, Volume 99, Issue 5, pp 433-442

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.