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This work derives 5 methods to evaluate families of odd zeta values by combining a power of $\pi$ with Lambert series whose ratios of successive terms tend to $e^{-\pi\sqrt{a}}$ with integers $a\ge7$, outperforming Ramanujan's results with merit $a=4$. Families with $a=7$ and $a=8$ evaluate $\zeta(2n+1)$. Families with $a=9$ and $a=16$ evaluate $\zeta(4n+1)$ with faster convergence. A fifth family with $a=12$ evaluates $\zeta(6n+1)$ and gives the fastest convergence for $\zeta(6n+7)$. Members of three of the families were discovered empirically by Simon Plouffe. An intensive new search strongly suggests that there are no more than 5 families with integers $a\ge7$. There are at least 20 families that involve Lambert series with rational $a>4$. Quasi-modular transformations of Lambert series resolve rational sequences that were discovered empirically. Expansions of Lambert series in polylogarithms, familiar from quantum field theory, provide proofs of all known evaluations with rational merit $a>4$., Comment: 22 pages, including 20 families with rational merit

The Dickman function $\rho(u)$ gives the asymptotic probability that a large integer $N$ has no prime divisor exceeding $N^{1/u}$. We expand it in terms of rapidly computable multiple polylogarithms, as defined by Goncharov and intensively used for evaluations of Feynman integrals in quantum field theory. In parallel, we solve Buchstab's differential-delay equation, which concerns large integers $N$ divisible by no prime less than $N^{1/u}$. Discussion of the latter problem occurred in letters to the journal Nature during the second world war, from the physicists Frederick Lindemann and Wendell Furry. We recount how Furry evaluated a dilogarithm in reply to a puzzle resulting from Mertens' third theorem, raised by Lindemann. We refine Furry's analysis to include multiple polylogarithms of weights up to 200., Comment: 30 pages, 3 figures

This work deals with two types of Feynman integrals in perturbative quantum field theory: the 2-loop 2-point kite, with 5 arbitrary internal masses, and its completion by a sixth propagator, to give a 3-loop tetrahedral tadpole, with 6 arbitrary masses. These general-mass cases cover broken and unbroken gauge theories, based on the Lie algebras U(1), SU(2) and SU(3), for the electromagnetic, weak and strong interactions. The elliptic substructure of these integrals should not be regarded as an obstruction. Rather, it is a bonus, thanks to the arithmetic-geometric mean of Gauss. Compact formulae are given, to handle all cases. Zero-mass limits are carefully considered. Anomalous thresholds of triangles in the kite pose no problem. The number theory of tadpoles is investigated, with intriguing results., Comment: 16 pages, to appear in the proceedings of the 15th International Workshop on Lie Theory and Its Applications in Physics (LT-15), 19-25 June 2023, Varna, Bulgaria

Perturbative expansions in quantum field theory diverge for at least two reasons: the number of Feynman diagrams increases dramatically with the loop number and the process of renormalization may make the contribution of some diagrams large. We give an example of the second problem, from an ultra-violent renormalon of $\phi^3$ theory in 6 dimensions, where we can compute to very high loop-order. Taming this renormalon involves recent work on resurgence. This challenge is much more demanding than the corresponding problem for Yukawa theory in 4 dimensions., Comment: 8 pages, conference proceedings contribution to Loops and Legs in Quantum Field Theory 2022 - version to appear in PoS-LL2022

We perform an all-order resurgence analysis of a quantum field theory renormalon that contributes to an anomalous dimension in six-dimensional scalar $\phi^3$ theory and is governed by a third-order nonlinear differential equation. We augment the factorially divergent perturbative expansion associated to the renormalon by asymptotic expansions to all instanton orders, in a conjectured and well-tested formula. A distinctive feature of this renormalon singularity is the appearance of logarithmic terms, starting at second-instanton order in the trans-series. To highlight this and to illustrate our methods, we also analyze the trans-series for a closely related second-order nonlinear differential equation that exhibits a similarly resonant structure but lacks logarithmic contributions., Comment: 34 pages, v2: Discussion on ODE ambiguities and typos corrected. Accepted version to appear in Nuclear Physics B

