Mixing
Understanding Erdos' Proof on Weird Number Density
$begingroup$
In this 1974 paper, Paul Erdos proves that the set of weird numbers has positive density in the natural numbers.
THEOREM 5. The density of weird numbers is positive .
Proof. If $n$ is weird, then let $epsilon_n$be as in the proof of the lemma. Now, by the
lemma, if $t$ is an integer and $a-(t)/t < 1 + epsilon_n$ then $nt$ is weird . But the density of the
integers $t$ with $Q(t)/t < 1 + epsilon_n$ is positive for any $epsilon_n > 0$.
Actually, we proved a slightly stronger result . If $n$ is weird, then the density of
{$m; n I m$ and $m$ is weird} is positive .
The lemma he is referring to in his proof is the following.
LEMMA. If $n$ is weird, then there is an $epsilon_n > 0$ such that $nt$ is weird if
$$sum_{d|t}{frac{1}{d}}<1+epsilon_n$$
I understand the heuristics behind this theorem, it is easy to make more weird numbers from multiples of other weird numbers, however, I am confused about the following statement.
But the density of the
integers $t$ with $Q(t)/t < 1 + epsilon_n$ is positive for any $epsilon_n > 0$.
Could someone show/explain to me where this statement comes from? It seems to be the crux of the proof, however, I don't see where it comes from.
Thank you in advance.
number-theory asymptotics
asked Jan 29 at 0:54
GnumbertesterGnumbertester
6701114
$endgroup$
add a comment |
$begingroup$
In this 1974 paper, Paul Erdos proves that the set of weird numbers has positive density in the natural numbers.
THEOREM 5. The density of weird numbers is positive .
Proof. If $n$ is weird, then let $epsilon_n$be as in the proof of the lemma. Now, by the
lemma, if $t$ is an integer and $a-(t)/t < 1 + epsilon_n$ then $nt$ is weird . But the density of the
integers $t$ with $Q(t)/t < 1 + epsilon_n$ is positive for any $epsilon_n > 0$.
Actually, we proved a slightly stronger result . If $n$ is weird, then the density of
{$m; n I m$ and $m$ is weird} is positive .
The lemma he is referring to in his proof is the following.
LEMMA. If $n$ is weird, then there is an $epsilon_n > 0$ such that $nt$ is weird if
$$sum_{d|t}{frac{1}{d}}<1+epsilon_n$$
I understand the heuristics behind this theorem, it is easy to make more weird numbers from multiples of other weird numbers, however, I am confused about the following statement.
But the density of the
integers $t$ with $Q(t)/t < 1 + epsilon_n$ is positive for any $epsilon_n > 0$.
Could someone show/explain to me where this statement comes from? It seems to be the crux of the proof, however, I don't see where it comes from.
Thank you in advance.
number-theory asymptotics
asked Jan 29 at 0:54
GnumbertesterGnumbertester
6701114
$endgroup$
add a comment |
$begingroup$
In this 1974 paper, Paul Erdos proves that the set of weird numbers has positive density in the natural numbers.
THEOREM 5. The density of weird numbers is positive .
Proof. If $n$ is weird, then let $epsilon_n$be as in the proof of the lemma. Now, by the
lemma, if $t$ is an integer and $a-(t)/t < 1 + epsilon_n$ then $nt$ is weird . But the density of the
integers $t$ with $Q(t)/t < 1 + epsilon_n$ is positive for any $epsilon_n > 0$.
Actually, we proved a slightly stronger result . If $n$ is weird, then the density of
{$m; n I m$ and $m$ is weird} is positive .
The lemma he is referring to in his proof is the following.
LEMMA. If $n$ is weird, then there is an $epsilon_n > 0$ such that $nt$ is weird if
$$sum_{d|t}{frac{1}{d}}<1+epsilon_n$$
I understand the heuristics behind this theorem, it is easy to make more weird numbers from multiples of other weird numbers, however, I am confused about the following statement.
But the density of the
integers $t$ with $Q(t)/t < 1 + epsilon_n$ is positive for any $epsilon_n > 0$.
Could someone show/explain to me where this statement comes from? It seems to be the crux of the proof, however, I don't see where it comes from.
Thank you in advance.
number-theory asymptotics
asked Jan 29 at 0:54
GnumbertesterGnumbertester
6701114
$endgroup$
In this 1974 paper, Paul Erdos proves that the set of weird numbers has positive density in the natural numbers.
THEOREM 5. The density of weird numbers is positive .
Proof. If $n$ is weird, then let $epsilon_n$be as in the proof of the lemma. Now, by the
lemma, if $t$ is an integer and $a-(t)/t < 1 + epsilon_n$ then $nt$ is weird . But the density of the
integers $t$ with $Q(t)/t < 1 + epsilon_n$ is positive for any $epsilon_n > 0$.
Actually, we proved a slightly stronger result . If $n$ is weird, then the density of
{$m; n I m$ and $m$ is weird} is positive .
The lemma he is referring to in his proof is the following.
LEMMA. If $n$ is weird, then there is an $epsilon_n > 0$ such that $nt$ is weird if
$$sum_{d|t}{frac{1}{d}}<1+epsilon_n$$
I understand the heuristics behind this theorem, it is easy to make more weird numbers from multiples of other weird numbers, however, I am confused about the following statement.
But the density of the
integers $t$ with $Q(t)/t < 1 + epsilon_n$ is positive for any $epsilon_n > 0$.
Could someone show/explain to me where this statement comes from? It seems to be the crux of the proof, however, I don't see where it comes from.
Thank you in advance.
number-theory asymptotics
number-theory asymptotics
asked Jan 29 at 0:54
GnumbertesterGnumbertester
6701114
asked Jan 29 at 0:54
GnumbertesterGnumbertester
6701114
edited Jan 29 at 1:03
Gnumbertester
asked Jan 29 at 0:54
GnumbertesterGnumbertester
6701114
asked Jan 29 at 0:54
GnumbertesterGnumbertester
6701114
asked Jan 29 at 0:54
GnumbertesterGnumbertester
6701114
6701114
add a comment |
add a comment |
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