The sum of digits function in number fields: distribution in residue classes
Let \(\mathbf K\) be a number field, \(\mathbf{Z_K}\) its ring of integers and let \(b\in \mathbf{Z_K}\), \(\mathcal N=\{0,1,\dots,|N(b)|-1\}\), where \(N(b)\) is the norm of \(b\) over \(\mathbb Q\). The pair \((b, \mathcal N)\) is a number system in \(\mathbf{Z_K}\) if every \(z\in \mathbf{Z_K}\) admits a unique representation \[ z=c_0+c_1b+\dots+c_hb^h, \] where \(h\geq 0, c_0,c_1,\dots,c_h\in \mathcal N\) and \(c_h\neq 0\) if \(h\neq 0\). Furthermore, let \(\nu_b(z)=c_0+c_1+\dots+c_h\) denote the \(b\)-adic sum of digits function. Continuing the investigation of \(\nu_b(z)\) [see same author, Bull. Lond. Math. Soc. 30, 37--45 (1998; Zbl 0921.11051)], the author gives in the paper under review certain results on the distribution of this function in residue classes. For example, he proves that \(\nu_b(z)\) is equidistributed in residue classes modulo \(m\) in any ideal \(s\). This generalizes a result of \textit{A. O. Gelfond} [Acta Arith. 13, 259--265 (1968; Zbl 0155.09003)].
scientific article; zbMATH DE number 1271768