Multiplicative functions dictated by Artin symbols (Q2855863)

The author considers completely multiplicative complex functions which come from a fixed finite Galois (not necessarily abelian) extension \(K/\mathbb{Q}\) via Artin symbols \(\bigl(\frac{K/\mathbb{Q}}{p}\bigr)\) of (in \(K/\mathbb{Q}\) unramified) rational primes \(p\), where \(\bigl(\frac{K/\mathbb{Q}}{p}\bigr)\) means the conjugacy class in \(\text{Gal}(K/\mathbb{Q})\) of Frobenius automorphisms \(\bigl(\frac{K/\mathbb{Q}}{\mathfrak{p}}\bigr)\) of prime ideals \(\mathfrak{p}\) of \(K\) above \(p\). More precisely, let \(\mathcal{S}_K\) be the set of functions \(f:\mathbb{Z}^+\to\mathbb{C}\) such thatNEWLINENEWLINE{\parindent=0.5cm \begin{itemize}\item[(i)] \(f\) is completely multiplicative; \item[(ii)] \(|f(p)|\leq1\) for each prime \(p\), with equality holding if \(p\) splits completely in \(K/\mathbb{Q}\); \item[(iii)] \(f(p_1)=f(p_2)\) if \(p_1\) and \(p_2\) are primes unramified in \(K/\mathbb{Q}\) having the same Artin symbol \(\bigl(\frac{K/\mathbb{Q}}{p_1}\bigr)=\bigl(\frac{K/\mathbb{Q}}{p_2}\bigr)\). NEWLINENEWLINE\end{itemize}} The main result of the paper under review consists in the following theorem: if \(f\in\mathcal{S}_K\) satisfies \(\sum_{n\leq X}f(n)=O_f(X^{1/2-\delta})\) as \(X\to\infty\) for some fixed \(\delta>0\), then there is a Dirichlet character \(\chi\) of conductor dividing the discriminant of \(K\) such that \(f(p)=\chi(p)\) for all primes \(p\) unramified in \(K/\mathbb{Q}\).NEWLINENEWLINEThe proof of the theorem is based on a modification of \textit{K. Soundararajan}'s approach given in [Expo. Math. 23, No. 1, 65--70 (2005; Zbl 1167.11034)].