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On the values of the Epstein zeta function

Published online by Cambridge University Press:  18 May 2009

John Roderick Smart
John Roderick Smart
Affiliation:
University of Wisconsin, Madison, Wisconsin
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Let ζ(s) = σn-s (Res >1) denote the Riemann zeta function; then, as is well known, , where Bm denotes the mth Bernoulli number, In this paper we investigate the possibility of similar evaluations of the Epstein zeta function ζq(s) at the rational integers s = k> 2. Let

be a positive definite quadratic form and

where the summation is over all pairs of integers except (0, 0). In attempting to evaluate ζq(k) we are guided by Kronecker's first limit formula [11]

where γ is Euler's constant,

is the Dedekind eta-function, and τ is the complex number in the upper half plane, ℋ, associated with Q by the formula

On the basis of (1.3) we would expect a formula involving functions of τ. This formula is stated in Theorem 1, (2.13).

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 14 , Issue 1 , February 1973 , pp. 1 - 12
Copyright
Copyright © Glasgow Mathematical Journal Trust 1973

References

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