiple Zeta Value in all 3 cases:
Asymptotically, the limit as *N* goes to infinity, and in general.

Furthermore, this construction provides for a positive resolution to Goncharov's conjecture
(see [7] pg. 30). The Selberg integral functional can be extended to map the special values to
depth *m* multiple polylogarithms on *X*."

D. Wohl, "Analysis of zeta functions,
Multiple zeta values, and related integrals" (preprint 06/02)

[Abstract:] "In this work, we begin to uncover the architecture of the general family of
zeta functions and multiple zeta values as they appear in the theory of integrable systems and
conformal field theory. One of the key steps in this process is to recognize the roles that
zeta functions play in various arenas using transform methods. Other logical connections are
provided by the the appearance of the Drinfeld associator, Hopf algebras, and techniques of
conformal field theory and braid groups. These recurring themes are subtly linked in a vast
scheme of a logically woven tapestry.

An immediate application of this framework is to provide an answer to a question of
Kontsevich regarding the appearance of Drinfeld type integrals and in particular, multiple zeta
values in: a) Drinfeld's work on the *KZ* equation and the associator; b)
Etingof-Kazhdan's quantization of Poisson-Lie algebras; c) Tamarkin's proof of formality
theorems; d) Kontsevich's quantization of Poisson manifolds. Combinatorial arguments relating
Feynman diagrams to Selberg integrals, multiple zeta values, and finally Poisson manifolds
provide an additional step in this framework. Along the way, we provide additional insight
into the various papers and theorems mentioned above. This paper represents an overall
introduction to work currently in progress. More details to follow. See our paper math.QA/0206030
for a proof of the Connes-Kreimer Conjecture."

A.I. Solomon, G.E.H. Duchamp, P. Blasiak, A. Horzela, K.A. Penson, "Hopf
algebra structure of a model quantum field theory" (Talk presented
by first-named author at 26th International Colloquium on Group Theoretical Methods in Physics, New York, June 2006. See cs.OH/0609107 for follow-up talk delivered by second-named author.)

[abstract:] "Recent elegant work on the structure of Perturbative Quantum Field Theory (PQFT) has revealed an astonishing interplay
between analysis (Riemann Zeta functions), topology (Knot theory), combinatorial graph theory (Feynman Diagrams) and algebra (Hopf structure).
The difficulty inherent in the complexities of a fully-fledged field theory such as PQFT means that the essential beauty of the relationships
between these areas can be somewhat obscured. Our intention is to display some, although not all, of these structures in the context of a simple
zero-dimensional field theory; i.e. a quantum theory of non-commuting operators which do not depend on spacetime. The combinatorial properties of
these boson creation and annihilation operators, which is our chosen example, may be described by graphs, analogous to the Feynman diagrams of PQFT,
which we show possess a Hopf algebra structure. Our approach is based on the partition function for a boson gas. In a subsequent note in these
Proceedings we sketch the relationship between the Hopf algebra of our simple model and that of the PQFT algebra."

I. Gálvez-Carrillo, R.M. Kaufmann and A. Tonks, "Three Hopf algebras and their common simplicial and categorical background" (preprint 07/2016)

[abstract:] "We consider three a priori totally different setups for Hopf algebras from number theory, mathematical physics and algebraic topology. These are the Hopf algebras of Goncharov for multiple zeta values, that of Connes–Kreimer for renormalization, and a Hopf algebra constructed by Baues to study double loop spaces. We show that these examples can be successively unified by considering simplicial objects, cooperads with multiplication and Feynman categories at the ultimate level. These considerations open the door to new constructions and reinterpretation of known constructions in a large common framework."

A. Abdesselam, A. Chandra and G. Guadagni, "Rigorous quantum field theory functional integrals over the $p$-adics I: Anomalous dimensions" (preprint 01/2013)

