winding of planar brownian curves the authors compute the joint probability for a closed brownian curve to wind n times around a prescribed point and to enclose a given algebraic area an estimate from below of the arithmetic area is obtained

some new lattice sums including an exact result for the electrostatic potential within the nacl lattice the electrostatic potential at point 1 6 1 6 1 6 in the unit cell of side length 2 of the nacl lattice is found to be square root 3 an n dimensional generalisation of a dipole sum evaluated exactly by glasser and zucker 1980 in the case n 2 is given

windings of the 2d free rouse chain we study the long time dynamical properties of a chain of harmonically bound brownian particles this chain is allowed to wander everywhere in the plane we show that the scaling variables for the occupation times tj areas aj and winding angles j j 1 n labels the particles take the same general form as in the usual brownian motion we also compute the asymptotic joint laws p tj p aj p j and discuss the correlations occurring in those distributions

the random walk winding number problem convergence to a diffusion process with excluded area the winding number of a random walk in the plane is the net angle through which the walker encircles the origin the author discusses the moments and asymptotic form of the winding number distribution for walks of constant step size l the author shows that the root mean square winding number theta rms grows logarithmically with the number of steps n as n to infinity in contrast with the n1 2 growth of the radial distribution the corresponding diffusion process however has infinite theta rms and does not provide a good approximation for the random walk distribution instead a diffusion process considered recently by rudnick and hu 1987 where the area surrounding the origin has been removed out to a radius r provides the correct asymptotic distribution the author finds that the optimum radius for convergence of the finite step and diffusion distributions is precisely r e 2l

the two loop sunrise graph in two space time dimensions with arbitrary masses in terms of elliptic dilogarithms we present the two loop sunrise integral with arbitrary non zero masses in two space time dimensions in terms of elliptic dilogarithms we find that the structure of the result is as simple and elegant as in the equal mass case only the arguments of the elliptic dilogarithms are modified these arguments have a nice geometric interpretation

the two loop sunrise integral around four space time dimensions and generalisations of the clausen and glaisher functions towards the elliptic case we present the result for the finite part of the two loop sunrise integral with unequal masses in four space time dimensions in terms of the o e0 part and the o e1 part of the sunrise integral around two space time dimensions the latter two integrals are given in terms of elliptic generalisations of clausen and glaisher functions interesting aspects of the result for the o e1 part of the sunrise integral around two space time dimensions are the occurrence of depth two elliptic objects and the weights of the individual terms

winding angle distribution for planar random walk polymer ring entangled with an obstacle and all that spitzer edwards prager frisch model revisited using a general green function formulation we re derive both i spitzer and his followers results for the winding angle distribution of the planar brownian motion and ii edwards prager frisch results on the statistical mechanics of a ring polymer entangled with a straight bar in the statistical mechanics part we consider both cases of quenched and annealed topology among new results we compute exactly the expectation value of the surface area of the locus of points such that each of them has linking number n with a given closed random walk trajectory ring polymer we also consider the generalizations of the problem for the finite diameter disc like obstacle and winding within a cavity

analytic and algorithmic aspects of generalized harmonic sums and polylogarithms in recent three loop calculations of massive feynman integrals within quantum chromodynamics qcd and e g in recent combinatorial problems the so called generalized harmonic sums in short s sums arise they are characterized by rational or real numerator weights also different from 1 in this article we explore the algorithmic and analytic properties of these sums systematically we work out the mellin and inverse mellin transform which connects the sums under consideration with the associated poincare iterated integrals also called generalized harmonic polylogarithms in this regard we obtain explicit analytic continuations by means of asymptotic expansions of the s sums which started to occur frequently in current qcd calculations in addition we derive algebraic and structural relations like differentiation with respect to the external summation index and different multi argument relations for the compactification of s sum expressions finally we calculate algebraic relations for infinite s sums or equivalently for generalized harmonic polylogarithms evaluated at special values the corresponding algorithms and relations are encoded in the computer algebra package harmonicsums

systematic investigation of two dimensional static array sums we discuss general properties of doubly periodic sums over the square lattice linking phased bloch type sums in the direct lattice with displaced sums in the reciprocal lattice using the poisson summation formula we discuss cardinal points where the sums reduce to a single product of dirichlet l functions and exhibit all cardinal points for the square lattice we introduce a new method for evaluating lattice sums and illustrate this by solving low order systems of sums of integer order 2 3 4 5 for the last case the analytic expressions for the sums involve complementary l functions or alternatively l functions with complex characters

