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Math-5655 · mathematics · Mathematics

Erdős Problem #446

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Statement

Statement and result.
Let $\delta(n)$ be the natural density of integers with a divisor in $(n,2n)$, and $\delta_r(n)$ the density of those with exactly $r$ such divisors. The known answer is \[ \delta(n)\asymp \frac1{(\log n)^\alpha(\log\log n)^{3/2}},\qquad \alpha=1-\frac{1+\log\log2}{\log2}, \tag{1} \] and, for each fixed $r\geq1$, \[ \delta_r(n)\gg_r\delta(n). \tag{2} \] In particular $\delta_1(n)=o(\delta(n))$ is false. We give a complete deduction from Ford's interval-divisor theorems, including the open-endpoint convention and the existence of all densities.
Exact finite formulas.
Let $D_n=\{n+1,\ldots,2n-1\}$, with integer $n\geq2$, and $L=\operatorname{lcm}(D_n)$. The number $T_n(m)=|\{d\in D_n:d\mid m\}|$ is periodic modulo $L$. Hence all the stated natural densities exist and are rational. Inclusion-exclusion gives \[ \delta(n)= \sum_{\varnothing\ne S\subseteq D_n} \frac{(-1)^{|S|+1}}{\operatorname{lcm}(S)}. \tag{3} \] More generally, binomial inversion gives, for $r\geq1$, \[ \delta_r(n)= \sum_{\substack{S\subseteq D_n\\|S|\geq r}} \frac{(-1)^{|S|-r}\binom{|S|}{r}}{\operatorname{lcm}(S)}. \tag{4} \] To verify $4$, for any integer with $t$ divisors in $D_n$, its contribution before averaging is $\sum_{j=r}^{t}(-1)^{j-r}\binom jr\binom tj$. This is $1$ if $t=r$ and zero otherwise, by $\binom tj\binom jr=\binom tr\binom{t-r}{j-r}$. Averaging a divisibility event over one period gives the reciprocal of its least common multiple, proving both formulas.
The explicit external input.
In The distribution of integers with a divisor in a given interval, Ford defines $\varepsilon(y,z)$ and $\varepsilon_r(y,z)$ using the half-open interval $(y,z]$. His Corollary 2 gives $1$ for $\varepsilon(n,2n)$. His Corollary 7 gives $\varepsilon_r(n,2n)\gg_r\varepsilon(n,2n)$ for each fixed $r$. These deep theorems are external inputs; their analytic proofs are not reconstructed here.
Transfer to the exact statement.
The only difference between $(n,2n)$ and $(n,2n]$ is the divisor $2n$. Thus the indicator for having any divisor, and the indicator for having exactly $r$ divisors, can change only on multiples of $2n$. Consequently \[ |\delta(n)-\varepsilon(n,2n)|\leq\frac1{2n},\qquad |\delta_r(n)-\varepsilon_r(n,2n)|\leq\frac1{2n}. \tag{5} \] The error $1/n$ is negligible compared with the function in $1$. Applying the external bounds and then $5$ proves $1$--$2$. In particular there is a fixed $c>0$ such that $\delta_1(n)/\delta(n)\geq c$ for all sufficiently large $n$; the ratio cannot tend to zero.
A useful check on the interpretation.
The period average of $T_n$ is exactly \[ \sum_{d=n+1}^{2n-1}\frac1d=\log2+o(1). \] Conditioning on $T_n\geq1$, the mean is this sum divided by $\delta(n)$, and therefore tends to infinity. This is consistent with $2$: a positive conditional proportion can have exactly one divisor while the conditional mean diverges. Conflating these two statements would lead to the incorrect conjecture.
Sources and scope.
Ford, Annals of Mathematics 168 $2008$, 367--433, Corollaries 2 and 7: https://annals.math.princeton.edu/2008/168-2/p01. Current statement: https://www.erdosproblems.com/446, checked 24 September 2026. This is a complete deduction relative to those explicitly identified established theorems.
Completion estimate.
Both requested conclusions follow, with density existence and endpoint errors checked.

Restatement from Math-5655-A1 r1, with the source manuscript’s qualifications.

