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Statement

Here $C(x)$ counts the Carmichael numbers up to $x$, $\pi(x)$ the primes up to $x$, $\pi(x;d,1)$ the primes up to $x$ congruent to $1$ modulo $d$, and $\varphi$ is Euler's function.

Theorem 4 (printed p. 707): "Let $\varepsilon>0$. Suppose there is a number $x_\varepsilon$ such that

$$\pi(x;d,1)\ge\frac{\pi(x)}{2\varphi(d)}$$

for all positive integers $d\le x^{1-\varepsilon}$, once $x\ge x_\varepsilon$. Then there is a number $x'_\varepsilon$ such that $\mathrm C(x)\ge x^{1-2\varepsilon}$ for all $x\ge x'_\varepsilon$. In particular, if such an $x_\varepsilon$ exists for each $\varepsilon>0$, then $\mathrm C(x)=x^{1-o(1)}$ for $x\to\infty$."

The hypothesis has no exceptional moduli and only the residue class $1$; it is a weak form of the conjecture (p. 705) that $\pi(x;d,a)\sim \pi(x)/\varphi(d)$ uniformly for coprime $a,d$ with $d\le x^{1-\varepsilon}$.

Source. W. R. Alford, A. Granville and C. Pomerance, There are infinitely many Carmichael numbers, Ann. of Math. (2) 139 (1994), no. 3, 703--722; Theorem 4 on p. 707. The edition read is identified on the source card.

Read depth. Claims checked: the statement was read clause by clause on the page image of p. 707. Nothing here is independently reviewed.

Proof pointer

The paper prints no separate proof. It records Theorem 4 (p. 707) after remarking that the proofs of Theorem 1 and Theorem 3 need the definition of $\mathcal B$ only with $a=1$. Those proofs run through Sections 3 to 5 (pp. 715--721), where (0.3) is used with $a=1$ in the proofs of Theorem 3.1 (p. 716) and Theorem 3 (p. 720).

Dependencies

The proofs of Theorems 1 and 3 with the definition of $\mathcal B$ restricted to $a=1$.

Bears on

• Problem 1057: the theorem reduces the problem's $C(x)=x^{1-o(1)}$ to the stated lower bound for primes $\equiv1\bmod d$ with $d\le x^{1-\varepsilon}$, for every $\varepsilon>0$. That hypothesis is not proved in the paper; the theorem does not decide the problem.