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A Minimum Problem for the Epstein Zeta-Function

Published online by Cambridge University Press:  18 May 2009

R. A. Rankin
Affiliation:
The University, Birmingham, 15
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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.

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 1953

References

“On the number of points of a given lattice in a random hypersphere.” (To appear in the Quarterly Journal.)

For the general theory of Zh(s) see Deuring, Max, “Zetafunktionen quadratischer FormenJ. reine angew. Math. 172 (1935), 226252.Google Scholar

Deuring (loc. cit.) gives a somewhat similar formula for the function G (x, y), but with a different form of remainder and without the explicit numerical constants which are essential for our purpose.

Cambridge, 1922.

“On the number of lattice points inside a random oval,” Quart. J. Math., Oxford Ser. 19 (1948), 126.Google Scholar