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Statement

Theorem 5 (printed p. 708): "For each number $\alpha$ in the range $0<\alpha<25/144$, there is a computable number $x(\alpha)$ such that $\mathrm C(x)\ge x^\alpha$ for all $x\ge x(\alpha)$."

Here $C(x)$ counts the Carmichael numbers up to $x$. The range is strict at $25/144=(5/12)^2$. The paper adds (p. 708) that computing a numerical value of $x(\alpha)$ for a specific $\alpha$ may be difficult.

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; the effectivity discussion on p. 707 and Theorem 5 on p. 708. The edition read is identified on the source card.

Read depth. Claims checked: the statement and the derivation sketched on p. 707 were read clause by clause on the page images of pp. 707--708. Nothing here is independently reviewed.

Proof pointer

The paper derives it in prose on p. 707. The proof of Theorem 1 is effective: given numerical $\gamma_1(E)$, $x_1(E)$ and $x_2(B)$ it yields $x_0(E,B)$. The proof that every $B<5/12$ is in $\mathcal B$ (Section 2) is effective, and through the proof of Theorem 3 so are $\gamma_1(E)$ and $x_1(E)$ for every $0<E<5/12$. Friedlander's larger members of $\mathcal E$ rest on the ineffective Bombieri--Vinogradov theorem, so the effective exponent is $EB$ with $E,B<5/12$.

Dependencies

Theorems 1 and 3 and the effective Theorem 2.1 (p. 712).

Bears on

• Problem 1057: an effective lower bound $C(x)\ge x^\alpha$ for $\alpha<25/144$, weaker in exponent than Theorem 1's $x^{2/7}$; it does not decide the problem.