Moments of Averages of Ramanujan Sums over Number Fields

TL;DR

This paper studies moments of Ramanujan sums over number fields by defining $C_{\mathcal{J}}({\mathcal{I}})$ on the ideal lattice and analyzing associated Dirichlet series. Assuming the Generalized Lindelöf Hypothesis, it proves a sharp first-moment formula $\sum_{0<\mathcal{N}(\mathcal{I})\le y}\sum_{0<\mathcal{N}(\mathcal{J})\le x} C_{\mathcal{J}}({\mathcal{I}})=\rho_{\mathbb{K}} y + O\left(x y^{1/2+\epsilon}\log x\right)$ for $y>x^2$, and a detailed second-moment theory showing main terms of order $yx^2$ (with extra $x^4$-type contributions in some regimes) and explicit constants involving $\zeta_{\mathbb{K}}(2)$ and $\zeta_{\mathbb{K}}(0)$. The work also proves unconditional second-moment bounds for cyclotomic fields and develops a general framework for arbitrary number fields, combining generalized divisor-function averages, contour integration, and zeta-function bounds. Together, these results illuminate how Ramanujan sums distribute over the arithmetic of number fields and extend known results from $\mathbb{Q}$ to broader algebraic settings, with potential applications to Ramanujan expansions and related mean-value problems in analytic number theory.

Abstract

Assuming the generalized Lindelöf hypothesis, we provide asymptotic formulas for the mean values of the first and second moments of Ramanujan sums over any number field. Additionally, unconditionally, we estimate the second moment of Ramanujan sums over cyclotomic number fields.

Theorems & Definitions (18)

• Conjecture 1.1: Generalized Lindelöf hypothesis
• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5