, Volume 99, Issue 5, pp 433-442
Date: 16 Nov 2012

The average size of Ramanujan sums over quadratic number fields

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Abstract

This article is concerned with Ramanujan sums \({c_{\mathcal{I}_1}(\mathcal{I}),}\) where \({\mathcal{I},\mathcal{I}_1}\) are integral ideals in an arbitrary quadratic number field \({\mathbb{Q}(\sqrt{d}).}\) In particular, the asymptotic behavior of sums of \({c_{\mathcal{I}_1}(\mathcal{I}),}\) over both \({\mathcal{I}}\) and \({c_{\mathcal{I}_1}(\mathcal{I}),}\) is investigated.

The author gratefully acknowledges support from the Austrian Science Fund (FWF) under project Nr. P20847-N18.