Hooley 1965 difference between consecutive numbers prime

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theorem_1: Hooley's theorem that, as n tends to infinity through a sequence along which n/phi(n) tends to infinity, the number of gaps Delta_i < cn/phi(n) between consecutive integers prime to n is phi(n){1 + o(1)}(1 - e^{-c}), uniformly for c in any fixed range bounded at either end by positive constants.

theorem_2: Hooley's theorem, stated without proof, that for 0 <= alpha < 2 the sum of the alpha-th powers of the gaps between consecutive integers prime to n is {1 + o(1)} Gamma(alpha+1) n (n/phi(n))^{alpha-1} as n tends to infinity through a sequence along which n/phi(n) tends to infinity.

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Hooley, Christopher, On the difference between consecutive numbers prime to $n$: II. Publ. Math. Debrecen 12 (1965), 39--49, doi:10.5486/pmd.1965.12.1-4.06. No copyright or license line is printed on the scan's rendered first or last page; the journal site's record for the article shows only the site-wide footer "© 2026, Publicationes Mathematicae, Debrecen, Hungary" and names no license (https://publi.math.unideb.hu/paper/2953, read 2026-10-02), every other right reserved.

Let a_1 < ... < a_{phi(n)} be the integers up to n coprime to n and Delta_i = a_{i+1} - a_i their gaps. Theorem 1, the paper's main result, shows that as n tends to infinity through a sequence along which n/phi(n) also tends to infinity, the number f_n(c) of gaps Delta_i < cn/phi(n) satisfies f_n(c) = phi(n){1 + o(1)}(1 - e^{-c}) uniformly for c in any fixed range bounded at both ends by positive constants: the normalized gap Delta_i/(n/phi(n)) has in the limit the gamma distribution with parameter 1 (the exponential law), so the distribution of Delta_i phi(n)/n becomes essentially independent of n; this confirms a conjecture of Erdős, who had raised it for n a product of the consecutive primes 2·3···p. Theorem 2, stated without proof as a consequence of Theorem 1 by the methods of the earlier paper, part I (Acta Arith. 8 (1963), 295--299), turns part I's bound (A) sum Delta_i^alpha = O(n (n/phi(n))^{alpha-1}), proved there for 1 <= alpha < 2, into the asymptotic formula sum Delta_i^alpha = {1 + o(1)} Gamma(alpha+1) n (n/phi(n))^{alpha-1} for 0 <= alpha < 2 along the same sequences. The method starts from the exclusion-principle formula (2) of Section 3, which expresses f_n(c) through the counts N_r(n,y) of sets of r reduced residues a_{i_1} < ... < a_{i_r} with a_{i_r} - a_{i_1} < y = cn/phi(n), then estimates N_r in Sections 4 to 9, with careful uniformity bookkeeping described in the notation of Section 2. For problem 235 the paper supplies the limiting exponential/gamma distribution of the normalized gaps between integers coprime to n.

Source: https://publi.math.unideb.hu/paper/2953.

Bears on. #235: Theorem 1 (p. 49) gives the limiting proportion $1-e^{-c}$ of gaps below $cn/\varphi(n)$ as $n/\varphi(n)\to\infty$, for $c$ in any fixed range bounded at either end by positive constants, and the paper presents it as proving Erdős's conjecture for $n$ a product of consecutive primes, the problem's $N_k$; the problem counts gaps up to $cN_k/\varphi(N_k)$ for every $c\ge0$, and its page and claim page record how its statement is read from the theorem.

Results.

• Theorem 1 (p. 49): as $n\to\infty$ with $n/\varphi(n)\to\infty$, the number of gaps $\Delta_i<cn/\varphi(n)$ is $\varphi(n)\{1+o(1)\}(1-e^{-c})$, uniformly for $c$ in any fixed range bounded at either end by positive constants; its page also points to formulas (1), (2), (22) and (23) of the proof.
• Theorem 2 (p. 49), stated without proof: for $0\le\alpha<2$, $\sum_{i=1}^{\varphi(n)-1}\Delta_i^{\alpha}=\{1+o(1)\}\Gamma(\alpha+1)\,n\,(n/\varphi(n))^{\alpha-1}$ along the same sequences.

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