Joint distribution of completely \(q\)-additive functions in residue classes
An arithmetic function \(f\) is said to be completely \(q\)-additive, if \(f(0)=0\), and \(f(n)=\sum_{i\geq 0}f(a_i)\) for \(n=\sum_{i\geq0}a_iq^i\). The authors study the joint distribution of completely \(q_j\)-additive functions in residue classes. Namely, if \(f_i\) is \(q_i\)-additive (\(i=1,\dots,\ell\)) with pairwise primes \(q_i\), then the cardinality of the set \[ \{1\leq n\leq N\mid f_j(a_jn+b_j)\equiv c_j\pmod{m_j},\,1\leq j\leq\ell\} \] is asymptotically equal to \(cN+O(N^{1-\delta})\) (if it is not zero). Here, \(c\) and \(\delta\) are explicitly given. Similar result holds in short intervals, i.e., when \(M\leq n\leq N+M\). The above theorems are significant generalizations of the results of \textit{D.-H. Kim} [J. Number Theory 74, No. 2, 307--336 (1999; Zbl 0920.11067)].
scientific article; zbMATH DE number 6516893
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