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Statement

Setting (p. 12). For $B>0$, with $D$ running over fundamental discriminants,

$$\beta(B)=\liminf_{\lvert D\rvert\to\infty}\frac1{(\log\lvert D\rvert)^B} \sum_{n\le(\log\lvert D\rvert)^B}\Big(\frac Dn\Big),$$

and $\alpha(B)$ is the corresponding $\limsup$. $\rho$ is the Dickman-de Bruijn function and $\sigma$ the solution of the integral equation (1.5) attached to $\chi$ (p. 7; see the Theorem 3 page). The paper notes (pp. 12--13) that Theorem 1 gives $\beta(B)\ge\delta_1$ for all $B$, and, with the Hall-Montgomery example, $\beta(B)=\delta_1$ for $B\le1$; Theorem 9 answers Mark Watkins's question whether $\beta(B)<0$ for all $B$.

Theorem 9 (p. 13, quoted). "Given $u\ge1$, let $\mathcal C(u)$ denote the set of all measurable functions $\chi$ such that $\chi(t)=1$ for $t\le1$, $\chi(t)\in[-1,1]$ for $1\le t\le u$, and $\chi(t)=0$ for $t>u$. Define

$$\gamma(B)=\min_{u\ge1}\ \min_{\chi\in\mathcal C(u)}\sigma(Bu),$$

for all $B>0$, where $\sigma$ refers to the solution to (1.5). Then $\beta(B)\le\gamma(B)$ for all $B>0$, where $-\rho(B)\le\gamma(B)<0$."

After the theorem (p. 13) the paper states, without proof in this paper, that under the Generalized Riemann Hypothesis it can show $\beta(B)\ge\gamma(B/2)$, and that it believes $\beta(B)=\gamma(B)$ for all $B$.

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), printed page $=$ PDF page: the definitions on p. 12, Theorem 9 and the remarks after it on p. 13, Section 9 on pp. 55--58. The published pagination differs and was not compared. The edition read is identified on the source card.

Read depth. Claims checked: the definitions, the statement and the remarks after it were read clause by clause on the page images, and Section 9 was read for what it proves: the theorem, not the conditional lower bound. The proof was not checked step by step.

Proof pointer

Section 9 (pp. 55--58). Proposition 9.1 (p. 55) shows that, averaged over the fundamental discriminants $0<D\le X$ in a suitable residue class, the sum of $\left(\frac Dn\right)$ over $n\le(\log X)^B$ equals the partial sum of a prescribed completely multiplicative $f$ with $f(p)=\pm1$ for $p\le z$ and $f(p)=0$ for $p>z$, up to an error $O((\log X)^B/z)$; the non-square moduli are handled by the Pólya-Vinogradov inequality (p. 56). With $z=\frac14\log X$ this gives (p. 57) a fundamental discriminant $D$ with $X/\log X\ll D\le X$ whose normalized sum up to $(\log D)^B$ is at most that of $f$ plus $o(1)$, display (9.4). Choosing $f$ by the converse of Proposition 1 (p. 7) to follow a given $\chi\in\mathcal C(u)$ makes the right side $\sigma(Bu)+o(1)$, which gives $\beta(B)\le\gamma(B)$ (p. 57). The bound $\lvert\sigma(Bu)\rvert\le\rho(B)$, display (9.5), gives $\gamma(B)\ge-\rho(B)$. For $\gamma(B)<0$ (pp. 57--58), the paper takes $\chi=1$ on $[0,1]$, $-1$ on $[1,2]$ and $0$ beyond, and shows from (9.5) that the corresponding solution changes sign infinitely often, so it is negative at arbitrarily large arguments.

Dependencies

Proposition 1 and its converse (p. 7) and Proposition 9.1 (p. 55) of the same paper.

Bears on

No Erdős problem page of the corpus cites this theorem.