[abstract:] "In this article we provide the complete proof of the result announced in arXiv:1210.7717 about the construction of scale invariant non-Gaussian generalized stochastic processes over three dimensional $p$-adic space. The construction includes that of the associated squared field and our result shows this squared field has a dynamically generated anomalous dimension which rigorously confirms a prediction made more than forty years ago, in an essentially identical situation, by K. G. Wilson. We also prove a mild form of universality for the model under consideration. Our main innovation is that our rigourous renormalization group formalism allows for space dependent couplings. We derive the relationship between mixed correlations and the dynamical systems features of our extended renormalization group transformation at a nontrivial fixed point. The key to our control of the composite field is a partial linearization theorem which is an infinite-dimensional version of the Koenigs Theorem in holomorphic dynamics. This is akin to a nonperturbative construction of a nonlinear scaling field in the sense of F. J. Wegner infinitesimally near the critical surface. Our presentation is essentially self-contained and geared towards a wider audience. While primarily concerning the areas of probability and mathematical physics we believe this article will be of interest to researchers in dynamical systems theory, harmonic analysis and number theory. It can also be profitably read by graduate students in theoretical physics with a craving for mathematical precision while struggling to learn the renormalization group."

L. N. Lipatov, A. Sabio Vera, V. N. Velizhanin and G.G. Volkov, "Reflexive numbers and Berger
graphs from Calabi-Yau spaces" (preprint 01/05)

[abstract:] "A novel relation between number theory and recently found Berger graphs is studied.
The Berger graphs under investigation are constructed mainly for the CY_3 space. The method of analysis
is based on the slice classification of CY_l polyhedra in the so-called Universal Calabi-Yau algebra. The
concept of reflexivity in these polyhedra is reviewed and translated into the theory of reflexive numbers. A
new approach based on recurrence relations and Quantum Field Theory methods is applied to the simply-laced
and quasi-simply-laced subsets of the reflexive numbers. In the correspondence between the reflexive vectors
and Berger graphs the role played by the generalized Coxeter labels is shown to be important. We investigate
the positive roots of some of the Berger graphs to guess the algebraic structure hidden behind them."

V.N. Velizhanin, "Twist-2 at five loops: Wrapping corrections without wrapping computations" (preprint 11/2013)

[abstract:] "Using known all-loop results from the BFKL and generalized double-logarithmic equations and large spin limit we have computed the five-loop anomalous dimension of twist-2 operators without consideration of any wrapping effects. One part of the anomalous dimension was calculated in a usual way with the help of Asymptotic Bethe Ansatz. The rest part, related with the wrapping effects, was reconstructed from known constraints **with the help of methods from number theory**."

V. Gayral, B. Iochum and D.V. Vassilevich, "Heat kernel and number theory
on NC-torus" (preprint 07/2006)

[abstract:] "The heat trace asymptotics on the noncommutative torus, where generalized Laplacians are made out of left and right
regular representations, is fully determined. It turns out that this question is very sensitive to the number-theoretical aspect of the
deformation parameters. The central condition we use is of a Diophantine type. More generally, the importance of number theory is made
explicit on a few examples. We apply the results to the spectral action computation and revisit the UV/IR mixing phenomenon for a scalar
theory. Although we find non-local counterterms in the NC $\phi^4$ theory on $\T^4$, we show that this theory can be made renormalizable
at least at one loop, and may be even beyond."

S.K. Ashok, F. Cachazo, E. Dell'Aquila, "Children's drawings from Seiberg-Witten curves",
*Communications in Number Theory and Physics* **1** no. 2 (2007) 237-305

[abstract:] "We consider $N=2$ supersymmetric gauge theories perturbed by tree level superpotential terms near isolated singular points in the Coulomb moduli
space. We identify the Seiberg-Witten curve at these points with polynomial equations used to construct what Grothendieck called "dessins d'enfants" or "children's
drawings" on the Riemann sphere. From a mathematical point of view, the dessins are important because the absolute Galois group $Gal(\bar{Q}/Q)$ acts faithfully
on them. We argue that the relation between the dessins and Seiberg-Witten theory is useful because gauge theory criteria used to distinguish branches of $N=1$
vacua can lead to mathematical invariants that help to distinguish dessins belonging to different Galois orbits. For instance, we show that the confinement index
defined in hep-th/0301006 is a Galois invariant. We further make some conjectures on the relation between
Grothendieck's programme of classifying dessins into Galois orbits and the physics problem of classifying phases of $N=1$ gauge theories. "

T. Huber, "The Sudakov form factor to three loops in $N=4$ super Yang–Mills" (preprint 10/2012, from proceedings of the 11th DESY workshop "Loops and Legs in Quantum Field Theory", April 2012, Wernigerode, Germany)