winding of the plane brownian motion on ecrit le mouvement brownien dans un format complexe x a 1b et on note son invariance conforme ce mouvement brownien x dans le plan perfore est releve a m 3 et on etudie si ce mouvement est recurseur

iterated elliptic and hypergeometric integrals for feynman diagrams we calculate 3 loop master integrals for heavy quark correlators and the 3 loop quantum chromodynamics corrections to the parameter they obey non factorizing differential equations of second order with more than three singularities which cannot be factorized in mellin n space either the solution of the homogeneous equations is possible in terms of 2f1 gaus hypergeometric functions at rational argument in some cases integrals of this type can be mapped to complete elliptic integrals at rational argument this class of functions appears to be the next one arising in the calculation of more complicated feynman integrals following the harmonic polylogarithms generalized polylogarithms cyclotomic harmonic polylogarithms square root valued iterated integrals and combinations thereof which appear in simpler cases the inhomogeneous solution of the corresponding differential equations can be given in terms of iterative integrals where the new innermost letter itself is not an iterative integral a new class of iterative integrals is introduced containing letters in which multiple definite integrals appear as factors for the elliptic case we also derive the solution in terms of integrals over modular functions and also modular forms using q product and series representations implied by jacobi s i functions and dedekind s function the corresponding representations can be traced back to polynomials out of lambert eisenstein series having representations also as elliptic polylogarithms a q factorial 1 k logarithms and polylogarithms of q and their q integrals due to the specific form of the physical variable x q for different processes different representations do usually appear numerical results are also presented

the winding angle distribution of an ordinary random walk the winding angle about a selected point of a random walk quantifies what is perhaps the simplest manifestation of entanglement a phenomenon of great importance in the study of polymers the distribution of winding angles is studied here for ordinary or non self avoiding walks new results highlight the crucial influence of the exclusion of a finite region about the selected point on the winding angle distribution the results of this analysis are compared with simulations of random walks on a lattice the agreement between the two is very good there are however small and as yet unexplained discrepancies

belyi functions for hyperbolic hypergeometric to heun transformations abstract a complete classification of belyi functions for transforming certain hypergeometric equations to heun equations is given the considered hypergeometric equations have the local exponent differences 1 k 1 l 1 m that satisfy k l m n and the hyperbolic condition 1 k 1 l 1 m 1 there are 366 galois orbits of belyi functions giving the considered non parametric hypergeometric to heun pull back transformations their maximal degree is 60 which is well beyond reach of standard computational methods to obtain these belyi functions we developed two efficient algorithms that exploit the implied pull back transformations

the andrews sellers family of partition congruences in 1994 james sellers conjectured an infinite family of ramanujan type congruences for 2 colored frobenius partitions introduced by george e andrews these congruences arise modulo powers of 5 in 2002 dennis eichhorn and sellers were able to settle the conjecture for powers up to 4 in this article we prove sellers conjecture for all powers of 5 in addition we discuss why the andrews sellers family is significantly different from classical congruences modulo powers of primes

large deflection computation of the plastica abstract a nonlinear finite element method is presented for the computation of the large deflection of an inextensible rod bent beyond the linear elastic range of its material an orthogonal trajectory accession method is used to trace the displacement load curves for a rod flexed by an axial force a circular ring crushed by diametrical forces a c ring opened by an end force and a column collapsing under its own weight

third order braid invariants the author analyses the topological invariants of three braided curves a t b t and c t 3 braids are represented as a single phase curve gamma t in a two dimensional configuration space this configuration space consists of a set of triangular regions connected at their vertices the curve gamma t passes through a vertex whenever a t b t and c t are collinear the sequence of vertices completely describes the braid up to uniform twists the length t of this sequence can be employed as a measure of topological complexity the energy of a set of braided magnetic flux tubes is expected to be proportional to t2 w2 where w is the total winding number or signed crossing number of the braid second order winding numbers are integrals of closed 1 forms like d theta ab the author presents a third order winding number psi gamma which is also an integral of a closed 1 form but which depends on relations between all three curves

some thoughts about random walks on figure eight abstract in this paper we discuss the issues of recurrence and transience of random walks in multiply connected spaces and in particular in the presence of knots such problems are of interest in polymer physics since almost all sufficiently long linear polymer chains are quasi knotted at least at theta solvent quality conditions and hence the transience recurrence provides mathematically rigorous definition of the entanglement concept these problems are also of biological interest since they clarify the role of topology in problems involving molecular recognition obtained results extend and generalize the results of earlier published rapid communication kholodenko phys rev e 58 1998 r5213 they also provide some answers to problems which were just mentioned in our earlier published report kholodenko phys reports 298 1998 251