Record values are determined for the order $|\Sha|$ of the Tate--Shafarevich group of an elliptic curve $E$, computed analytically by the Birch--Swinnerton-Dyer conjecture, and for the Goldfeld--Szpiro ratio $G=|\Sha|/\sqrt{N}$, where $N$ is the conductor of $E$. The curves have rank zero and are isogenous to quadratic twists of Frey curves constructed from coprime positive integers $(a,b,c)$ with $a+b=c$ and $c>r^{1.4}$, where the radical $r$ is the product of the primes dividing $abc$. Curves with $|\Sha|>250000^2$ and $G>12$ are found in 20 isogeny classes. Three curves have $G>150$. The largest value of $|\Sha|$ is $1937832^2>3.755\times10^{12}$. This is more than 3.5 times the previous record, which had been computed at a cost about 600 times greater than that for the new record. The primes 25913, 27457, 36929 and 49253 are identified as divisors of $|\Sha|$ values., Comment: 17 pages

Integer relation algorithms can convert numerical results for Feynman integrals to exact evaluations, when one has reason to suspect the existence of reductions to linear combinations of a basis, with rational or algebraic coefficients. Once a tentative reduction is obtained, confidence in its validity is greatly increased by computing more decimal digits of the terms and verifying the stability of the result. Here we give examples of how the PSLQ and LLL algorithms have yielded remarkable reductions of Feynman integrals to multiple polylogarithms and to the periods and quasi-periods of modular forms. Moreover, these algorithms have revealed quadratic relations between Feynman integrals. A recent application concerning black holes involves quadratic relations between combinations of Feynman integrals with algebraic coefficients., Comment: 21 pages

Eta quotients on $\Gamma_0(6)$ yield evaluations of sunrise integrals at 2, 3, 4 and 6 loops. At 2 and 3 loops, they provide modular parametrizations of inhomogeneous differential equations whose solutions are readily obtained by expanding in the nome $q$. Atkin-Lehner transformations that permute cusps ensure fast convergence for all external momenta. At 4 and 6 loops, on-shell integrals are periods of modular forms of weights 4 and 6 given by Eichler integrals of eta quotients. Weakly holomorphic eta quotients determine quasi-periods. A Rademacher sum formula is given for Fourier coefficients of an eta quotient that is a Hauptmodul for $\Gamma_0(6)$ and its generalization is found for all levels with genus 0, namely for $N = 1,2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, 25$. There are elliptic obstructions at $N = 11, 14, 15, 17, 19, 20, 21, 24, 27, 32, 36, 49,$ with genus 1. We surmount these, finding explicit formulas for Fourier coefficients of eta quotients in thousands of cases. We show how to handle the levels $N=22, 23, 26, 28, 29, 31, 37, 50$, with genus 2, and the levels $N=30,33,34,35,39,40,41,43,45,48,64$, with genus 3. We also solve examples with genera $4,5,6,7,8,13$., Comment: 26 pages

In a recent study of how the output voltage of a Hall plate is affected by the shape of the plate and the size of its contacts, Udo Ausserlechner has come up with a remarkable double integral that can be viewed as a generalization of the classical elliptic "AGM" integral. Here we discuss transformation properties of the integral, which were experimentally observed by Ausserlechner, as well as its analytical and arithmetic features including connections with modular forms., Comment: 15 pages

Several powerful techniques for evaluating massless scalar Feynman diagrams are developed, viz: the solution of recurrence relations to evaluate diagrams with arbitrary numbers of loops in $n=4-2\omega$ dimensions; the discovery and use of symmetry properties to restrict and compute Taylor series in $\omega$; the reduction of triple sums over Chebyshev polynomials to products of Riemann zeta functions; the exploitation of conformal invariance to avoid four-dimensional Racah coefficients. As an example of the power of these techniques we evaluate all of the 216 diagrams, with 5 loops or less, which give finite contributions of order $1/k^2$ or $1/k^4$ to a propagator of momentum $k$ in massless four-dimensional scalar field theories. Remarkably, only 5 basic numbers are encountered: $\zeta(3)$, $\zeta(5)$, $\zeta(7)$, $\zeta(9)$ and the value of the most symmetrical diagram, which is calculated to 14 significant figures. It is conceivable that these are the only irrationals appearing in 6-loop beta functions. En route to these results we uncover and only partially explain many remarkable relations between diagrams., Comment: report of The Open University, Milton Keynes, England (UK), 1985, copy provided by John Gracey, http://cds.cern.ch/record/164890, LaTeX reproduction by Erik Panzer

