ON THE EPSTEIN ZETA FUNCTION

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NAN-YUE ZHANG
KENNETH S. WILLIAMS

Abstract




The Epstein zeta function $Z(s)$ is defined for Re$s > 1$ by


\[Z(s)=\sum_{m,n=-\infty, (m, n)\neq (0, 0)}^\infty \frac{1}{(am^2−bnm+cn^2)^s}\]





where $a$, $b$, $c$ are real numbers with $a>0$ and $b^2 - 4ac<0$. $Z(s)$ can be continued analytically to the whole complex plane except for a simple pole at $s =1$. Simple proofs of the functional equation and of the Kronecker "Grenz-formel" for $Z (s)$ are given. The value of $Z(k)$($k =2, 3, \cdots$) is determined in terms of infinite series of the form


\[\sum_{n=1}^\infty\frac{\cot^r n\pi\tau}{n^{2k-1}} (r=1, 2, \cdots, k)\]


where $\tau=(b+\sqrt{b^2-4ac})/2a$.







Article Details

How to Cite
ZHANG, N.-Y., & WILLIAMS, K. S. (1996). ON THE EPSTEIN ZETA FUNCTION. Tamkang Journal of Mathematics, 26(2), 165–176. https://doi.org/10.5556/j.tkjm.26.1995.4394
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Papers

References

P. T. Bateman and E. Grosswald, "On Epstein's zeta function," Acta Arith., 9 (1964), 365-373.

N. N. Lebedev, Special Functions and their Applications, Prentice-Hall Inc., Englewood Cliffs, N. J. (1965).

Y. Motohashi, "A new proof of the limit formula of Kronecker," Proc. Japan. Acad., 44 (1968),614-616.

A. Selberg and S. Chowla, "On Epstein's zeta function," J. Reine Angew. Math., 227 (1967),86-110.

C. L. Siegel, Advanced Analytic Number Theory, Tata Institute of Fundamental Research, Bombay (1980).

J. R. Smart, "On the values of the Epstein zeta function," Glasgow Math. J., 14 (1973), 1-12.

P. R. Taylor, "The functional equation for Epstein's zeta function," Quart. J. Math. Oxford Ser., 11 (1940), 177-182.

E. C. Titchmarsh, The Theory of the Zeta-function, Clarendon Press, Oxford (1986).

I. J. Schwatt, An Introduction to the Operations with Series, Cheesea Publishing Company, New York (1961).