A Minimum Problem for the Epstein Zeta-Function

In some recent work by D. G. Kendall and the author † on the number of points of a lattice which lie in a random circle the mean value of the variance emerged as a constant multiple of the value of the Epstein zeta-function Z(s) associated with the lattice, taken at the point s= . Because of the connexion with the problems of closest packing and covering it seemed likely that the minimum value of Z ( ) would be attained for the hexagonal lattice; it is the purpose of this paper to prove this and to extend the result to other real values of the variable s .