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
(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
be a positive definite quadratic form
(2)
where
with ,
...
is a symmetric matrix. Then the Epstein zeta
function can be defined as
(3)
where
and
are arbitrary vectors, the sum
runs over a -dimensional
lattice, and is omitted if is a lattice vector (Glasser and
Zucker 1980, p. 69).
For a point lattice with covolume 1, Luo
and Wei (2026) proved that the Epstein zeta function is minimized for every uniquely, up to an orthogonal
transformation, by the rescaled D4
lattice.
Their computer-assisted proof uses certified intervalbounds. Luo and Wei (2026) state that ChatGPT assisted with
review of proof arguments and citations and with preparation
of the computational verification materials.
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. USA35, 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.
matrix 
of a positive definite real quadratic
form and 
a complex variable with ![Re[rho]>n/2](/images/equations/EpsteinZetaFunction/Inline4.svg)
(where ![Re[z]](/images/equations/EpsteinZetaFunction/Inline5.svg)
denotes the real part) is
defined by
(Glasser and Zucker 1980, p. 68).
be a positive definite quadratic form
with 
,
...
is a symmetric matrix. Then the Epstein zeta
function can be defined as
and 
are arbitrary vectors, the sum
runs over a 
-dimensional
lattice, and 
is omitted if 
is a lattice vector (Glasser and
Zucker 1980, p. 69).
with covolume 1, Luo
and Wei (2026) proved that the Epstein zeta function 
is minimized for every 
uniquely, up to an orthogonal
transformation, by the rescaled D4
lattice 
.
Their computer-assisted proof uses certified interval bounds. Luo and Wei (2026) state that ChatGPT assisted with
review of proof arguments and citations and with preparation
of the computational verification materials.
