# Article

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Keywords:
Epstein zeta function; Riemann theta function; variance of volume estimate; Rankin-Sobolev problem
Summary:
Values of the Epstein zeta function of a positive definite matrix and the knowledge of matrices with minimal values of the Epstein zeta function are important in various mathematical disciplines. Analytic expressions for the matrix theta functions of integral matrices can be used for evaluation of the Epstein zeta function of matrices. As an example, principal coefficients in asymptotic expansions of variance of the lattice point count in the random ball are calculated for some lattices.
References:
[1] Conway, H., Sloane, N. J. A.: Sphere Packings, Lattices and Groups. Springer, New York (1998).
[2] Delone, B. N., Ryshkov, S. S.: A contribution to the theory of the extrema of a multi-dimensional $\zeta$-function. Dokl. Akad. Nauk SSSR 173 (1967), 991-994. MR 0220676
[3] Janáček, J.: Variance of periodic measure of bounded set with random position. Comment. Math. Univ. Carolinae 47 (2006), 473-482. MR 2281006 | Zbl 1150.62315
[4] Kendall, D. G., Rankin, R. A.: On the number of points of a given lattice in a random hypersphere. Quarterly J. Math., 2nd Ser. 4 (1953), 178-189. DOI 10.1093/qmath/4.1.178 | MR 0057484 | Zbl 0052.14503
[5] Rankin, R. A.: A minimum problem for the Epstein zeta function. Proc. Glasgow. Math. Assoc. 1 (1953), 149-158. DOI 10.1017/S2040618500035668 | MR 0059300 | Zbl 0052.28005
[6] Sobolev, S. L.: Formulas for mechanical cubatures in $n$-dimensional space. Dokl. Akad. Nauk SSSR 137 (1961), 527-530. MR 0129548 | Zbl 0196.49202

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