[abstract:] "We review the results for the Sudakov form factor in $N=4$ super Yang–Mills theory up to the three-loop level. At each loop order, the form factor is expressed as a linear combination of only a handful scalar integrals, with small integer coefficients. Working in dimensional regularisation, the expansion coefficients of each integral exhibit homogeneous transcendentality in the Riemann zeta-function. We find that the logarithm of the form factor reproduces the correct values of the cusp and collinear anomalous dimensions. Moreover, the form factor in $N=4$ super Yang–Mills can be related to the leading transcendentality pieces of the QCD quark and gluon form factor. Finally, we comment briefly on the ultraviolet properties of the $N=4$ form factor in $D>4$ dimensions."

M. Banagl, "Positive topological quantum field theories" (preprint 03/2013)

[abstract:] "We propose a new notion of positivity for topological field theories (TFTs), based on S. Eilenberg's concept of completeness for semirings. We show that a complete ground semiring, a system of fields on manifolds and a system of action functionals on these fields determine a positive TFT. The main feature of such a theory is a semiring-valued topologically invariant state sum that satisfies a gluing formula. The abstract framework has been carefully designed to cover a wide range of phenomena. For instance, we derive Polya's counting theory in combinatorics from state sum identities in a suitable positive TFT. Several other concrete examples are discussed, among them Novikov signatures of fiber bundles over spacetimes and **arithmetic functions in number theory**. In the future, we will employ the framework presented here in constructing a new differential topological invariant that detects exotic smooth structures on spheres."

J. Andrade, "Hilbert–Pólya conjecture, zeta-functions and bosonic quantum field theories" (preprint 05/2013)

[abstract:] "The original Hilbert and Pólya conjecture is the assertion that the non-trivial zeros of the Riemann zeta function can be the spectrum of a self-adjoint operator. So far no such operator was found. However the suggestion of Hilbert and Pólya, in the context of spectral theory, can be extended to approach other problems and so it is natural to ask if there is a quantum mechanical system related to other sequences of numbers which are originated and motivated by number theory.

In this paper we show that the functional integrals associated with a hypothetical class of physical systems described by self-adjoint operators associated with bosonic fields whose spectra is given by three different sequence of numbers cannot be constructed. The common feature of the sequence of numbers considered here, which causes the impossibility of zeta regularization, is that the various Dirichlet series attached to such sequences – such as those which are sums over "primes" of $(\mathrm{norm}\ P)^{-s}$ have a natural boundary, i.e., they cannot be continued beyond the line $\Re(s)=0$. The main argument is that once the regularized determinant of a Laplacian is meromorphic in $s$, it follows that the series considered above cannot be a regularized determinant. In other words we show that the generating functional of connected Schwinger functions of the associated quantum field theories cannot be constructed."

Mike
Hoffman's notes
on multiple zeta values, referring to QFT

M. Hoffman, "Algebras of multiple zeta values,
quasi-symmetric functions, and Euler sums" (a talk given at UQAM, 05/98)

J.-F. Burnol,
"The Explicit Formula and a
propagator" (preprint, 09/98)

[abstract:] "I give a new derivation of the Explicit
Formula for the general number field *K*, which treats all primes in exactly the same way, whether they are
discrete or archimedean, and also ramified or not. In another token, I
advance a probabilistic interpretation of Weil's positivity criterion, as
opposed to the usual geometrical analogies or goals. But in the end, I
argue that the new formulation of the Explicit Formula signals a specific
link with Quantum Fields, as opposed to the
Hilbert-Pólya operator idea (which leads rather to Quantum Mechanics)."

[excerpt:] "There is an essential concept from Quantum Physics which I believe will
play an important role in the problem. This is the notion of correlation functions of
a quantum field. Well-known advances in the field of knot invariants and many other
geometric arenas have shown that these correlation functions can be used to represent
intersection products or linking numbers and many other geometrical things. On the
other hand they have of course a probabilistic interpretation (being complex amplitudes)
and rigorous mathematical developments of constructive quantum field theory have as
their goal the construction of probability measures on spaces of distributions, as
considered above.

For these reasons, and other more precise reflections on the subject, moving towards
quantum fields is an urgent goal. In this context it is reassuring that the "-log(|*x*|)"
formulation of the Explicit Formula enables a few
additional comments.

