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On Epstein’s zeta-function. (English. Russian original) Zbl 1201.11042

J. Math. Sci., New York 122, No. 6, 3679-3684 (2004); translation from Zap. Nauchn. Semin. POMI 286, 169-178 (2002).
Summary: Let \(Q(x_1,x_2,x_3)= x_1^2+ x_2^2+ x_3^2\), and let \(\zeta(s;Q)\) be Epstein’s zeta-function of the form \(Q\). Here the author proves that for \(|t|> C> 0\) one has the estimate \(\zeta(1+it;Q)\ll |t|^{1/4+\varepsilon}\). In the proof estimates of exponential sums are used involving the function \(r_3(n)\), the number of representations of an integer \(n\) as sums of squares of integers.

MSC:

11E45 Analytic theory (Epstein zeta functions; relations with automorphic forms and functions)
11M41 Other Dirichlet series and zeta functions
11L07 Estimates on exponential sums
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