Order of the Epstein zeta-function in the critical strip
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 \(\Re s=(k-1)/2\) 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 \(\Re s=\frac 12\). 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 \textit{D. R. Heath-Brown} [Acta Arith. 49, No. 4, 323-339 (1988; Zbl 0583.12011), ibid. 77, No. 4, 405 (1996)].
scientific article; zbMATH DE number 1777284