Original problem on erdosproblems.com

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-5655-A1 · No claim recorded · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Math-5655

Erdős Problem 446

Statement and result.
Let $\delta(n)$ be the natural density of integers with a divisor in $(n,2n)$, and $\delta_r(n)$ the density of those with exactly $r$ such divisors. The known answer is \[ \delta(n)\asymp \frac1{(\log n)^\alpha(\log\log n)^{3/2}},\qquad \alpha=1-\frac{1+\log\log2}{\log2}, \tag{1} \] and, for each fixed $r\geq1$, \[ \delta_r(n)\gg_r\delta(n). \tag{2} \] In particular $\delta_1(n)=o(\delta(n))$ is false. We give a complete deduction from Ford's interval-divisor theorems, including the open-endpoint convention and the existence of all densities.
Exact finite formulas.
Let $D_n=\{n+1,\ldots,2n-1\}$, with integer $n\geq2$, and $L=\operatorname{lcm}(D_n)$. The number $T_n(m)=|\{d\in D_n:d\mid m\}|$ is periodic modulo $L$. Hence all the stated natural densities exist and are rational. Inclusion-exclusion gives \[ \delta(n)= \sum_{\varnothing\ne S\subseteq D_n} \frac{(-1)^{|S|+1}}{\operatorname{lcm}(S)}. \tag{3} \] More generally, binomial inversion gives, for $r\geq1$, \[ \delta_r(n)= \sum_{\substack{S\subseteq D_n\\|S|\geq r}} \frac{(-1)^{|S|-r}\binom{|S|}{r}}{\operatorname{lcm}(S)}. \tag{4} \] To verify $4$, for any integer with $t$ divisors in $D_n$, its contribution before averaging is $\sum_{j=r}^{t}(-1)^{j-r}\binom jr\binom tj$. This is $1$ if $t=r$ and zero otherwise, by $\binom tj\binom jr=\binom tr\binom{t-r}{j-r}$. Averaging a divisibility event over one period gives the reciprocal of its least common multiple, proving both formulas.
The explicit external input.
In The distribution of integers with a divisor in a given interval, Ford defines $\varepsilon(y,z)$ and $\varepsilon_r(y,z)$ using the half-open interval $(y,z]$. His Corollary 2 gives $1$ for $\varepsilon(n,2n)$. His Corollary 7 gives $\varepsilon_r(n,2n)\gg_r\varepsilon(n,2n)$ for each fixed $r$. These deep theorems are external inputs; their analytic proofs are not reconstructed here.
Transfer to the exact statement.
The only difference between $(n,2n)$ and $(n,2n]$ is the divisor $2n$. Thus the indicator for having any divisor, and the indicator for having exactly $r$ divisors, can change only on multiples of $2n$. Consequently \[ |\delta(n)-\varepsilon(n,2n)|\leq\frac1{2n},\qquad |\delta_r(n)-\varepsilon_r(n,2n)|\leq\frac1{2n}. \tag{5} \] The error $1/n$ is negligible compared with the function in $1$. Applying the external bounds and then $5$ proves $1$--$2$. In particular there is a fixed $c>0$ such that $\delta_1(n)/\delta(n)\geq c$ for all sufficiently large $n$; the ratio cannot tend to zero.
A useful check on the interpretation.
The period average of $T_n$ is exactly \[ \sum_{d=n+1}^{2n-1}\frac1d=\log2+o(1). \] Conditioning on $T_n\geq1$, the mean is this sum divided by $\delta(n)$, and therefore tends to infinity. This is consistent with $2$: a positive conditional proportion can have exactly one divisor while the conditional mean diverges. Conflating these two statements would lead to the incorrect conjecture.
Sources and scope.
Ford, Annals of Mathematics 168 $2008$, 367--433, Corollaries 2 and 7: https://annals.math.princeton.edu/2008/168-2/p01. Current statement: https://www.erdosproblems.com/446, checked 24 September 2026. This is a complete deduction relative to those explicitly identified established theorems.
Completion estimate.
Both requested conclusions follow, with density existence and endpoint errors checked. COMPLETION: 100%

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