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The spectrum of multiplicative functions. (English) Zbl 1036.11042
Let \(S\) be a subset of the unit disc \(U\) and \({\mathcal F}(S)\) denote the set of completely multiplicative functions \(f\) with \(f(p)\in S\) for all primes \(p\). Define \[ \Gamma_N(S)= \left\{N^{-1} \sum_{n\leq N}f(n): f \in{\mathcal F}(S)\right\},\;\Gamma(S)= \lim_{N\to\infty} \Gamma_N(S), \] where \(z\in\Gamma(S)\) means that \(z_N\to z\) as \(N\to\infty\) with \(z_N\in\Gamma_N(S)\). The set \(\Gamma(S)\) is called the spectrum of \(S\), and the aim of this important, wide ranging, comprehensive and clearly written paper is to study \(\Gamma(S)\). It can be assumed that \(1\in S\) and that \(S\) is closed; for \(\Gamma(S)=\{0\}\) if \(1\notin S\), and \(\Gamma (\overline S)=\Gamma(S)\).
In [Proc. Edinb. Math. Soc., II. Ser. 39, 581–588 (1996; Zbl 0876.11045)], R. R. Hall proved that for real-valued \(f\in{\mathcal F}([-1,1])= {\mathcal F}\), \(c=\inf\{x^{-1}\sum_{n \leq x}f(n):x\geq 1, \;f\in{\mathcal F}\}>-1\). Using the example that for \(q\) prime \(f(q)=\begin{cases} 1 & \text{ for } q\leq x^{1/(1+\sqrt e)}\\ -1& \text{ for }x^{1/(1+\sqrt e)}\leq q\leq x\end{cases}\) he showed that \(c\leq \delta_1\) where \(\delta_1=- 0.656999\dots\), and conjectured that \(c= \delta_1\), as did H. L. Montgomery independently. In the current paper, the authors prove this conjecture by showing that \(\Gamma([-1,1])\) is the interval \([\delta_1,1]\).
For other closed sets \(S\subseteq U\) with \(1\in S\), the structure of the spectrum of \(S\) is more complicated and less well understood. The authors investigate this structure and show that \(\Gamma (S)\) may be described in terms of Euler products and solutions of certain integral equations. For \(f\in{\mathcal F}(S)\), let \[ \Theta(f,x)= \prod_{p \leq x}\left( 1+ \frac {f(p)}{p} +\frac{f(p^2)} {p^2}+\dots \right) \left(1-\frac 1p \right); \] then \(\Theta (f,x) \in\Gamma(S)\) for all \(x\). Let \(\Gamma_\Theta(S)= \lim_{x\to \infty} \{\Theta(f,x): f\in{\mathcal F}(S)\}\), which is a closed subset of \(\Gamma (S)\) called the Euler product of the spectrum of \(S\). If \(f\) has a mean value \(\Theta(f, \infty)\), then \(\Theta(f,\infty) \in\Gamma(S)\). Conditions under which a function \(f\in{\mathcal F}(S)\), \(S\subseteq U\), has a mean value have been established by several authors, including A. Wintner, E. Wirsing, G. Halász, and Corollary 2 of this paper provides another answer to this problem. For \(V \subseteq U\), the angle of \(V\) is defined by \[ \text{Ang} (V)= \sup\{ |\arg(1-v) |:v \in V,\;v\neq 1\}; \] the authors deduce that if \(\text{Ang}(S)<\frac {\pi}{2}\), then every \(f\in{\mathcal F}(S)\) has a mean value and \(\Gamma_\Theta(S)= \{\Theta(f, \infty):f\in{\mathcal F}(S)\}\).
In general, and even in the case \(S=[-1,1]\), \(\Gamma (S)\) contains elements not in \(\Gamma_\Theta(S)\). To each member of a certain class of measurable functions taking values in the convex hull of \(S\) is associated the solution \(\sigma:[0,\infty)\to U\) of a given integral equation; let \(\Lambda(S)\) consist of all elements \(\sigma(u)\). For \(J,K \subseteq U\), let \(J\times K=\{jk:j\in J\), \(k\in K\}\). The authors establish the structure theorem \(\Gamma(S)= \Gamma_\theta(S) \times\Lambda (S)\) for any closed subset \(S\subseteq U\) with \(1\in S\). They also derive several other results that throw further light on the sets \(\Gamma(S)\), \(\Gamma_\Theta(S)\), \(\Lambda(S)\); these involve another set \({\mathcal E}(S)\) consisting of ‘spirals’ connecting 1 to 0. In particular the authors deduce from more general results that \(\Gamma(S)\) is connected and that, if the convex hull of \(S\) contains a real point other than 1, then \(\Gamma_\Theta(S)={\mathcal E}(S)\), \(\Gamma_\Theta(S)\) and \(\Gamma (S)\) are starlike, and \(\Gamma(S)= \Lambda(S)\). They conjecture that the sets \(S\), \(\Gamma (S)\) have the same angle, and Theorem 6 supports this conjecture when the angle of \(S\) is near 0 or \(\frac{\pi} {2}\).
Other sections of the paper concern \(m\)th power residues, in particular quadratic residues, and generalizations of the spectrum, for example to the logarithmic spectrum. The proofs use a variety of techniques, and the pathway to establishing each main result is described at the end of section 1.

MSC:
11N37 Asymptotic results on arithmetic functions
11L40 Estimates on character sums
11A15 Power residues, reciprocity
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