New series representation for the madelung constant

Abstract

A new series for the Madelung constant M is derived on the basis of a representation given by Crandall [Exp. Math. 8 (1999), 367]. We are able to write it in the form M = C+S, where S is a rapidly convergent series, and the constant C is fundamental: C = -1/8 - ln 2/4π - 4π/3 + 1/2√2 + τs(1/8) τ(3/8)/π3/2√2 ≈ -1.747564594 . . . The remarkable result is that even if S is discarded, the constant C alone gives ten significant figures of M. This result advances the state of the art in the discovery of what Crandall has termed "close calls" to an exact Madelung evaluation. We present related identities and discuss how this fundamental ten-digit accuracy might be improved further.