Indeed as is well-known "-log(|*x*|)" is the propagator of the free Boson field in
2 dimensions (here we look at the complex place and |*x*| so that we have both
holomorphic and anti-holomorphic sectors). But let us rather consider the non-archimedean
completion *K*_{v}, of the number field *K*. here too "-log(|*x*|)"
can be seen as a propagator associated to an action..."

[conclusion:] "This paper finds its roots in the conviction that the Riemann
Hypothesis has a lot to do with (suitably envisioned) Quantum Fields.
The belief in a possible link between the Riemann Hypothesis and
Quantum Mechanics seems to be widespread and is a modern formulation
of the Hilbert-Pólya operator approach. I believe that techniques and
philosophy more organic to Quantum Fields will be most relevant."

These notes provide a good introduction to Burnol's
programme of research.

D. Spector, "Supersymmetry and the Möbius
inversion function", *Communications in Mathematical Physics*
**127** (1990) 239.

"We show that the Möbius inversion function of number theory can be
interpreted as the operator (-1)^{F} in quantum field
theory...We will see in this paper that the function...has a very natural
interpretation. In the proper context, it is equivalent to
(-1)^{F}, the operator that distinguishes fermionic from
bosonic states and operators, with the fact that $mu(n) = 0$
when *n* is not squarefree being equivalent to the Pauli exclusion
principle..."

Although nothing particularly new is added to number theory or supersymmetric
QFT, and the claim about the PNT is spurious, this paper
introduces a 'number theoretical QFT' (analogous to B. Julia's
'number theoretical gas') which provides a surprising natural 'physical' framework for
certain manipulations on arithmetic functions. A standard calculation of a Witten index
produces the Möbius function, and Euler's totient function and Möbius
inversion emerge easily.

I. Ya. Aref'eva, I.V. Volovich, "Quantization of the Riemann
zeta-function and cosmology" (preprint 12/2006)

[abstract:] "Quantization of the Riemann zeta-function is proposed. We treat the Riemann zeta-function as a symbol of a
pseudodifferential operator and study the corresponding classical and quantum field theories. This approach is motivated by the theory of
p-adic strings and by recent works on stringy cosmological models. We show that the Lagrangian for the zeta-function field is equivalent
to the sum of the Klein-Gordon Lagrangians with masses defined by the zeros of the Riemann zeta-function. Quantization of the mathematics
of Fermat-Wiles and the Langlands program is indicated. The Beilinson conjectures on the values of L-functions of motives are interpreted
as dealing with the cosmological constant problem. Possible cosmological applications of the zeta-function field theory are discussed."

B. Dragovich, "On
*p*-adic and adelic generalization of quantum field theory", *Nuclear Physics* B -
Proceedings Supplements **102-103** (2001) 150-155

"A brief review of *p*-adic and adelic quantum mechanics is presented, and a new approach to *p*-adic
and adelic quantum field theory is proposed. Path integral method turns out to be generic for
quantum dynamics on both archimedean and nonarchimedean spaces."

B. Dragovich, "Some Lagrangians with zeta function nonlocality" (preprint, 05/2008)

[abstract:] "Some nonlocal and nonpolynomial scalar field models originated from $p$-adic string theory are considered. Infinite number of
spacetime derivatives is governed by the Riemann zeta function through d'Alembertian $\Box$ in its argument. Construction of the corresponding
Lagrangians begins with the exact Lagrangian for effective field of $p$-adic tachyon string, which is generalized replacing $p$ by arbitrary natural number $n$
and then taken a sum of over all $n$. Some basic classical field properties of these scalar fields are obtained. In particular, some trivial solutions of the
equations of motion and their tachyon spectra are presented. Field theory with Riemann zeta function nonlocality is also interesting in its own right."

B. Dragovich, "Zeta nonlocal scalar fields" (preprint, 04/2008)

[abstract:] "We consider some nonlocal and nonpolynomial scalar field models originated from p-adic string theory. Infinite number
of spacetime derivatives is determined by the operator valued Riemann zeta function through d'Alembertian $\Box$ in its argument. Construction
of the corresponding Lagrangians L starts with the exact Lagrangian $\mathcal{L}_p$ for effective field of p-adic tachyon string, which is
generalized replacing p by arbitrary natural number n and then taken a sum of $\mathcal{L}_n$ over all n. The corresponding new objects we call
zeta scalar strings. Some basic classical field properties of these fields are obtained and presented in this paper. In particular, some solutions of the
equations of motion and their tachyon spectra are studied. Field theory with Riemann zeta function dynamics is interesting in its own right as well."

