The distribution of integers with a divisor in a given interval.

*(English)*Zbl 1181.11058This 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.

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.

Reviewer: Roger Heath-Brown (Oxford)

##### MSC:

11N25 | Distribution of integers with specified multiplicative constraints |