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Statement

Setting as in Theorem 1: $a_1<\cdots<a_{\varphi(n)}$ are the integers not exceeding $n$ that are prime to $n$.

Theorem 2 (p. 49). For $0\le\alpha<2$,

$$\sum_{i=1}^{\varphi(n)-1}(a_{i+1}-a_i)^{\alpha} =\{1+o(1)\}\,\Gamma(\alpha+1)\,n\Bigl(\frac{n}{\varphi(n)}\Bigr)^{\alpha-1}$$

as $n\to\infty$ through a sequence of values for which $n/\varphi(n)\to\infty$. By note (iv) of Section 2 (p. 40), the $o(1)$ here need not be uniform in $\alpha$.

The theorem is stated without proof. The paper says (pp. 39, 49) that it can be deduced from Theorem 1 by the methods of part I, C. Hooley, On the difference of consecutive numbers prime to $n$, Acta Arith. 8 (1963), 295--299, where the bound (A) of p. 39, $\sum_{i=1}^{\varphi(n)-1}\Delta_i^{\alpha}=O\{n(n/\varphi(n))^{\alpha-1}\}$, was shown for $1\le\alpha<2$; a footnote on p. 39 adds that (A) also holds for $0\le\alpha<1$ by Hölder's inequality. Theorem 2 replaces (A) by an asymptotic formula when $n/\varphi(n)\to\infty$.

Proof pointer

No proof is given in the paper; see the deduction it names above.

Read depth

Claims checked: the statement and the surrounding remarks on pp. 39, 40 and 49 were read on the page images of the print. No proof was checked, since the paper gives none.

Dependencies

Theorem 1 of the paper and the methods of part I (Acta Arith. 8 (1963), 295--299), which the corpus does not hold.

Source. C. Hooley, 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; the edition read is named on the source card.

Bears on

• Problem 235: context only. Theorem 2 gives the $\alpha$-th moments, for $0\le\alpha<2$, of the same normalized gaps whose limiting distribution the problem asks about; it does not state that distribution, which is Theorem 1.