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Statement

Theorem 2 (p. 4, quoted). "For integers $m\ge2$, define

$$\gamma_m=\liminf_{x\to\infty}\inf_\ell\frac1x \sum_{\substack{n\le x\\ n\equiv a^m\ (\mathrm{mod}\ \ell)}}1, \qquad\text{and}\qquad \gamma_m'=\liminf_{x\to\infty}\inf_\ell\frac1{\log x} \sum_{\substack{n\le x\\ n\equiv a^m\ (\mathrm{mod}\ \ell)}}\frac1n.$$

Then $\gamma_2=\delta_0$, $\gamma_2'=1/2$, and for $m\ge3$,

$$0<\gamma_m\le\rho(m)\left(=\frac1{m^{m+o(m)}}\right)<\frac1{2^{m-1}} \le\gamma_m'\le\min_{\beta\ge0}\frac1{e^\beta}\sum_{k=0}^\infty \frac{\beta^{km}}{(km)!}\left(\sim\frac1{e^{m/e}}\right).$$

Here $\rho(u)$ is the Dickman-de Bruijn function, defined by $\rho(u)=1$ for $0\le u\le1$, and $u\rho'(u)=-\rho(u-1)$ for all $u\ge1$."

The sums count the integers $n\le x$ that are $m$-th power residues mod $\ell$; the sentence introducing the theorem (p. 4) speaks of a prime modulus $\ell$. The constant $\delta_0=0.171500\ldots$ is the quadratic residue constant of Corollary 1. The paper adds (p. 4) that the exact values of $\gamma_m$ and $\gamma_m'$ are unknown for every $m\ge3$, and reports the numerical bounds $\gamma_3'\le0.3245$, $\gamma_4'\le0.2187$, $\gamma_5'\le0.14792$ and $\gamma_6'\le0.1003$ from minimizing over $\beta$. In words: for each $m\ge2$ there is $\pi_m>0$ such that, for $x$ sufficiently large and all primes $p$, more than $\pi_m\%$ of the integers up to $x$ are $m$-th power residues mod $p$.

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: Theorem 2 and the remarks after it on p. 4, Section 2 on pp. 13--18. The published pagination differs and was not compared. The edition read is identified on the source card.

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

Proof pointer

Section 2 (pp. 13--18). The paper says (p. 13) that $\gamma_2=\delta_0$ is clear from the introduction, where it follows from Corollary 1. The upper bound $\gamma_m\le\rho(m)$ takes, by the Chebotarev density theorem, a prime $\ell\equiv1\pmod m$ with a character of order $m$ equal to $1$ on primes up to $x^{1/m}$ and to $e^{2\pi i/m}$ on the primes from there to $x$, so that the $m$-th power residues up to $x$ are the $x^{1/m}$-smooth integers (p. 13). Positivity of $\gamma_m$ iterates Proposition 2.2 (p. 14), which rests on Hall's Lemma 1$'$ and Hildebrand's Lemma 2.1 (p. 13), and is completed on p. 15. Section 2b (pp. 15--18) treats the logarithmic proportions, the lower bound coming from the zero-sum count of Lemma 2.3 (p. 16) through Corollary 2.5 (p. 17).

Dependencies

Corollary 1 for $\gamma_2=\delta_0$; Lemma 1$'$ (Hall, p. 5), Lemma 2.1 (Hildebrand, p. 13), Proposition 2.2 (p. 14), Lemmas 2.3 and 2.4 and Corollary 2.5 (pp. 16--17) of the same paper.

Bears on

No Erdős problem page of the corpus cites this theorem.