B.D.B. Roth, "A
general approach to quantum fields and strings on adeles",
*Physics Letters* B **213** No. 3 (1988) 263-268

[abstract:] "We develop a general framework for constructing quantum theories on the space of adeles. We consider
complementary approaches utilizing both the functional integral and generalized Poincaré (translation) invariant
Hilbert space methods. As a starting point, we seek to construct the most general quantum theory on the rational numbers.
Very general arguments land us in the adeles. Quantum amplitudes generically will not factorize into separate *p*-adic
sectors. The vacuum amplitude, in particular, factorizes only in a linear regime. We conclude with implications for the
cosmological constant and adelic string sigma models."

The following three items contain reviews of work done on *p*-adic
and adelic approaches to QFT:

L. Brekke and P. Freund, "*p*-adic numbers in physics", *Physics Reports*
**233**, (1993) 1-66.

V.S. Vladimirov, I.V. Volovich,
E.I. Zelenov, *
**p*-Adic Analysis and Mathematical Physics (World Scientific
Publishing, 1994)

A. Khrennikov,
p*-Adic Valued Distributions in Mathematical Physics*,
(Kluwer, 1994).

The literature covered includes:

S. Albeverio, A. Khrennikov, and R. Cianci, "A representation of quantum
field Hamiltonian in a *p*-adic Hilbert space", *Theor. Mathem.
Physics* **112** no. 3 (1997) 355-374.

S. Albeverio and A. Khrennikov, "A regularization of quantum field
Hamiltonians with the aid of *p*-adic numbers", *Acta Appl. Math.*
**50** (1998) 225-251.

A. Khrennikov, "Statistical interpretation of *p*-adic valued quantum
field theory", *Dokl. Akad. Nauk* **328**, no. 1 (1993) 46-50.

I. Ya. Aref'eva, "Physics at the Planck length and *p*-adic field theory", from
*Differential Geometry Methods in Theoretical Physics*, eds. LL. Chau and W. Nahm (Plenum, 1990)

V.A. Smirnov, "Renormalization in *p*-adic quantum field theory", *Mod. Phys. Lett.* **A6** (1991) 1421-1427

V.A. Smirnov, "*p*-Adic Feynman amplitudes", Preprint MPI-Ph/91-48 (1991) 1-13

B. Grossman, "Adelic conformal field theory", *Physics Letters* **B215** (1988) 14-19

B. Grossman, "Arithmetic directions in topological quantum field theory and strings", pr3print, Rockerfeller University DOE/ER40 325-52-Task.B. (1988)

E. Yu. Lerner and M.D. Missarov, "Scalar models in *p*-adic quantum field theory
and hierarchical models", *Theor. math. Phys.* **78** (1989) 248-257.

H. Nishino, Y. Okada and M.R. Ubriaco, "Effective field theory and a *p*-adic
string", *Phys. Rev.* **D40** (1989) 1153-1157.

M.R. Ubriaco, "Fermions on the field of *p*-adic numbers", *Phys. Rev.* **D41** (1990) 2631-2636

M.R. Ubriaco, "Field quantization in nonarchimedean field theory", preprint LTP-012-UPR

E. Elizalde, "Spectral
zeta functions in non-commutative spacetime", *Nucl. Phys. Proc. Suppl.* **104** (2002)
157-160

"Formulas for the most general case of the zeta function associated to a
quadratic+linear+constant form (in {\bf Z}) are given. As examples, the spectral zeta functions
$\zeta_\alpha (s)$ corresponding to bosonic ($\alpha =2$) and to fermionic ($\alpha =3$)
quantum fields living on a noncommutative, partially toroidal spacetime are investigated.
Simple poles show up at *s* = 0, as well as in other places (simple or double, depending on the
number of compactified, noncompactified, and noncommutative dimensions of the spacetime). This
poses a challenge to the zeta-function regularization procedure."

