Does the average primeness of natural numbers tend to zero?
$begingroup$
This question was posted in MSE. It got many upvotes but no answer hence posting it in MO.
A number is either prime or composite, hence primality is a binary concept. Instead I wanted to put a value of primality to every number using some function $f$ such that $f(n) = 1$ iff $n$ is a prime otherwise, $0 < f(n) < 1$ and as the number divisors of $n$ increases, $f(n)$ decreases on average. Thus $f(n)$ is a measure of the degree of primeness of $n$ where 1 is a perfect prime and 0 is a hypothetical perfect composite. Hence $frac{1}{N}sum_{r le N} f(r)$ can be interpreted as a the average primeness of the first $N$ integers.
After trying several definitions and going through the ones in literature, I came up with:
Define $f(n) = dfrac{2s_n}{n-1}$ for $n ge 2$, where $s_n$ is the
standard deviation of the divisors of $n$.
One reason for using standard deviation was that I was already studying the distribution of the divisors of a number.
Question 1: Does the average primeness tend to zero? i.e. does the following hold?
$$
lim_{N to infty} frac{1}{N}sum_{r = 2}^N f(r) = 0
$$
Question 2: Is $f(n)$ injective over composites? i.e., do there exist composites $3 < m < n$ such that $f(m) = f(n)$?
My progress
$f(4.35times 10^8) approx 0.5919$ and decreasing so the limit if it exists must be between 0 and 0.5919.- For $2 le i le n$, computed data shows that the minimum value of $f(i)$ occurs at the largest highly composite number $le n$.
Note: Here standard deviation of $x_1, x_2, ldots , x_n$ is defined as $sqrt frac{sum_{i=1}^{n} (x-x_i)^2}{n}$. Also notice that even if we define standard deviation as $sqrt frac{sum_{i=1}^{n} (x-x_i)^2}{n-1}$ our questions remain unaffected because in this case in the definition of $f$, we will be multiplying with $sqrt 2$ instead of $2$ to normalize $f$ in the interval $(0,1)$.
nt.number-theory real-analysis analytic-number-theory prime-numbers
$endgroup$
|
show 4 more comments
$begingroup$
This question was posted in MSE. It got many upvotes but no answer hence posting it in MO.
A number is either prime or composite, hence primality is a binary concept. Instead I wanted to put a value of primality to every number using some function $f$ such that $f(n) = 1$ iff $n$ is a prime otherwise, $0 < f(n) < 1$ and as the number divisors of $n$ increases, $f(n)$ decreases on average. Thus $f(n)$ is a measure of the degree of primeness of $n$ where 1 is a perfect prime and 0 is a hypothetical perfect composite. Hence $frac{1}{N}sum_{r le N} f(r)$ can be interpreted as a the average primeness of the first $N$ integers.
After trying several definitions and going through the ones in literature, I came up with:
Define $f(n) = dfrac{2s_n}{n-1}$ for $n ge 2$, where $s_n$ is the
standard deviation of the divisors of $n$.
One reason for using standard deviation was that I was already studying the distribution of the divisors of a number.
Question 1: Does the average primeness tend to zero? i.e. does the following hold?
$$
lim_{N to infty} frac{1}{N}sum_{r = 2}^N f(r) = 0
$$
Question 2: Is $f(n)$ injective over composites? i.e., do there exist composites $3 < m < n$ such that $f(m) = f(n)$?
My progress
$f(4.35times 10^8) approx 0.5919$ and decreasing so the limit if it exists must be between 0 and 0.5919.- For $2 le i le n$, computed data shows that the minimum value of $f(i)$ occurs at the largest highly composite number $le n$.
Note: Here standard deviation of $x_1, x_2, ldots , x_n$ is defined as $sqrt frac{sum_{i=1}^{n} (x-x_i)^2}{n}$. Also notice that even if we define standard deviation as $sqrt frac{sum_{i=1}^{n} (x-x_i)^2}{n-1}$ our questions remain unaffected because in this case in the definition of $f$, we will be multiplying with $sqrt 2$ instead of $2$ to normalize $f$ in the interval $(0,1)$.
nt.number-theory real-analysis analytic-number-theory prime-numbers
$endgroup$
$begingroup$
From the linked question it seems that $s_n$ grows faster than $n$ so that $f(n)$ doesn't go to zero.
$endgroup$
– lcv
2 days ago
1
$begingroup$
@lcv No $s_n$ doesn't grow faster than $n$. What are you looking at?
$endgroup$
– Nilos
2 days ago
1
$begingroup$
"...I wanted to have a continuous function...". In what topology is $f$ continuous? If you put discrete topology on natural numbers, then any function is continuous so you probably have something else in mind.
