A characterization of some unimodular multiplicative functions (Q2714299)

An arithmetic function \(g(n)\neq 0\) is said to be a unimodular multiplicative function if \(g(n)\) is multiplicative and \(g(n)\) satisfies the condition \(|g(n)|=1\) for all positive integers \(n\). The author studies unimodular multiplicative functions \(g_1\), \(g_2\) for which \(g_1(an+b)-dg_2(cn)\) tends to zero in some sense. A typical result of this article is the following. NEWLINENEWLINENEWLINETheorem 3: Let \(a\), \(b\), \(c\) be positive integers and let \(d\neq 0\) be a complex number. Then \(g_1\), \(g_2\) unimodular multiplicative functions satisfy the conditions \(g_1(an+b)-dg_2(cn)=o(1)\) where \(n\to \infty\) if and only if there are a real number \(\tau\) and \(G_1\), \(G_2\) unimodular multiplicative functions such that \(g_1(n)=n^{i\tau}G_1(n)\), \(g_2(n)=n^{i\tau}G_2(n)\) and \(G_1(an+b)-d\frac{c^{i\tau}}{a^{i\tau}}G_2(cn)=0\) hold for all positive integers. NEWLINENEWLINENEWLINEFor more information on these latter results, see the paper of \textit{I. Kátai} [Acta. Math. Hung. 58, No. 3/4, 343-350 (1991; Zbl 0746.11036)] and \textit{E. Wirsing, Y. Tang} and \textit{P. Shao} [J. Number Theory 56, 391-395 (1996; Zbl 0845.11030)].