# Smooth $$L^2$$ L 2 distances and zeros of approximations of Dedekind zeta functions

Smooth $$L^2$$ L 2 distances and zeros of approximations of Dedekind zeta functions We consider a family of approximations of the Dedekind zeta function $$\zeta _K(s)$$ ζ K ( s ) of a number field $$K/\mathbb {Q}$$ K / Q . Weighted $$L^2$$ L 2 -norms of the difference of two such approximations of $$\zeta _K(s)$$ ζ K ( s ) are computed. We work with a weight which is a compactly supported smooth function. Mean square estimates for the difference of approximations of $$\zeta _K(s)$$ ζ K ( s ) can be obtained from such weighted $$L^2$$ L 2 -norms. Some results on the location of zeros of a family of approximations of Dedekind zeta functions are also derived. These results extend results of Gonek and Montgomery on families of approximations of the Riemann zeta-function. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Manuscripta Mathematica Springer Journals

# Smooth $$L^2$$ L 2 distances and zeros of approximations of Dedekind zeta functions

, Volume 154 (2) – Jan 17, 2017
29 pages

/lp/springer_journal/smooth-l-2-l-2-distances-and-zeros-of-approximations-of-dedekind-zeta-wfAbg6Mxyd
Publisher
Springer Berlin Heidelberg
Subject
Mathematics; Mathematics, general; Algebraic Geometry; Topological Groups, Lie Groups; Geometry; Number Theory; Calculus of Variations and Optimal Control; Optimization
ISSN
0025-2611
eISSN
1432-1785
D.O.I.
10.1007/s00229-016-0911-6
Publisher site
See Article on Publisher Site

### Abstract

We consider a family of approximations of the Dedekind zeta function $$\zeta _K(s)$$ ζ K ( s ) of a number field $$K/\mathbb {Q}$$ K / Q . Weighted $$L^2$$ L 2 -norms of the difference of two such approximations of $$\zeta _K(s)$$ ζ K ( s ) are computed. We work with a weight which is a compactly supported smooth function. Mean square estimates for the difference of approximations of $$\zeta _K(s)$$ ζ K ( s ) can be obtained from such weighted $$L^2$$ L 2 -norms. Some results on the location of zeros of a family of approximations of Dedekind zeta functions are also derived. These results extend results of Gonek and Montgomery on families of approximations of the Riemann zeta-function.

### Journal

Manuscripta MathematicaSpringer Journals

Published: Jan 17, 2017

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