$endgroup$
– Aknazar Kazhymurat
2 days ago
$begingroup$
I have verified that $f$ is injective over composites less than 10,000,000.
$endgroup$
– Matt F.
2 days ago
$begingroup$
@AknazarKazhymurat I have reworded that line. Hope it is clearer now?
$endgroup$
– Nilos
2 days ago
|
show 4 more comments
$begingroup$
This question was posted in MSE. It got many upvotes but no answer hence posting it in MO.
A number is either prime or composite, hence primality is a binary concept. Instead I wanted to put a value of primality to every number using some function $f$ such that $f(n) = 1$ iff $n$ is a prime otherwise, $0 < f(n) < 1$ and as the number divisors of $n$ increases, $f(n)$ decreases on average. Thus $f(n)$ is a measure of the degree of primeness of $n$ where 1 is a perfect prime and 0 is a hypothetical perfect composite. Hence $frac{1}{N}sum_{r le N} f(r)$ can be interpreted as a the average primeness of the first $N$ integers.
After trying several definitions and going through the ones in literature, I came up with:
Define $f(n) = dfrac{2s_n}{n-1}$ for $n ge 2$, where $s_n$ is the
standard deviation of the divisors of $n$.
One reason for using standard deviation was that I was already studying the distribution of the divisors of a number.
Question 1: Does the average primeness tend to zero? i.e. does the following hold?
$$
lim_{N to infty} frac{1}{N}sum_{r = 2}^N f(r) = 0
$$
Question 2: Is $f(n)$ injective over composites? i.e., do there exist composites $3 < m < n$ such that $f(m) = f(n)$?
My progress
$f(4.35times 10^8) approx 0.5919$ and decreasing so the limit if it exists must be between 0 and 0.5919.- For $2 le i le n$, computed data shows that the minimum value of $f(i)$ occurs at the largest highly composite number $le n$.
Note: Here standard deviation of $x_1, x_2, ldots , x_n$ is defined as $sqrt frac{sum_{i=1}^{n} (x-x_i)^2}{n}$. Also notice that even if we define standard deviation as $sqrt frac{sum_{i=1}^{n} (x-x_i)^2}{n-1}$ our questions remain unaffected because in this case in the definition of $f$, we will be multiplying with $sqrt 2$ instead of $2$ to normalize $f$ in the interval $(0,1)$.
nt.number-theory real-analysis analytic-number-theory prime-numbers
$endgroup$
This question was posted in MSE. It got many upvotes but no answer hence posting it in MO.
A number is either prime or composite, hence primality is a binary concept. Instead I wanted to put a value of primality to every number using some function $f$ such that $f(n) = 1$ iff $n$ is a prime otherwise, $0 < f(n) < 1$ and as the number divisors of $n$ increases, $f(n)$ decreases on average. Thus $f(n)$ is a measure of the degree of primeness of $n$ where 1 is a perfect prime and 0 is a hypothetical perfect composite. Hence $frac{1}{N}sum_{r le N} f(r)$ can be interpreted as a the average primeness of the first $N$ integers.
After trying several definitions and going through the ones in literature, I came up with:
Define $f(n) = dfrac{2s_n}{n-1}$ for $n ge 2$, where $s_n$ is the
standard deviation of the divisors of $n$.
One reason for using standard deviation was that I was already studying the distribution of the divisors of a number.
Question 1: Does the average primeness tend to zero? i.e. does the following hold?
$$
lim_{N to infty} frac{1}{N}sum_{r = 2}^N f(r) = 0
$$
Question 2: Is $f(n)$ injective over composites? i.e., do there exist composites $3 < m < n$ such that $f(m) = f(n)$?
My progress
$f(4.35times 10^8) approx 0.5919$ and decreasing so the limit if it exists must be between 0 and 0.5919.- For $2 le i le n$, computed data shows that the minimum value of $f(i)$ occurs at the largest highly composite number $le n$.
Note: Here standard deviation of $x_1, x_2, ldots , x_n$ is defined as $sqrt frac{sum_{i=1}^{n} (x-x_i)^2}{n}$. Also notice that even if we define standard deviation as $sqrt frac{sum_{i=1}^{n} (x-x_i)^2}{n-1}$ our questions remain unaffected because in this case in the definition of $f$, we will be multiplying with $sqrt 2$ instead of $2$ to normalize $f$ in the interval $(0,1)$.
nt.number-theory real-analysis analytic-number-theory prime-numbers
nt.number-theory real-analysis analytic-number-theory prime-numbers
edited 2 days ago
Nilos
asked 2 days ago
NilosNilos
1,4711834
1,4711834
$begingroup$
From the linked question it seems that $s_n$ grows faster than $n$ so that $f(n)$ doesn't go to zero.
