×

The growth rate of the Dedekind zeta-function on the critical line. (English) Zbl 0583.12011

Acta Arith. 49, No. 4, 323-339 (1988); acknowledgment of priority ibid. 77, No. 4, 405 (1996).
Let \(K\) be an algebraic number field of degree n and let \(\zeta_ K(s)\) be its Dedekind zeta-function. This paper concerns bounds for \(\zeta_ K(+it)\) with respect to t. (The bounds are not uniform in K.) The trivial estimate is \(O(t^{n/4})\). For \(K={\mathbb{Q}}\) one has the classical estimate \(O(t^{1/6+\epsilon})\) for any \(\epsilon >0\), due to Hardy and Littlewood. When \(K\) is abelian one may obtain an analogous result \[ (*)\quad \zeta_ K(+it)=O(t^{n/6+\epsilon}) \] merely by factoring \(\zeta_ K(s)\) into Dirichlet L-functions. The object of the present paper is to prove (*) in the nonabelian case. This is done by the n- dimensional version of van der Corput’s method. Using existing techniques this would have been cumbersome, if not impossible. However a considerable simplification is introduced by using weighted sums and integrals. This avoids in particular the complicated conditions previously imposed on the regions of summations and integrations.

MSC:

11R42 Zeta functions and \(L\)-functions of number fields
11L40 Estimates on character sums
11M06 \(\zeta (s)\) and \(L(s, \chi)\)

Citations:

Zbl 0451.12007
PDFBibTeX XMLCite
Full Text: DOI EuDML