G. Cognola, E. Elizalde and S. Zerbini,
"Fluctuations
of quantum fields via zeta function regularization", *Phys. Rev.* **D65** (2002)

[abstract:] "Explicit expressions for the expectation values and
the variances of some observables, which are bilinear quantities in
the quantum fields on a *D*-dimensional manifold, are derived making use
of zeta function regularization. It is found that the variance,
related to the second functional variation of the effective action,
requires a further regularization and that the relative regularized
variance turns out to be 2/*N*, where *N* is the number of the fields,
thus being independent on the dimension *D*. Some illustrating examples
are worked through."

E. Elizalde, "Some uses of zeta-regularization
in quantum gravity and cosmology", *Grav. Cosmol.* **8** (2002) 43-48

"This is a short guide to some uses of the zeta-function regularization procedure as a a basic mathematical tool for quantum field theory
in curved space-time (as is the case of Nambu-Jona-Lasinio models), in quantum gravity models (in different dimensions), and also in
cosmology, where it appears e.g. in the calculation of possible 'contributions' to the cosmological constant coming through manifestations
of the vacuum energy density."

E. Elizalde,
*Ten Physical Applications of Spectral Zeta Functions* (Lecture Notes in
Physics. New Series M, Monographs, M35) (Springer-Verlag, 1995)

"Zeta-function regularization is a powerful method in perturbation theory.
This book is meant as a guide for the student of this subject. Everything is
explained in detail, in particular the mathematical difficulties and tricky
points, and several applications are given to show how the procedure works in
practice (e.g. Casimir effect, gravity and string theory, high-temperature
phase transition, topological symmetry breaking). The formulas some of which
are new can be used for accurate numerical calculations. The book is to be
considered as a basic introduction and a collection of exercises for those
who want to apply this regularization procedure in practice."

E. Elizalde, S.D. Odintsov, A. Romeo and S. Zerbini,
*Zeta
Regularization Techniques With Applications* (World Scientific, 1994)

"This book is the result of several years of work by the authors on different
aspects of zeta functions and related topics. The aim is twofold. On one hand, a
considerable number of useful formulas, essential for dealing with the different
aspects of zeta-function regularization (analytic continuation, asymptotic
expansions), many of which appear here, in book format, for the first time are
presented. On the other hand, the authors show explicitly how to make use of such
formulas and techniques in practical applications to physical problems of very
different nature. Virtually all types of zeta functions are dealt with in the book."

V. Moretti and D. Iellici, "Zeta-function regularization and one-loop renormalization of field fluctuations in curved space-times", *Phys. Lett.* **B425** (1998) 33-40

"A method to regularize and renormalize the fluctuations of a quantum field in a curved background in the zeta-function approach is
presented. The method produces finite quantities directly and finite scale-parametrized counterterms at most. These finite counterterms
are related to the presence of a particular pole of the effective-action zeta function as well as to the heat kernel coefficients. The method
is checked in several examples obtaining known or reasonable results. Finally, comments are given for as it concerns the recent proposal
by Frolov *et.al.* to get the finite Bekenstein–Hawking entropy from Sakharov's induced gravity theory."

N. Makhaldiani, "Renormdynamics, multiparticle production, negative binomial distribution and Riemann zeta function" (preprint 12/2010)
[abstract:] "Renormdynamic equations of motion and their solutions are given. New equation for NBD distribution and Riemann zeta function invented. Explicit forms of the z-Scaling functions are constructed."

J.S. Dowker, "Sphere Renyi entropies" (preprint 12/2012)

[abstract:] "I give some scalar field theory calculations on a $d$-dimensional lune of arbitrary angle, evaluating, numerically, the effective action which is expressed as a simple quadrature, for conformal coupling. Using this, the entanglement and Renyi entropies are computed. Massive fields are also considered and a renormalisation to make the (one-loop) effective action vanish for infinite mass is suggested and used, not entirely successfully. However a universal coefficient is derived from the large mass expansion. For the round sphere, I show how to convert the quadrature form of the conformal Laplacian determinant into the more usual sum of Riemann zeta functions (and $\log 2$)."