asymptotic laws of planar brownian motion on presente des resultats recents sur la distribution asymptotique des nombres de passages et d enroulements

rogers ramanujan identities in the hard hexagon model the hard hexagon model in statistical mechanics is a special case of a solvable class of hard square type models in which certain special diagonal interactions are added the sublattice densities and order parameters of this class are obtained and it is shown that many rogers ramanujan type identities naturally enter the working

the iterated structure of the all order result for the two loop sunrise integral we present a method to compute the laurent expansion of the two loop sunrise integral with equal non zero masses to arbitrary order in the dimensional regularisation e this is done by introducing a class of functions generalisations of multiple polylogarithms to include the elliptic case and by showing that all integrations can be carried out within this class of functions

brownian charges around loops let x t n be a poisson sequence of independent brownian motions in d d 3 let be a compact oriented submanifold of d of dimensiond 2 and volume l let t be the sum of the windings of x s n 0 s t around then t t converges in law towards a cauchy variable of parameter l 2 a similar result is valid when the winding is replaced by the integral of a harmonic 1 form in d

dirichlet l series with real and complex characters and their application to solving double sums a description of the properties of l series with complex characters is given by using these together with the more familiar l series with real characters it is shown how certain two dimensional lattice sums which previously could not be put into closed form may now be expressed in this way

lattice sums arising from the poisson equation in recent times attention has been directed to the problem of solving the poisson equation either in engineering scenarios computational or in regard to crystal structure theoretical herein we study a class of lattice sums that amount to poisson solutions namely the n dimensional forms by virtue of striking connections with jacobi function values we are able to develop new closed forms for certain values of the coordinates rk and extend such analysis to similar lattice sums a primary result is that for rational x y the natural potential 2 x y is where a is an algebraic number various extensions and explicit evaluations are given such work is made possible by number theoretical analysis symbolic computation and experimental mathematics including extensive numerical computations using up to 20 000 digit arithmetic

the kite integral to all orders in terms of elliptic polylogarithms we show that the laurent series of the two loop kite integral in d 4 2e space time dimensions can be expressed in each order of the series expansion in terms of elliptic generalisations of multiple polylogarithms using differential equations we present an iterative method to compute any desired order as an example we give the first three orders explicitly

orthogonal trajectory accession to the nonlinear equilibrium curve abstract an alternative to the riks wempner crisfield iterative correction scheme is presented that does not require an explicit displacement load accession path to the nonlinear equilibrium curve nor a known equilibrium point its symmetry with respect to the displacement and load assures success in rounding limit points as well as turning points

self conjugate t core partitions sums of squares and p blocks of an we prove that if t is an integer with t 8 or t 10 then every integer n 2 has a self conjugate t core partition this result has consequences in the representation theory of alternating groups and has a version as a theorem about the representation of integers by sums of squares we also give an infinite sequence of integers that have no self conjugate 9 core partition

eta quotients and theta functions abstract the jacobi triple product identity gives a closed form for many infinite product generating functions that arise naturally in combinatorics and number theory of particular interest is its application to dedekind s eta function z defined via an infinite product giving it as a certain kind of infinite sum known as a theta function using the theory of modular forms we classify all eta quotients that are theta functions

windings of random walks l auteur considere une suite de variables aleatoires bornees a valeurs dans r 2 ayant l identite comme matrice de covariance et un vecteur moyen nul il definit une marche aleatoire s a partir des sommes s des 21 premiers termes de la suite et il note n l angle total parcouru par s autour de l origine jusqu a un temps n il demontre sous des conditions de regularite legeres sur la loi du premier element de la suite un resultat sur la convergence de 2 n log n

the asymptotic joint distribution of windings of planar brownian motion let z z t t 0 be a complex valued brownian motion starting at zo and consider the real valued winding processes 0i 02 0n where 0j i is the continuous total angle wound by z about zj up to time t and z z zn are n 1 distinct points in the complex plane c let c i c n jb b be n 2 mutually independent one dimensional processes where the first n are standard cauchy processes and the last 2 are standard brownian motions let r inf t l u local time at 0 of b up to time t wj cj u b n j l n