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., Comment: 48 pages

I define multiple Watson values (MWVs) as iterated integrals, on the interval $x\in[0,1]$, of the 6 differential forms $A=d\log(x)$, $B=-d\log(1-x)$, $T=-d\log(1-z_1x)$, $U=-d\log(1-z_2x)$, $V=-d\log(1-z_3x)$ and $W=-d\log(1-z_4x)$, where $z_1=\gamma^2$, $z_2=\gamma/(1+\gamma)$, $z_3=\gamma^2/(1-\gamma)$ and $z_4=\gamma=2\sin(\pi/14)$ solves the cubic $(1-\gamma^2)(1-\gamma)=\gamma$. Following a suggestion by Pierre Deligne, I conjecture that the dimension of the space of ${\mathbb Z}$-linearly independent MWVs of weight $w$ is the number $D_w$ generated by $1/(1-2x-x^2-x^3)=1+\sum_{w>0}D_w x^w$. This agrees with 6639 integer relation searches, of dimensions up to $D_5+1=85$, performed at 2000-digit precision, for $w<6$., Comment: 7 pages

Multiple Landen values (MLVs) are defined as iterated integrals on the interval $x\in[0,1]$ of the differential forms $A=d\log(x)$, $B=-d\log(1-x)$, $F=-d\log(1-\rho^2x)$ and $G=-d\log(1-\rho x)$, where $\rho=(\sqrt{5}-1)/2$ is the golden section. I conjecture that the dimension of the space of ${\mathbb Z}$-linearly independent MLVs of weight $w$ is a tribonacci number $T_w$, generated by $1/(1-x-x^2- x^3)=1+\sum_{w>0}T_w x^w$, and that a basis is provided by all the words in the $\{A,G\}$ sub-alphabet that neither end in $A$ nor contain $A^3$. For $w<9$, I construct a much more efficient basis, for a MLV datamine, where no prime greater than 11 occurs in the denominators of 3,357,257 coefficients of rational reduction of 49,151 MLVs. Numerical data for 40 primitives then enable fast evaluation of all of these MLVs to 20,000 digits. The datamine provides reductions of Ap\'ery-type sums $A_w=\sum_{n>0}(-1)^{n+1}n^{-w}/{2n\choose n}$ and 6 ladder-combinations of depth-1 polylogarithms ${\rm Li}_w(\rho^p)=\sum_{n>0}\rho^{pn}n^{-w}$ with $p\in\{1,2,3,4,6,8,10,12,20,24\}$ and coefficients given by Landen, Coxeter and Lewin at $w=2$. I prove that the former evaluate to MLVs and conjecture that the latter do. Comparison is made between the properties of MLVs and multiple polylogarithms at roots of unity, encountered in the quantum field theory of the standard model of particle physics., Comment: 31 pages: version 2 includes a link to letter by Pierre Deligne

Multiple Deligne values (MDVs) are iterated integrals on the interval $x\in[0,1]$ of the differential forms $A=d\log(x)$, $B=-d\log(1-x)$ and $D=-d\log(1-\lambda x)$, where $\lambda$ is a primitive sixth root of unity. MDVs of weight 11 enter the renormalization of the standard model of particle physics at 7 loops, via a counterterm for the self-coupling of the Higgs boson. A recent evaluation by Erik Panzer exhibited the alarming primes 50909 and 121577 in the denominators of rational coefficients that reduce this counterterm to a Lyndon basis suggested by ideas from Pierre Deligne. Oliver Schnetz has studied this problem, using a method from Francis Brown. This gave 2111, 14929, 24137, 50909 and 121577 as factors of the denominator of the coefficient of $\pi^{11}/\sqrt{3}$. Here I construct a basis such that no denominator prime greater than 3 appears in the result. This is achieved by building a datamine of 13,369,520 rational coefficients, with tame denominators, for the the reductions of 118,097 MDVs with weights up to 11. Then numerical data for merely 53 primitives enables very fast evaluation of all of these MDVs to 20000 digits. In the course of this Aufbau, six conjectures for MDVs are formulated and stringently tested., Comment: 24 pages, 2 figures

Single-scale Feynman diagrams yield integrals that are periods, namely projective integrals of rational functions of Schwinger parameters. Algebraic geometry may therefore inform us of the types of number to which these integrals evaluate. We give examples at 3, 4 and 6 loops of massive Feynman diagrams that evaluate to Dirichlet $L$-series of modular forms and examples at 6, 7 and 8 loops of counterterms that evaluate to multiple zeta values or polylogarithms of the sixth root of unity. At 8 loops and beyond, algebraic geometry informs us that polylogs are insufficient for the evaluation of terms in the beta-function of $\phi^4$ theory. Here, modular forms appear as obstructions to polylogarithmic evaluation., Comment: 8 pages

