Integers without large prime factors in short intervals and arithmetic progressions

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Glyn Harman, Integers without large prime factors in short intervals and arithmetic progressions, Acta Arithmetica 91, no. 3 (1999), 279–289. The selected text is the eleven-page journal PDF.

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Scope

Theorem 1 treats intervals of length $x^{1/2}$. Theorem 2, printed p. 280, considers $y=x^\gamma$ for $3/7<\gamma<1/2$. It defines $\theta(\gamma)$ by an infimum over known exponent pairs and, subject to $\theta(\gamma)<\gamma$, obtains

$$\Psi(x+y,u)-\Psi(x,u)\gg y \qquad\text{for }u\geq x^{e^{-1/2}\theta(\gamma)+\epsilon},$$

where $\epsilon>0$. Here $\Psi(x,u)$ counts positive integers at most $x$ whose prime factors are all at most $u$. The simpler choice $\theta=(3-5\gamma)/2$ in display (1.4) is explicitly conditional on the exponent-pair conjecture. The theorem does not state the unconditional result throughout the displayed $\gamma$ range without the exponent-pair condition. Theorem 3 treats arithmetic progressions with separate hypotheses.

Read status: claims checked. The introductory definitions and Theorems 1–3 were read at statement depth. Theorem 2 and its conditionality were checked against the printed p. 280 image. No proof is verified here.

Bears on. Problem 962: the interval estimates are prospective inputs for a uniform bound on blocks without a sufficiently smooth integer. The original exponent-pair and smoothness conditions must be matched to that application.