an arc length method including line searches and accelerations this paper describes a method for introducing line searches into the arc length solution procedure such line searches may be used at each iteration to calculate an optimum scalar step length which scales the normal iterative vector in practice a loose tolerance is provided so that on many iterations the line searches are avoided however on difficult iterations the line searches are shown to lead to a substantial improvement in the convergence characteristics a simple single parameter acceleration is also developed using line search concepts the new arc length method is applied to both the geometrically nonlinear analysis of shallow shells and the materially nonlinear analysis of reinforced concrete beams and slabs significant improvements are demonstrated in relation to the standard arc length method

nested sums expansion of transcendental functions and multiscale multiloop integrals expansion of higher transcendental functions in a small parameter are needed in many areas of science for certain classes of functions this can be achieved by algebraic means these algebraic tools are based on nested sums and can be formulated as algorithms suitable for an implementation on a computer examples such as expansions of generalized hypergeometric functions or appell functions are discussed as a further application we give the general solution of a two loop integral the so called c topology in terms of multiple nested sums in addition we discuss some important properties of nested sums in particular we show that they satisfy a hopf algebra

positive harmonic functions on abelian covers abstract we prove the existence of positive harmonic functions and green s functions on certain abelian covers of non compact finite volume riemann surfaces these results are obtained by studying the asymptotic distribution of the lattice in hyperbolic space generated by the fundamental group of the covering surface

on the length of the homotopic brownian word in the thrice punctured sphere we consider the word associated to the homotopic class of the brownian path properly closed in the thrice punctured sphere we prove that its length has almost surely the same behaviour as a totally asymmetric cauchy process on the line more precisely the liminf has the same normalization in tlog t and the limsup can be described by the same integral test they are the brownian motion counterparts of some levy and khintchine results on continued fraction expansions

asymptotic laws for the winding angles of planar brownian motion using constrained path integrals we study the winding angle distribution of a two dimensional brownian motion around a given point by a careful analysis of the spectral properties of some schrodinger like hamiltonians we obtain a generalization of the messulam yor law various limiting cases are considered

algebraic formulas for the coefficients of half integral weight harmonic weak maass forms abstract we prove that the coefficients of certain weight 1 2 harmonic maass forms are traces of singular moduli for weak maass forms to prove this theorem we construct a theta lift from spaces of weight 2 harmonic weak maass forms to spaces of weight 1 2 vector valued harmonic weak maass forms on mp 2 z a result which is of independent interest we then prove a general theorem which guarantees with bounded denominator when such maass singular moduli are algebraic as an example of these results we derive a formula for the partition function p n as a finite sum of algebraic numbers which lie in the usual discriminant 24 n 1 ring class field

the two loop sunrise graph with arbitrary masses we discuss the analytical solution of the two loop sunrise graph with arbitrary non zero masses in two space time dimensions the analytical result is obtained by solving a second order differential equation the solution involves elliptic integrals and in particular the solutions of the corresponding homogeneous differential equation are given by periods of an elliptic curve

parallel integer relation detection techniques and applications let x1 x2 xn be a vector of real numbers an integer relation algorithm is a computational scheme to find the n integers ak if they exist such that a1x1 a2x2 anxn 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

random walk on the speiser graph of a riemann surface on considere la determination du type conforme d une surface de recouvrement de la sphere de riemann a n points on montre comment definir une marche aleatoire sur les sommets du graphe de speiser de sorte que la marche aleatoire soit transitoire si et seulement si la surface est hyperbolique

analysis of pslq an integer relation finding algorithm let k be either the real complex or quaternion number system and let o k be the corresponding integers let x x 1 x n be a vector in k n the vector x has an integer relation if there exists a vector m m 1 m n o k n m 0 such that m 1 x 1 m 2 x 2 m n x n 0 in this paper we define the parameterized integer relation construction algorithm pslq where the parameter can be freely chosen in a certain interval beginning with an arbitrary vector x x 1 x n k n iterations of pslq will produce lower bounds on the norm of any possible relation for x thus pslq can be used to prove that there are no relations for x of norm less than a given size let m x be the smallest norm of any relation for x for the real and complex case and each fixed parameter in a certain interval we prove that pslq constructs a relation in less than o n 3 n 2 logm x iterations

a systematic way of converting infinite series into infinite products it is shown how to display q series infinite products systematic ways of expanding q series and q products into other series and products are also given a method for determining the independence or otherwise of different q series is discussed

the winding angle of planar self avoiding walks the problem of understanding the asymptotic statistical behaviour of the winding angle on of a self avoiding walk of n steps on a planar lattice is posed exact series expansion data for the square lattice up to n 21 are reported these data and monte carlo estimates up to n g 170 steps are fitted well by a logarithmic growth law for 06 the ratio oh appears to approach a limit of 2 9 to 3 2 which is close to the gaussian value 3 heuristic scaling arguments are consistent with simple logarithmic growth and also illuminate the logarithmic behaviour known rigorously for free brownian motion

