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Epstein Zeta Function

The Epstein zeta function for an n×n matrix S of a positive definite real quadratic form and rho a complex variable with Re[rho]>n/2 (where Re[z] denotes the real part) is defined by

Z_n(S,rho)=1/2sum^'_(a in Z^n)(a^TSa)^(-rho),
(1)

where the sum is over all column vectors with integer coordinates and the prime means the summation excludes the origin (Terras 1973). Epstein (1903) derived the analytic continuation, functional equation, and so-called Kronecker limit formula for this function.

Epstein (1903) defined this function in the course of an effort to find the most general possible function satisfying a functional equation similar to that satisfied by the Riemann zeta function zeta(z) (Glasser and Zucker 1980, p. 68).

A slightly different notation is used in theoretical chemistry, where the Epstein zeta function arises in connection with lattice sums. Let q(l) be a positive definite quadratic form

q(l)=a_(11)l_1^2+...+a_(dd)l_d^2+2(a_(12)l_1l_2+...),
(2)

where A=(a_(ij)) with i,j=1, ...d is a symmetric matrix. Then the Epstein zeta function can be defined as

Z|g; h|(q;s)=sum_(l)(e^(-2piih·l))/(||q(l+g)||^(s/2)),
(3)

where g and h are arbitrary vectors, the sum runs over a d-dimensional lattice, and l=-g is omitted if g is a lattice vector (Glasser and Zucker 1980, p. 69).

For a point lattice L subset R^4, write E(L,s)=sum_(v in L\{0})||v||^(-2s) for s>2. Thus E(L,s)=2Z_4(S,s) when S is the Gram matrix of a basis of L. For L with covolume 1, Luo and Wei (2026) proved that this Epstein zeta function is minimized for every s>2 uniquely, up to an orthogonal transformation, by the rescaled D4 lattice D_4=2^(-1/4)D_4.

See also

D4 Lattice, Grenz-Formel, Lattice Sum, Riemann Zeta Function, Zeta Function

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References

Bateman, P. T. and Grosswald, E. "On Epstein's Zeta Function." Acta Arith. 9, 365-373, 1964.Chowla, S. and Selberg, A. "On Epstein's Zeta Function (I)." Proc. Nat. Acad. Sci. USA 35, 371-374, 1949.Deuring, M. F. "On Epstein's Zeta Function." Ann. Math. 38, 585-593, 1937.Epstein, P. "Zur Theorie allgemeiner Zetafunktionen. I." Math. Ann. 56, 614-644, 1903.Glasser, M. L. and Zucker, I. J. "Lattice Sums in Theoretical Chemistry." In Theoretical Chemistry: Advances and Perspectives, Vol. 5 (Ed. H. Eyring). New York: Academic Press, pp. 67-139, 1980.Hecke, E. Mathematische Werke. Göttingen, Germany: Vandenhoeck & Ruprecht, 1959.Luo, S. and Wei, J. "On Minima of Theta and Epstein Zeta Functions in Dimension Four." 29 Sep 2026. https://arxiv.org/abs/2609.37615.Selberg, A. and Chowla, S. "On Epstein's Zeta-Function." J. reine angew. Math. 227, 86-110, 1967.Shanks, D. "Calculation and Applications of Epstein Zeta Functions." Math. Comput. 29, 271-287, 1975.Siegel, C. L. Lectures on Advanced Analytic Number Theory. Bombay, India: Tata Institute of Fundamental Research, 1961.Taylor, P. R. "The Functional Equation for Epstein's Zeta-Function." Quart. J. Math. 11, 177-182, 1940.Terras, A. A. "Bessel Series Expansions of the Epstein Zeta Function and the Functional Equation." Trans. Amer. Math. Soc. 183, 477-486, 1973.

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Epstein Zeta Function

Cite this as:

Weisstein, Eric W. "Epstein Zeta Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EpsteinZetaFunction.html

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