The Hexagonal Lattice and the Epstein Zeta Function

2016 ◽  
pp. 127-140 ◽  
Author(s):  
Andreas Henn
Keyword(s):  
Zeta Function ◽  
Hexagonal Lattice ◽  
Epstein Zeta Function

scholarly journals A Minimum Problem for the Epstein Zeta-Function

1953 ◽  
Vol 1 (4) ◽  
pp. 149-158 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Zeta Function ◽  
Recent Work ◽  
Hexagonal Lattice ◽  
Mean Value ◽  
Minimum Problem ◽  
Constant Multiple ◽  
Packing And Covering ◽  
Closest Packing ◽  
Epstein Zeta Function ◽  
The Mean

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.

scholarly journals Minimal theta functions

1988 ◽  
Vol 30 (1) ◽  
pp. 75-85 ◽  
Author(s):  
Hugh L. Montgomery
Keyword(s):  
Functional Equation ◽  
Quadratic Form ◽  
Zeta Function ◽  
Theta Functions ◽  
Binary Quadratic Form ◽  
Positive Definite ◽  
Epstein Zeta Function ◽  
Image Position

Let be a positive definite binary quadratic form with real coefficients and discriminant b2 − 4ac = −1.Among such forms, let . The Epstein zeta function of f is denned to beRankin [7], Cassels [1], Ennola [5], and Diananda [4] between them proved that for every real s > 0,We prove a corresponding result for theta functions. For real α > 0, letThis function satisfies the functional equation(This may be proved by using the formula (4) below, and then twice applying the identity (8).)

scholarly journals A lemma about the Epstein zeta-function

1964 ◽  
Vol 6 (4) ◽  
pp. 198-201 ◽  
Author(s):  
Veikko Ennola
Keyword(s):  
Quadratic Form ◽  
Zeta Function ◽  
Positive Definite ◽  
Simple Pole ◽  
Positive Definite Quadratic Form ◽  
Epstein Zeta Function ◽  
Image Position

Let h (m, n) = αm2 + 2δmn + βn2 be a positive definite quadratic form with determinant αβ–δ2 = 1. It may be put in the shapewith y > 0. We write (for s > 1)The function Zn(s) may be analytically continued over the whole s-plane. Its only singularity is a simple pole with residue π at s = 1.

scholarly journals On a divisor problem related to the Epstein zeta-function, II

2011 ◽  
Vol 131 (9) ◽  
pp. 1734-1742
Author(s):  
Guangshi Lü ◽  
Jie Wu ◽  
Wenguang Zhai
Keyword(s):  
Zeta Function ◽  
Divisor Problem ◽  
Epstein Zeta Function

Local minima of the Epstein zeta-function

1989 ◽  
Vol 45 (1) ◽  
pp. 83-88
Author(s):  
S. Sh. Shushbaev
Keyword(s):  
Zeta Function ◽  
Local Minima ◽  
Epstein Zeta Function
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