ising n fold integrals as diagonals of rational functions and integrality of series expansions we show that the n fold integrals n of the magnetic susceptibility of the ising model as well as various other n fold integrals of the ising class or n fold integrals from enumerative combinatorics like lattice green functions correspond to a distinguished class of functions generalizing algebraic functions they are actually diagonals of rational functions as a consequence the power series expansions of the analytic at x 0 solutions of these linear differential equations derived from geometry are globally bounded which means that after just one rescaling of the expansion variable they can be cast into series expansions with integer coefficients we also give several results showing that the unique analytical solution of calabi yau odes and more generally picard fuchs linear odes with solutions of maximal weights are always diagonals of rational functions besides in a more enumerative combinatorics context generating functions whose coefficients are expressed in terms of nested sums of products of binomial terms can also be shown to be diagonals of rational functions we finally address the question of the relations between the notion of integrality series with integer coefficients or more generally globally bounded series and the modularity of odes

computation and theory of extended mordell tornheim witten sums we consider some fundamental generalized mordell tornheim witten mtw zeta function values along with their derivatives and explore connections with multiple zeta values mzvs to achieve this we make use of symbolic integration high precision numerical integration and some interesting combinatorics and special function theory our original motivation was to represent unresolved constructs such as eulerian log gamma integrals we are able to resolve all such integrals in terms of a mtw basis we also present for a substantial subset of mtw values explicit closed form expressions in the process we signicantly extend methods for high precision numerical computation of polylogarithms and their derivatives with respect to order

asymptotic winding angle distributions for planar brownian motion with drift we study the asymptotic winding angle distribution of a two dimensional brownian curve around a given point the influence of boundaries and or drifts on this distribution is addressed in particular we show that for some special drifts the limiting distribution is given in terms of levy laws

finiteness of simple holomorphic eta quotients of a given weight abstract we provide a simplified proof of zagier s conjecture mersmann s theorem which states that of any particular weight there are only finitely many holomorphic eta quotients none of which is an integral rescaling of another eta quotient or a product of two holomorphic eta quotients other than 1 and itself

transformations of some gauss hypergeometric functions this paper presents explicit algebraic transformations of some gauss hypergeometric functions specifically the transformations considered apply to hypergeometric solutions of hypergeometric differential equations with the local exponent differences 1 k 1 l 1 m such that k l m are positive integers and 1 k 1 l 1 m 1 all algebraic transformations of these gauss hypergeometric functions are considered we show that apart from classical transformations of degree 2 3 4 6 there are several other transformations of degree 6 8 9 10 12 18 24 besides we present an algorithm to compute relevant belyi functions explicitly

a cubic counterpart of jacobi s identity and the agm we produce exact cubic analogues of jacobi s celebrated theta function identity and of the arithmetic geometric mean iteration of gauss and legendre the iteration in question is a n 1 a n 2b n 3 and b n 1 formula cannot be replicated the limit of this iteration is identified in terms of the hypergeometric function f 1 3 2 3 1 which supports a particularly simple cubic transformation

asymptotic winding angle distribution for two dimensional levy flights the author studies the winding angle theta of planar levy flights around a given point say the origin o in particular the author calculates the variance theta 2 and show that after a long time t it is proportional to in t for brownian random walks theta 2 varies as in t 2 in the same conditions the author shows that the variable x theta in t is approximately distributed according to a gaussian law finally the authors compare results with computer simulations

iterated binomial sums and their associated iterated integrals we consider finite iterated generalized harmonic sums weighted by the binomial 2kk in numerators and denominators a large class of these functions emerges in the calculation of massive feynman diagrams with local operator insertions starting at 3 loop order in the coupling constant and extends the classes of the nested harmonic generalized harmonic and cyclotomic sums the binomially weighted sums are associated by the mellin transform to iterated integrals over square root valued alphabets the values of the sums for n and the iterated integrals at x 1 lead to new constants extending the set of special numbers given by the multiple zeta values the cyclotomic zeta values and special constants which emerge in the limit n of generalized harmonic sums we develop algorithms to obtain the mellin representations of these sums in a systematic way they are of importance for the derivation of the asymptotic expansion of these sums and their analytic continuation to n c the associated convolution

