Order of the Epstein zeta-function in the critical strip. (English. Russian original) Zbl 1007.11019

Fomenko, O. M.

J. Math. Sci., New York 110, No. 6, 3150-3163 (2002); translation from Zap. Nauchn. Semin. POMI 263, 205-225 (2000).

Let \(Q(x_1,\dots,x_k)\) be a positive definite quadratic form of \(k\geq 2\) variables, and let \(\zeta(s,Q)\) be its Epstein zeta-function. The growth rate of \(\zeta(s,Q)\) on the line \(\operatorname{Re} s=(k-1)/2\) is investigated. For \(k\geq 4\) and for an integral form \(Q\), the problem is reduced to a similar problem on the behavior of the Dirichlet \(L\)-series on the line \(\operatorname{Re} s=\frac 12\). For \(k=3\), the diagonal form over \(\mathbb R\) is investigated by the van der Corput method. For \(k=2\), the known result of Titchmarsh is re-proved by using a variant of the van der Corput method given by D. R. Heath-Brown [Acta Arith. 49, No. 4, 323-339 (1988; Zbl 0583.12011), ibid. 77, No. 4, 405 (1996)].

Reviewer: Veikko Ennola (Turku)

MSC:

11E45	Analytic theory (Epstein zeta functions; relations with automorphic forms and functions)
11M41	Other Dirichlet series and zeta functions

Keywords:

quadratic form; Epstein zeta-function; critical line; growth rate

Citations:

Zbl 0583.12011