K.V. Shajesh, I. Brevik, I. Cavero-Peláez and P. Parashar, "Self-similar plates: Casimir energies" (07/2016)

[abstract:] "We construct various self-similar configurations using parallel $\delta$-function plates and show that it is possible to evaluate the Casimir interaction energy of these configurations using the idea of self-similarity alone. We restrict our analysis to interactions mediated by a scalar field, but the extension to electromagnetic field is immediate. Our work unveils an easy and powerful method that can be easily employed to calculate the Casimir energies of a class of self-similar configurations. As a highlight, in an example, we determine the Casimir interaction energy of a stack of parallel plates constructed by positioning $\delta$-function plates at the points constituting the Cantor set, a prototype of a fractal. This, to our knowledge, is the first time that the Casimir energy of a fractal configuration has been reported. Remarkably, the Casimir energy of some of the configurations we consider turn out to be positive, and a few even have zero Casimir energy. For the case of positive Casimir energy that is monotonically decreasing as the stacking parameter increases the interpretation is that the pressure of vacuum tends to inflate the infinite stack of plates. We further support our results, derived using the idea of self-similarity alone, by rederiving them using the Green's function formalism. These expositions gives us insight into the **connections between the regularization methods used in quantum field theories and regularized sums of divergent series in number theory**."

G. Moore, "Arithmetic and attractors" (preprint 07/03)

[abstract:] "We study relations between some topics in number theory and supersymmetric black holes. These relations are
based on the "attractor mechanism" of *N*=2 supergravity. In IIB string compactification this mechanism singles out certain
"attractor varieties". ' We show that these attractor varieties are constructed from products of elliptic curves with complex multiplication
for *N*=4 and *N*=8 compactifications. The heterotic dual theories are related to rational conformal field theories. In the
case of *N*=4 theories U-duality inequivalent backgrounds with the same horizon area are counted by the class number of a
quadratic imaginary field. The attractor varieties are defined over fields closely related to class fields of the quadratic imaginary field.
We discuss some extensions to more general Calabi-Yau compactifications and explore further connections to arithmetic including
connections to Kronecker's Jugendtraum and the theory of modular heights. The paper also includes a short review of the attractor
mechanism. A much shorter version of the paper summarizing the main points is the companion note entitled "Attractors and Arithmetic""

R. Padma and H. Gopalkrishna Gadiyar,
"Renormalisation and the density of prime pairs" (preprint 06/98)

"Ideas from physics are used to show that the prime pairs have the
density conjectured by Hardy and Littlewood. The proof involves dealing
with infinities like in quantum field theory."

J. Fine, "Some notes on the inverse problem for braids"
(preprint 08/2006)

[abstract:] "The Kontsevich integral $Z$ associates to each braid $b$ (or more generally knot $k$) invariants $Z_i(b)$ lying in
finite dimensional vector spaces, for $i = 0,1,2,...$. These values are not yet known, except in special cases. The inverse problem
is that of determining $b$ from its invariants $Z_i(b)$. In this paper we study the case of braids on two strands, which is already
sufficient to produce interesting and unexpected mathematics. In particular, we find connections with number theory, numerical analysis
and field theory in physics. However, we will carry this study out with an eye to the more general case of braids on $n$ strands. We
expect that solving the inverse problem even for $n=3$ will present real difficulties. Most of the concepts in this paper also apply to
knots, but to simplify the exposition we will rarely mention this.
The organisation and bulk of the writing of this paper predates its most significant results. We hope later to present better and
develop further these results."

Here is an excerpt from a posting by on the sci.physics newsgroup (02/98)
by Dan Piponi:

"In (bosonic) string theory via the operator formalism you find an
infinite looking zero point energy just like in QED except that you get
a sum that looks like:

Now the naive thing to do is the same: subtract off this zero point
energy. However later on you get into complications. In fact (if I
remember correctly) you must replace this infinity with -1/12 (of all
things!) to keep things consistent.

Now it turns out there is a nice mathematical kludge that allows you to
see 1+2+3+4+... as equalling -1/12. What you do is rewrite it as

But even more amazingly is that you can get the -1/12 by a completely
different route - using the path integral formalism rather than the
operator formalism. This -1/12 is tied up in a deep way with the
geometry of string theory so it's a lot more than simply a trick to keep
the numbers finite.

However I don't know if the equivalent operation in QED is tied up with
the same kind of interesting geometry."