The L-loop 4-point ladder diagram of massless phi^3 theory is finite when all 4 legs are off-shell and is given in terms of polylogarithms with orders ranging from L to 2L. We obtain the exact solution of the linear Dyson-Schwinger equation that sums these ladder diagrams and show that this sum vanishes exponentially fast at strong coupling., Comment: 5 pages, 1 figure, presented at "Loops and Legs in Quantum Field Theory 2010", Woerlitz, Germany, April 2010

One of the most effective techniques of experimental mathematics is to compute mathematical entities such as integrals, series or limits to high precision, then attempt to recognize the resulting numerical values. Recently these techniques have been applied with great success to problems in mathematical physics. Notable among these applications are the identification of some key multi-dimensional integrals that arise in Ising theory, quantum field theory and in magnetic spin theory., Comment: 18 pages, 2 figures

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., Comment: 24 pages, LaTeX

The Dickman function F(alpha) gives the asymptotic probability that a large integer N has no prime divisor exceeding N^alpha. It is given by a finite sum of generalized polylogarithms defined by the exquisite recursion L_k(alpha)=- int_alpha^{1/k} dx L_{k-1}(x/(1-x))/x with L_0(alpha)=1. The behaviour of these Dickman polylogarithms as alpha tends to 0 defines an intriguing series of constants, C_k. I conjecture that exp(gamma z)/Gamma(1-z) is the generating function for sum_{k\ge0} C_k z^k. I obtain high-precision evaluations of F(1/k), for integers k<11, and compare the Dickman problem with problems in condensed matter physics and quantum field theory., Comment: 11 pages, LaTeX

For square-free $N\equiv3$ mod 8 and $N$ coprime to 3, I show how to reduce the singular value $k_N$ to radicals, using a novel pair $[f,g]$ of real numbers that are algebraic integers of the Hilbert class field of $Q(\sqrt{-N})$. One is a class invariant of modular level 48, with a growth $g=\alpha(N)\exp(\pi\sqrt{N}/48)+o(1)$, where $\alpha(N)\in[-\sqrt2,\sqrt2]$ is uniquely determined by the residue of $N$ modulo 64. Hence $g$ is a very economical generator of the class field. For prime $N\equiv3$ mod 4, I conjecture that the Chowla--Selberg formula provides an algebraic {\em unit} of the class field and determine its minimal polynomial for the 155 cases with $N<2000$. For N=2317723, with class number $h(-N)=105$, I compute the minimal polynomial of $g$ in 90 milliseconds. Its height is smaller than the {\em cube} root of the height of the generating polynomial found by the double eta-quotient method of {\em Pari-GP}. I reduce the complete elliptic integral $K_{2317723}$ to radicals and values of the $\Gamma$ function, by determining the Chowla--Selberg unit and solving the septic, quintic and cubic equations that generate sub-fields of the class field. I conclude that the residue 3 modulo 8, initially discarded in elliptic curve primality proving, outperforms the residue 7., Comment: 15 pages, appendix added; more scholarly references

A proof is found for the elliptic integral evaluation of the Bessel moment $$M:=\int_0^\infty t I_0^2(t)K_0^2(t)K_0(2t) {\rm d}t ={1/12} {\bf K}(\sin(\pi/12)){\bf K}(\cos(\pi/12)) =\frac{\Gamma^6(\frac13)}{64\pi^22^{2/3}}$$ resulting from an angular average of a 2-loop 4-point massive Feynman diagram, with one internal mass doubled. This evaluation follows from contour integration of the Green function for a hexagonal lattice, thereby relating $M$ to a linear combination of two more tractable moments, one given by the Green function for a diamond lattice and both evaluated by using W.N. Bailey's reduction of an Appell double series to a product of elliptic integrals. Cubic and sesquiplicate modular transformations of an elliptic integral from the equal-mass Dalitz plot are proven and used extensively. Derivations are given of the sum rules $$\int_0^\infty(I_0(a t)K_0(a t)-\frac{2}{\pi} K_0(4a t) K_0(t))K_0(t) {\rm d}t=0$$ with $a>0$, proven by analytic continuation of an identity from Bailey's work, and $$\int_0^\infty t I_0(a t)(I_0^3(a t)K_0(8t)- \frac{1}{4\pi^2} I_0(t)K_0^3(t)) {\rm d}t=0$$ with $2\ge a\ge0$, proven by showing that a Feynman diagram in two spacetime dimensions generates the enumeration of staircase polygons in four dimensions., Comment: 13 pages, now includes staircase polygons and complex separatrices