nonlinear finite element computation of the equilibrium stability and motion of the extensional beam and ring abstract discrete gauss integration of the element total potential energy is applied to the formation of a cubic cubic c1 extensible completely nonlinear curved beam finite element the versatility accuracy effectiveness and robustness of the element and the newton raphson technique used to solve the nonlinear algebraic stiffness equation set up with it is numerically demonstrated by computations of the nonlinear equilibrium stability and motion of beams and rings

brownian motion and harmonic functions on the class surface of the thrice punctured sphere soit m la sphere perforee a 0 1 et equipee de la metrique de courbure constante 1 on definit un recouvrement intermediaire la surface de classe m 1 on montre que m 1 ne porte pas de fonctions harmoniques positives non constantes ceci signifie que le mouvement brownien sur m 1 a un seul mode d ecoulement vers l infini

a consistent equilibrium in a cross section of an elastic plastic beam abstract a phenomenon of inequality of equilibrium and constitutive internal forces in a cross section of elastic plastic beams is common to many finite element formulations it is here discussed in a rate independent elastic plastic beam context and a possible treatment is presented the starting point of our discussion is reissners finite strain beam theory and its finite element implementation the questions of the consistency of interpolations for displacements and rotations and the related locking phenomena are fully avoided by considering the rotation function of the centroid axis of a beam as the only unknown function of the problem approximate equilibrium equations are derived by the use of the distribution theory in conjunction with the collocation method the novelty of our formulation is an inclusion of a balance function that measures the error between the equilibrium and constitutive bending moments in a cross section an advantage of the present approach is that the locations where the balance of equilibrium and constitutive moments should be satisfied can be prescribed in advance in order to minimize the error explicit analytical expressions are used for the constitutive forces for a rectangular cross section and bilinear constitutive law they are given in appendix a the comparison between the results of the two finite element formulations the one using consistent and the other inconsistent equilibrium in a cross section is shown for a cantilever beam subjected to a point load the problem of high curvature gradients in a plastified region is also discussed and solved by using an adapted collocation method in which the coordinate system is transformed such to follow high gradients of curvature

picard fuchs equations for feynman integrals we present a systematic method to derive an ordinary differential equation for any feynman integral where the differentiation is with respect to an external variable the resulting differential equation is of fuchsian type the method can be used within fixed integer space time dimensions as well as within dimensional regularisation we show that finding the differential equation is equivalent to solving a linear system of equations we observe interesting factorisation properties of the d dimensional picard fuchs operator when d is specialised to integer dimensions

expansion around half integer values binomial sums and inverse binomial sums i consider the expansion of transcendental functions in a small parameter around rational numbers this includes in particular the expansion around half integer values i present algorithms which are suitable for an implementation within a symbolic computer algebra system the method is an extension of the technique of nested sums the algorithms allow in addition the evaluation of binomial sums inverse binomial sums and generalizations thereof

harmonic sums and polylogarithms generated by cyclotomic polynomials the computation of feynman integrals in massive higher order perturbative calculations in renormalizable quantum field theories requires extensions of multiply nested harmonic sums which can be generated as real representations by mellin transforms of poincare iterated integrals including denominators of higher cyclotomic polynomials we derive the cyclotomic harmonic polylogarithms and harmonic sums and study their algebraic and structural relations the analytic continuation of cyclotomic harmonic sums to complex values of n is performed using analytic representations we also consider special values of the cyclotomic harmonic polylogarithms at argument x 1 resp for the cyclotomic harmonic sums at n which are related to colored multiple zeta values deriving various of their relations based on the stuffle and shuffle alge bras and three multiple ar gument relations we also consider infinite generalized nested harmonic sums at roots of unity which are related to the infinite cyclotomic harmonic sums basis representations are derived for weight w 1 2 sums up to cyclotomy l 20

further relations amongst infinite series and products ii the evaluation of three dimensional lattice sums for pt i see ibid vol 20 p l13 7 1987 new relations amongst infinite series and infinite products are given a survey of these and previous results is presented these relations provide a way of evaluating certain three dimensional lattice sums in closed form over 50 such evaluations are displayed

holomorphic eta quotients of weight 1 2 abstract we give a short proof of zagier s conjecture mersmann s theorem which states that each holomorphic eta quotient of weight 1 2 is an integral rescaling of some eta quotient from zagier s list of fourteen primitive holomorphic eta quotients in particular given any holomorphic eta quotient f of weight 1 2 this result enables us to provide a closed form expression for the coefficient of q n in the q series expansion of f for all n we also demonstrate another application of the above theorem in extending the levels of the simple resp irreducible holomorphic eta quotients