The study of lattice sums began when early investigators wanted to go from mechanical properties of crystals to the properties of the atoms and ions from which they were built (the literature of Madelung's constant). A parallel literature was built around the optical properties of regular lattices of atoms (initiated by Lord Rayleigh, Lorentz and Lorenz). For over a century many famous scientists and mathematicians have delved into the properties of lattices, sometimes unwittingly duplicating the work of their predecessors. Here, at last, is a comprehensive overview of the substantial body of knowledge that exists on lattice sums and their applications. The authors also provide commentaries on open questions, and explain modern techniques which simplify the task of finding new results in this fascinating and ongoing field. Lattice sums in one, two, three, four and higher dimensions are covered.

Abstract. We study the densities of uniform random walks in the plane. A special focus is on the case of short walks with three or four steps and less completely those with five steps. As one of the main results, we obtain a hypergeometric representation of the density for four steps, which complements the classical elliptic representation in the case of three steps. It appears unrealistic to expect similar results for more than five steps. New results are also presented concerning the moments of uniform random walks and, in particular, their derivatives. Relations with Mahler measures are discussed.

We study the moments of the distance traveled by a walk in the plane with unit steps in random directions. While this historically interesting random walk is well understood from a modern probabilistic point of view, our own interest is in determining explicit closed forms for the moment functions and their arithmetic values at integers when only a small number of steps is taken. As a consequence of a more general evaluation, a closed form is obtained for the average distance traveled in three steps. This evaluation, as well as its proof, rely on explicit combinatorial properties, such as recurrence equations of the even moments (which are lifted to functional equations). The corresponding general combinatorial and analytic features are collected and made explicit in the case of 3 and 4 steps. Explicit hypergeometric expressions are given for the moments of a 3-step and 4-step walk and a general conjecture for even length walks is made.

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We investigate the moments of 3-step and 4-step uniform random walk in the plane. In particular, we further analyse a formula conjectured in [BNSW09] expressing 4-step moments in terms of 3-step moments. Diverse related results including hypergeometric and elliptic closed forms for W 4 (±1) are given and two new conjectures are recorded.

We consider the moments of products of complete elliptic integrals of the first and second kinds. In particular, we derive new results using elementary means, aided by computer experimentation and a theorem of W. Zudilin. Diverse related evaluations, and two conjectures, are also given.

We compute the critical L-values of some weight 3, 4, or 5 modular forms, by transforming them into integrals of the complete elliptic integral K . In doing so, we prove closed-form formulas for some moments of K 3 . Many of our L-values can be expressed in terms of Gamma functions, and we also obtain new lattice sum evaluations.

We resolve a family of recently observed identities involving 1/π using the theory of modular forms and hypergeometric series. In particular, we resort to a formula of Brafman which relates a generating function of the Legendre polynomials to a product of two Gaussian hypergeometric functions. Using our methods, we also derive some new Ramanujan-type series.