$endgroup$
– lcv
2 days ago
1
$begingroup$
@lcv No $s_n$ doesn't grow faster than $n$. What are you looking at?
$endgroup$
– Nilos
2 days ago
1
$begingroup$
"...I wanted to have a continuous function...". In what topology is $f$ continuous? If you put discrete topology on natural numbers, then any function is continuous so you probably have something else in mind.
$endgroup$
– Aknazar Kazhymurat
2 days ago
$begingroup$
I have verified that $f$ is injective over composites less than 10,000,000.
$endgroup$
– Matt F.
2 days ago
$begingroup$
@AknazarKazhymurat I have reworded that line. Hope it is clearer now?
$endgroup$
– Nilos
2 days ago
|
show 4 more comments
$begingroup$
From the linked question it seems that $s_n$ grows faster than $n$ so that $f(n)$ doesn't go to zero.
$endgroup$
– lcv
2 days ago
1
$begingroup$
@lcv No $s_n$ doesn't grow faster than $n$. What are you looking at?
$endgroup$
– Nilos
2 days ago
1
$begingroup$
"...I wanted to have a continuous function...". In what topology is $f$ continuous? If you put discrete topology on natural numbers, then any function is continuous so you probably have something else in mind.
$endgroup$
– Aknazar Kazhymurat
2 days ago
$begingroup$
I have verified that $f$ is injective over composites less than 10,000,000.
$endgroup$
– Matt F.
2 days ago
$begingroup$
@AknazarKazhymurat I have reworded that line. Hope it is clearer now?
$endgroup$
– Nilos
2 days ago
$begingroup$
From the linked question it seems that $s_n$ grows faster than $n$ so that $f(n)$ doesn't go to zero.
$endgroup$
– lcv
2 days ago
$begingroup$
From the linked question it seems that $s_n$ grows faster than $n$ so that $f(n)$ doesn't go to zero.
$endgroup$
– lcv
2 days ago
1
1
$begingroup$
@lcv No $s_n$ doesn't grow faster than $n$. What are you looking at?
$endgroup$
– Nilos
2 days ago
$begingroup$
@lcv No $s_n$ doesn't grow faster than $n$. What are you looking at?
$endgroup$
– Nilos
2 days ago
1
1
$begingroup$
"...I wanted to have a continuous function...". In what topology is $f$ continuous? If you put discrete topology on natural numbers, then any function is continuous so you probably have something else in mind.
$endgroup$
– Aknazar Kazhymurat
2 days ago
$begingroup$
"...I wanted to have a continuous function...". In what topology is $f$ continuous? If you put discrete topology on natural numbers, then any function is continuous so you probably have something else in mind.
$endgroup$
– Aknazar Kazhymurat
2 days ago
$begingroup$
I have verified that $f$ is injective over composites less than 10,000,000.
$endgroup$
– Matt F.
2 days ago
$begingroup$
I have verified that $f$ is injective over composites less than 10,000,000.
$endgroup$
– Matt F.
2 days ago
$begingroup$
@AknazarKazhymurat I have reworded that line. Hope it is clearer now?
$endgroup$
– Nilos
2 days ago
$begingroup$
@AknazarKazhymurat I have reworded that line. Hope it is clearer now?
$endgroup$
– Nilos
2 days ago
|
show 4 more comments
1 Answer
1
active
oldest
votes
$begingroup$
The answer to Question 1 is "yes".
To see this, notice that $s_n$ is at most square root of the average square of divisor, i.e.
$$
s_nleq sqrt{frac{sum_{dmid n}d^2}{sum_{dmid n} 1}}=sqrt{frac{sigma_2(n)}{sigma_0(n)}},
$$
where $sigma_k(n)$ is the sum of $k$-th powers of divisors of $n$. Now,
$$
sigma_2(n)=n^2sigma_{-2}(n),
$$
so
$$
sigma_2(n)<frac{pi^2}{6}n^2
$$
for all $n$. Therefore we have
$$
f(n)leq frac{2}{n-1} sqrt{frac{pi^2}{6}n^2/sigma_0(n)}leq frac{5.14}{sqrt{sigma_0(n)}}
$$
for all $n$. Now, almost all $nleq N$ have at least $0.5lnln N$ distinct prime factors. In particular, for almost all $nleq N$ we have $sigma_0(n)geq 0.5lnln N$. Therefore, our bound for $f(n)$ together with the trivial observation that $0leq f(n)leq 1$ gives
$$
sum_{nleq N} f(n)leq sum_{nleq N, sigma_0(n)geq 0.5lnln N} frac{5.14}{sqrt{sigma_0(n)}}+sum_{nleq N, sigma_0(n)<0.5lnln N} 1= o(N),
$$
as needed.
