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.

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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.

Euler considered sums of the form ∞ m=1 1 m s m−1 n=1 1 n t .Here natural generalizations of these sums namely [p, q] := [p, q](s, t) = ∞ m=1 χ p (m) m s m−1 n=1 χ q (n) n t ,are investigated, where χ p and χ q are characters, and s and t are positive integers. The cases when p and q are either 1, 2a, 2b or −4 are examined in detail, and closedform expressions are found for t = 1 and general s in terms of the Riemann zeta function and the Catalan zeta function-the Dirichlet series L −4 (s) = 1 −s − 3 −s + 5 −s − 7 −s + · · · . Some results for arbitrary p and q are obtained as well.In Memoriam: Between the submission and acceptance of this report we greatly regret that our esteemed colleague John Boersma passed away. This paper is dedicated to his memory.

The = (4/7r2) (2 + a)2 )K(k+)K(k-) [7] (2 + a2 + a2Z)Kk)Kk) [7 [3] [3a]in which k.2 =1 ±2za(2 + a2)1/2(a4 + 2a2 + z)'/2

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