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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
##### Keywords:
integral equations; Euler product; $$m$$th power residues
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