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Connected components of complex divisor functions. (English) Zbl 1420.11127

Summary: For any complex number \(c\), define the divisor function \(\sigma_c\colon \mathbb{N} \to \mathbb{C}\) by \(\sigma_c(n) = \sum_{d\mid n} d^c\). Let \(\overline{\sigma_c(\mathbb{N})}\) denote the topological closure of the range of \(\sigma_c\). Extending previous work of the current author and C. Sanna [Unif. Distrib. Theory 12, No. 2, 77–90 (2017; Zbl 1448.11011)], we prove that \(\overline{\sigma_c(\mathbb{N})}\) has nonempty interior and has finitely many connected components if \(\mathrm{Re}(c) \leq 0\) and \(c \neq 0\). We end with some open problems.

MSC:

11N64 Other results on the distribution of values or the characterization of arithmetic functions
11A25 Arithmetic functions; related numbers; inversion formulas
11B05 Density, gaps, topology

Citations:

Zbl 1448.11011
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References:

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