The order of the Epstein zeta-function in the critical strip

O. M. Fomenko

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 16, Tome 263 (2000), pp. 205-225

Résumé

Let $Q(x_1,\dots,x_k)$ be a positive quadratic form of $k\ge2$ variables and let $\zeta(s;Q)$ be the Epstein zeta-function of the form $Q$. The growth rate of $\zeta(s;Q)$ on the line $\operatorname{Re}s=(k-1)/2$ is investigated. For $k\ge4$ 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 $\operatorname{Re}s=1/2$. In the case $k=3$, the diagonal form over $\mathbb R$ 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.