We record what is known about the closed forms for various Bessel function moments arising in quantum field theory, condensed matter theory and other parts of mathematical physics. More generally, we develop formulae for integrals of products of six or fewer Bessel functions. In consequence, we are able to discover and prove closed forms for $c_{n,k}:=\int_0^\infty t^k K_0^n(t) {\rm d}t$ with integers $n=1,2,3,4$ and $k\ge0$, obtaining new results for the even moments $c_{3,2k}$ and $c_{4,2k}$. We also derive new closed forms for the odd moments $s_{n,2k+1}:=\int_0^\infty t^{2k+1}I_0^{}(t) K_0^{n-1}(t) {\rm d}t$ with $n=3,4$ and for $t_{n,2k+1}:=\int_0^\infty t^{2k+1}I_0^2(t) K_0^{n-2}(t) {\rm d}t$ with $n=5$, relating the latter to Green functions on hexagonal, diamond and cubic lattices. We conjecture the values of $s_{5,2k+1}$, make substantial progress on the evaluation of $c_{5,2k+1}$, $s_{6,2k+1}$ and $t_{6,2k+1}$ and report more limited progress regarding $c_{5,2k}$, $c_{6,2k+1}$ and $c_{6,2k}$. In the process, we obtain 8 conjectural evaluations, each of which has been checked to 1200 decimal places. One of these lies deep in 4- dimensional quantum field theory and two are probably provable by delicate combinatorics. There remains a hard core of five conjectures whose proofs would be most instructive, to mathematicians and physicists alike., Comment: 51 pages, 1 Postscript figure, uses amsmath.sty, added references

Historically, the polylogarithm has attracted specialists and non-specialists alike with its lovely evaluations. Much the same can be said for Euler sums (or multiple harmonic sums), which, within the past decade, have arisen in combinatorics, knot theory and high-energy physics. More recently, we have been forced to consider multidimensional extensions encompassing the classical polylogarithm, Euler sums, and the Riemann zeta function. Here, we provide a general framework within which previously isolated results can now be properly understood. Applying the theory developed herein, we prove several previously conjectured evaluations, including an intriguing conjecture of Don Zagier., Comment: 35 pages

Cohen, Lewin and Zagier found four ladders that entail the polylogarithms ${\rm Li}_n(\alpha_1^{-k}):=\sum_{r>0}\alpha_1^{-k r}/r^n$ at order $n=16$, with indices $k\le360$, and $\alpha_1$ being the smallest known Salem number, i.e. the larger real root of Lehmer's celebrated polynomial $\alpha^{10}+\alpha^9-\alpha^7-\alpha^6-\alpha^5-\alpha^4-\alpha^3+\alpha+1$, with the smallest known non-trivial Mahler measure. By adjoining the index $k=630$, we generate a fifth ladder at order 16 and a ladder at order 17 that we presume to be unique. This empirical integer relation, between elements of $\{{\rm Li}_{17}(\alpha_1^{-k})\mid0\le k\le630\}$ and $\{\pi^{2j}(\log\alpha_1)^{17-2j}\mid 0\le j\le8\}$, entails 125 constants, multiplied by integers with nearly 300 digits. It has been checked to more than 59,000 decimal digits. Among the ladders that we found in other number fields, the longest has order 13 and index 294. It is based on $\alpha^{10}-\alpha^6-\alpha^5-\alpha^4+1$, which gives the sole Salem number $\alpha<1.3$ with degree $d<12$ for which $\alpha^{1/2}+\alpha^{-1/2}$ fails to be the largest eigenvalue of the adjacency matrix of a graph., Comment: 18 pages, LaTeX

Let $\{x_1, x_2, ..., x_n\}$ be a vector of real numbers. An integer relation algorithm is a computational scheme to find the $n$ integers $a_k$, if they exist, such that $a_1 x_1 + a_2 x_2 + ... + a_n x_n= 0$. In the past few years, integer relation algorithms have been utilized to discover new results in mathematics and physics. Existing programs for this purpose require very large amounts of computer time, due in part to the requirement for multiprecision arithmetic, yet are poorly suited for parallel processing. This paper presents a new integer relation algorithm designed for parallel computer systems, but as a bonus it also gives superior results on single processor systems. Single- and multi-level implementations of this algorithm are described, together with performance results on a parallel computer system. Several applications of these programs are discussed, including some new results in number theory, quantum field theory and chaos theory., Comment: 18 pages, LaTeX