Granville 2001 spectrum multiplicative functions

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corollary_1: The sharp lower bound (delta_1 + o(1))x, delta_1 = -0.656999..., for the partial sums of a real completely multiplicative function with values in [-1,1], with equality exactly when f is asymptotically +1 on primes up to x^{1/(1+sqrt e)} and -1 on larger primes up to x in a 1/p-weighted sense.

theorem_1: Granville and Soundararajan's determination of the spectrum of [-1,1]: the limits of averages up to N of completely multiplicative functions with values in [-1,1], allowed to change with N, fill exactly the interval [delta_1, 1], where delta_1 = -0.656999... is an explicit constant.

theorem_2: Granville and Soundararajan's bounds for gamma_m and gamma'_m, the least natural and logarithmic proportions of m-th power residues up to x over all moduli l in the limit: gamma_2 = delta_0, gamma'_2 = 1/2, and for m >= 3 the natural proportion is positive and at most rho(m), and the logarithmic one is at least 1/2^{m-1}.

theorem_3: Granville and Soundararajan's Structure Theorem: for every closed subset S of the unit disc containing 1, the spectrum of S is the set of products of an Euler product value and a value of a solution of the integral equation (1.5) driven by a function with values in the convex hull of S.

theorem_5: Granville and Soundararajan's bound on the spectrum: for closed S in the unit disc with 1 in S, the spectrum is the whole disc exactly when the angle of S is pi/2, and otherwise lies in an explicit disc touching the unit circle only at 1.

theorem_9: Granville and Soundararajan's bound for beta(B), the liminf over fundamental discriminants D of the normalized sum of the Kronecker symbol (D/n) over n up to (log |D|)^B: it is at most a variational minimum gamma(B) of solutions of the integral equation (1.5), which is negative and at least -rho(B).

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Granville, Andrew and Soundararajan, K., The spectrum of multiplicative functions. Ann. of Math. (2) 153 (2001), no. 2, 407-470, DOI 10.2307/2661346 (volume, issue and DOI from the Crossref record and the journal's article page).

The copy read for this card is arXiv:math/9909190v1, dated 8 September 1999 on its first page, with 59 PDF pages. It is the version-1 preprint, not the published 2001 pagination above. Theorem and page locators used here refer to this version. The arXiv record carries no license field, so arXiv's assumed license applies (arXiv:math/9909190), every other right reserved.

The paper studies the spectrum Gamma(S) of a set S in the unit disc: the limits of sequences of averages (1/N) sum_{n<=N} f_N(n), where each f_N is completely multiplicative with f_N(p) in S for every prime p and may change with N (p. 2). For S = [-1,1] the mean values of single functions form only the Euler product spectrum [0,1] (pp. 5-6). Theorem 1 determines the spectrum of [-1,1] exactly as the interval [delta_1, 1] with delta_1 = 1 - 2 log(1+sqrt(e)) + 4 int_1^{sqrt e} log t/(t+1) dt = -0.656999..., proving the Hall-Montgomery conjecture that sum_{n<=x} f(n) >= (delta_1+o(1))x for real completely multiplicative f with |f| <= 1. Corollary 1 characterizes asymptotic equality through reciprocal-prime-weighted deviations from the extremal signs: +1 below x^{1/(1+sqrt e)} and -1 above. Applying this to the Legendre symbol gives that at least (delta_0+o(1))x integers below x are quadratic residues mod p, with delta_0 = (1+delta_1)/2 = 0.171500.... A Structure Theorem describes general spectra via Euler products and solutions of an integral equation, and further theorems bound angles, projections, connectedness and the logarithmic spectrum; Theorem 9 revisits quadratic residues and nonresidues, bounding beta(B) by an explicit variational quantity gamma(B) with -rho(B) <= gamma(B) < 0. Methods are Halasz-type mean value estimates, integral equations and inclusion-exclusion inequalities. The paper states no result about products of subsets of integers; what the corpus uses is the sharp lower bound of Corollary 1.

For the E0786 interface, the hypotheses matter: p. 2, footnote 1, defines complete multiplicativity as $f(mn)=f(m)f(n)$ for all positive integers $m,n$. The consequence (1.1) of Theorem 1 and Corollary 1, p. 3, require a real completely multiplicative $f:\mathbb N\to[-1,1]$. Tao's forum post 4061 on 2 February 2026 proposes using $(-1)^{F(n)}$ for an additive $F$ arising from a product-length set. The real-valued hypothesis needs justification if $F$ is only known to be real additive. The source does not itself state the E0786 subset-product problem. Neither the proposed transfer nor a sharp bound for either repetition convention has been proof-reviewed here.

Source: https://arxiv.org/abs/math/9909190, read as version 1.

Reading and proof scope. Complete printed/PDF pp. 1-3 were visually read for version identity, complete multiplicativity, the real-valued hypothesis, constants and equality condition. On 2026-10-07 pp. 4-13 were read on the page images for the statements of the Euler product spectrum, Theorem 3, Corollary 3, Theorem 5 and Theorem 9 used above, and section 9 (pp. 55-58) for what its proof covers. On 2026-10-08 the statements of Theorems 1, 2, 3, 5 and 9 and of Corollary 1 were read again clause by clause on pp. 2-13 for the result pages below, with the deductions on pp. 28-30 and the proof locations in sections 2, 4, 5, 7 and 9. No proof review of this paper was performed; the broader overview and uninspected results retain their earlier compilation scope.

Bears on. #786: a forum post (Tao, post 4061, 2 February 2026) proposes applying Corollary 1 to $(-1)^{F(n)}$ for an additive $F$ attached to the set, for the reading in which a factor may repeat; the paper does not state the problem or that transfer, which has not been checked here and does not reach the distinct-factor reading the problem page adopts. #121: background only. The problem page cites the paper for $F(N)=(1-c+o(1))N$, $c=0.1715\ldots$, on the different question with no odd number of elements multiplying to a square; the version read states no result about $F(N)$, and $0.1715\ldots$ is its quadratic residue constant $\delta_0$.

Results. Theorem 1 (p. 3), the spectrum of $[-1,1]$; Corollary 1 (p. 3), the sharp lower bound with its equality condition, and the quadratic residue bound of pp. 3-4; Theorem 2 (p. 4), densities of $m$-th power residues; Theorem 3 (p. 8), the Structure Theorem; Theorem 5 (p. 9), the disc containing the spectrum and its boundary points; Theorem 9 (p. 13), the bound $\beta(B)\le\gamma(B)$. Theorems 3', 4, 6, 7 and 8 and Corollaries 2-4 (pp. 6-12) describe the geometry of spectra and the logarithmic spectrum and are summarized above; no problem page uses them.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.