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The distribution of integers with a divisor in a given interval. (English) Zbl 1181.11058
This paper considers the number of integers $$n\leq x$$ having a divisor in a given interval $$(y,z]$$, denoted by $$H(x,y,z)$$. The principal result gives the order of magnitude of $$H(x,y,z)$$ uniformly in all parameters, irrespective of the size of $$z- y$$ relative to $$x$$ and $$y$$. Similarly one may write $$H_r(x,y, z)$$ for the number of integers $$n\leq x$$ with exactly $$r$$ divisors in $$(y, z]$$. For $$r= 1$$ the exact order of magnitude is found for any $$x$$, $$y$$, $$z$$ with $$z\leq x^{1/2-\varepsilon}$$, where $$\varepsilon$$ is any fixed positive constant. When $$r\geq 2$$ is fixed the order of magnitude of $$H_r(x,y,z)$$ is also determined for large $$y$$ and $$z$$ in the range $y+{y\over (\log y)^{\log 4-1-\varepsilon}}\leq z\leq\min\{y^C, x^{1/2-\varepsilon}\},$ for any fixed $$\varepsilon> 0$$ and $$C> 1$$. These questions have previously been investigated by R. R. Hall [Sets of multiples, Cambridge Tracts in Mathematics. 118. Cambridge: Cambridge University Press (1996; Zbl 0871.11001)] and by G. Tenenbaum [(*) Compos. Math. 51, 243–263 (1984; Zbl 0541.10038) and Acta Arith. 49, No. 2, 165–187 (1987; Zbl 0636.10038)] amongst others. This second paper of Tenenbaum makes two conjectures on the relative sizes of $$H(x,y,z)$$ and $$H_r(x,y,z)$$. One of these is established in the present paper apart from one small range of the variables. The other conjecture states that if $$r\geq 1$$ and $$\varepsilon> 0$$ are fixed then for $$y\leq x^{1/2-\varepsilon}$$ one has $H_r(x,y,z)= o(H(x,y,z))$ as $$z/y\to\infty$$. The present paper proves this completely.
It is not hard to show that both $$x^{-1}H(x,y,z)$$ and $$x^{-1}H_r(x,y,z)$$ tend to limits as $$x\to\infty$$. These limits are denoted by $$\varepsilon(y, z)$$ and $$\varepsilon_r(y, z)$$, respectively. It was conjectured by P. Erdős [Vestn. Leningr. Univ. 15, No. 13 (Ser. Mat. Mekh. Astron. No. 3), 41–49 (1960; Zbl 0104.26804)] that $$\varepsilon_1(y, 2y)/\varepsilon(y, 2y)\to 0$$ as $$y\to 0$$. This is now shown to be false. More generally, one has ${\varepsilon_r(y,\lambda y)\over \varepsilon(y,\lambda y)}\gg_{r,\lambda} 1$ for any fixed $$\lambda> 1$$ and $$r\in\mathbb N$$.
The paper presents a number of corollaries of the main theorems. We mention just one. If $$\rho(n)$$ denotes the largest divisor $$d$$ of $$n$$ with $$d\leq\sqrt{n}$$, then $$\sum_{n\leq x}\rho(n)$$ has exact order $x^{3/2}(\log x)^{-\delta}(\log\log r)^{-3/2}.$
The detailed estimates for $$H_r(x,y, z)$$ are quite complicated, but we shall describe here the results for $$H(x,y,z)$$, which will suffice to give the general flavour. Given $$z> y\geq 4$$ define $$\eta$$, $$u$$, $$\beta$$ and $$\xi$$ by $$z= e^ny= y^{1+u}$$, $$\eta= (\log y)^{-\beta}$$, and $\beta= \log 4- 1+{\xi\over\sqrt{\log\log y}},$ and set $$\delta= 1-{1+ \log\log 2\over\log 2}$$ and $G(\beta)= \begin{cases} 1+ {1+\beta\over\log 2}\log({1+\beta\over e\log 2}),\quad & 0\leq\beta\leq\log 4-1,\\ \beta,\quad & \beta\geq\log 4-1.\end{cases}$
In his paper (*), G. Tenenbaum showed that $$H(x,y,z)$$ has changes in behaviour around $$z- y^2$$, near $$z= 2y$$, and in the vicinity of $z= z_0(y):= y\exp\{(\log y)^{1-\log 4}\}.$ The main theorem of the paper then states that for $$1\leq y\leq y\leq x$$ one has firstly four preliminary cases, namely
$$H(x,y,z)= 0$$ for $$z< [y]+ 1$$;
$$H(x,y,z)= [x([y]+ 1)^{-1}]$$ for $$[y]+ 1\leq z< y+ 1$$;
$$H(x,y,z)$$ is of order $$1$$ for $$z> y+1$$ and $$x\leq 100000$$; and
$$H(x,y,z)$$ is of order $$x$$ for $$z> y+ 1$$, $$y\leq \max(100,\sqrt{2})$$ and $$x> 100000$$.
The main case is that in which $$x> 100000$$ and $$100\leq y\leq\min(z- 1,\sqrt{x})$$, and here there are three subranges. When $$y+ 1\leq z\leq z_0(y)$$ the function $$H(x,y,z)$$ has order of magnitude $$\eta x$$; for $$z_0(y)\leq z\leq 2y$$ it has order ${x\beta\over\max(1, -\xi)(\log y)^{G(\beta)}};$ and for $$2y\leq z\leq y^2$$ it has order of magnitude $$u^\delta(\log 2/u)^{-3/2}$$. Finally, when $$x> 100000$$, $$\sqrt{x}< y< z\leq x$$ and $$z\geq y+ 1$$ the function $$H(x,y,z)$$ has the same order of magnitude as $$H(x,x/z, x/y)$$ if $$x/y\geq x/z+ 1$$, and otherwise is of order $$\eta x$$. In all these cases, when we say “$$H(x,y,z)$$ has order of magnitude $$F(x,y,z)$$”, say, we mean that $$F(x,y,z)\ll H(x,y,z)\ll F(x,y, z)$$ with absolute implied constants.

MSC:
 11N25 Distribution of integers with specified multiplicative constraints
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