---

Statement

Setting (p. 2). For a subset $S$ of the closed unit disc $\mathbb U$, $\mathcal F(S)$ is the class of completely multiplicative functions $f$ (footnote 1: $f(mn)=f(m)f(n)$ for all positive integers $m,n$) with $f(p)\in S$ for every prime $p$. The paper sets

$$\Gamma_N(S)=\Big\{\frac1N\sum_{n\le N}f(n):f\in\mathcal F(S)\Big\}, \qquad \Gamma(S)=\lim_{N\to\infty}\Gamma_N(S),$$

where the limit of a sequence of sets $J_N$ is the set of $z$ for which some $z_N\in J_N$ satisfy $z_N\to z$. So a point of the spectrum $\Gamma(S)$ is a limit of averages $\frac1N\sum_{n\le N}f_N(n)$ in which the function $f_N$ may change with $N$.

Theorem 1 (p. 3, quoted). "The spectrum of the interval $[-1,1]$ is the interval $\Gamma([-1,1])=[\delta_1,1]$ where

$$\delta_1=1-2\log(1+\sqrt e)+4\int_1^{\sqrt e}\frac{\log t}{t+1}\,dt =-0.656999\ldots."$$

Consequence stated after the theorem (p. 3, display (1.1)). For every real-valued completely multiplicative $f$ with $\lvert f(n)\rvert\le1$, $\sum_{n\le x}f(n)\ge(\delta_1+o(1))x$. The paper records that Hall proved Heath-Brown's 1994 conjecture that some constant $c>-1$ works in place of $\delta_1$, and that Hall and, independently, Montgomery conjectured (1.1). The choice (1.2), $f(q)=1$ for primes $q\le x^{1/(1+\sqrt e)}$ and $f(q)=-1$ for primes $x^{1/(1+\sqrt e)}\le q\le x$, gives equality in (1.1), so $\delta_1$ cannot be raised. The sharp form with its equality condition is Corollary 1.

For contrast (p. 6), the Euler product spectrum of $[-1,1]$, which by Wirsing's theorem is the set of mean values of single functions in $\mathcal F([-1,1])$ (p. 5), is only $[0,1]$; the negative part of the spectrum comes from functions that change with $N$.

Source. Andrew Granville and K. Soundararajan, The spectrum of multiplicative functions, Ann. of Math. (2) 153 (2001), no. 2, 407--470; read as arXiv:math/9909190v1 (8 September 1999), whose printed page equals its PDF page: the setting on p. 2, Theorem 1, (1.1) and (1.2) on p. 3, Section 5 on pp. 29--40. The published pagination differs and was not compared. The edition read is identified on the source card.

Read depth. Claims checked: the setting, the statement and the remarks after it were read clause by clause on the page images. The proof was not checked.

Proof pointer

Section 5 (pp. 29--40). By the Structure Theorem (Theorem 3) and its variant Theorem 3$'$ (p. 9), $\Gamma([-1,1])=\Lambda([-1,1])$, the set of values $\sigma(u)$ of solutions of the integral equation (1.5) with $\chi$ taking values in $[-1,1]$. The inclusion $\Lambda([-1,1])\supset [\delta_1,1]$ comes from the choice $\chi(t)=1$ for $t\le1$ and $\chi(t)=-1$ for $t>1$, whose solution $\rho_-$ satisfies $u\rho_-'(u)=-2\rho_-(u-1)$, decreases on $[1,1+\sqrt e]$ and takes the value $\delta_1$ at $1+\sqrt e$ (p. 8). The reverse inclusion is Theorem 5.1 (p. 29): whenever $\int_0^{u_0}(1-\chi(t))/t\,dt=1$, one has $\lvert\sigma(u)\rvert\le \lvert\delta_1\rvert$ for $u\ge u_0$, while $\sigma$ stays non-negative before $u_0$.

Dependencies

The Structure Theorem (Theorem 3) and Theorem 3$'$, Proposition 1 and its converse (p. 7), and Theorem 5.1 (p. 29) of the same paper.

Bears on

• Problem 786: through (1.1), whose sharp form Corollary 1 is the input of the transfer proposed in a forum post for the reading with repeated factors allowed; see that page. The paper does not state the problem.
• Problem 121: background only, through the same lower bound; see Corollary 1.