Order of the Epstein Zeta-Function in the Critical Strip

Let Q(x₁,...,x_k) be a positive quadratic form of k2 variables and let (s,Q) be the Epstein zeta-function of the form Q. The growth rate of (s,Q) on the line Re s = (k–1)/2 is investigated. For k4 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 = 1/2. In the case k=3, the diagonal form over is investigated by the van der Corput method. For k=2, the known result due to Titchmarsh is re-proved by using a variant of the van der Corput method given by Heath-Brown. Bibliography: 9 titles.