Using contour integration method one can even prove something like
$$
sum_{nleq N} f(n)=O(N(ln N)^{1/sqrt{2}-1})
$$
$endgroup$
5
$begingroup$
That last expression reminds me an XKCD alt-text: "If you ever find yourself raising $log(text{anything})^{1/sqrt{2}}$, set down the marker and back away from the whiteboard; something has gone horribly wrong."
$endgroup$
– Michael Seifert
2 days ago
2
$begingroup$
@MichaelSeifert Bah, it was $log(anything)^e$ not $frac{1}{sqrt{2}}$. Log to the power of one over the sqrt of 2 is mundane; log of something to the power e is a sign of insanity.
$endgroup$
– Yakk
2 days ago
1
$begingroup$
@Yakk Yeah, but Randall also says that taking $pi$-th root of anything is insane. However, the paper "Mean values of multiplicative functions" by Montgomery and Vaughan, Theorem 5, contains $(log x)^{1/pi-1}$ and is totally fine! (P.S. Is inequality $n_p<p^{frac{1}{4sqrt{e}}+o(1)}$ ok?..)
$endgroup$
– Asymptotiac K
2 days ago
1
$begingroup$
@AsymptotiacK: Thanks for the good answer to Question 1. I think that in general if $f(n)$ is any function who value decreases from 1 to zero as its defined measure of primness decreases then the mean value of $f$ must tend to zero because as we go higher up the number line, for any $k ge 2$, the probability of of finding numbers with $le k$ factors should decrease
$endgroup$
– Nilotpal Kanti Sinha
yesterday
add a comment |
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1 Answer
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1 Answer
1
active
oldest
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active
oldest
votes
active
oldest
votes
$begingroup$
The answer to Question 1 is "yes".
To see this, notice that $s_n$ is at most square root of the average square of divisor, i.e.
$$
s_nleq sqrt{frac{sum_{dmid n}d^2}{sum_{dmid n} 1}}=sqrt{frac{sigma_2(n)}{sigma_0(n)}},
$$
where $sigma_k(n)$ is the sum of $k$-th powers of divisors of $n$. Now,
$$
sigma_2(n)=n^2sigma_{-2}(n),
$$
so
$$
sigma_2(n)<frac{pi^2}{6}n^2
$$
for all $n$. Therefore we have
$$
f(n)leq frac{2}{n-1} sqrt{frac{pi^2}{6}n^2/sigma_0(n)}leq frac{5.14}{sqrt{sigma_0(n)}}
$$
for all $n$. Now, almost all $nleq N$ have at least $0.5lnln N$ distinct prime factors. In particular, for almost all $nleq N$ we have $sigma_0(n)geq 0.5lnln N$. Therefore, our bound for $f(n)$ together with the trivial observation that $0leq f(n)leq 1$ gives
$$
sum_{nleq N} f(n)leq sum_{nleq N, sigma_0(n)geq 0.5lnln N} frac{5.14}{sqrt{sigma_0(n)}}+sum_{nleq N, sigma_0(n)<0.5lnln N} 1= o(N),
$$
as needed.
Using contour integration method one can even prove something like
$$
sum_{nleq N} f(n)=O(N(ln N)^{1/sqrt{2}-1})
$$
$endgroup$
5
$begingroup$
That last expression reminds me an XKCD alt-text: "If you ever find yourself raising $log(text{anything})^{1/sqrt{2}}$, set down the marker and back away from the whiteboard; something has gone horribly wrong."
$endgroup$
– Michael Seifert
2 days ago
2
$begingroup$
@MichaelSeifert Bah, it was $log(anything)^e$ not $frac{1}{sqrt{2}}$. Log to the power of one over the sqrt of 2 is mundane; log of something to the power e is a sign of insanity.
$endgroup$
– Yakk
2 days ago
1
$begingroup$
@Yakk Yeah, but Randall also says that taking $pi$-th root of anything is insane. However, the paper "Mean values of multiplicative functions" by Montgomery and Vaughan, Theorem 5, contains $(log x)^{1/pi-1}$ and is totally fine! (P.S. Is inequality $n_p<p^{frac{1}{4sqrt{e}}+o(1)}$ ok?..)
$endgroup$
– Asymptotiac K
2 days ago
1
$begingroup$
@AsymptotiacK: Thanks for the good answer to Question 1. I think that in general if $f(n)$ is any function who value decreases from 1 to zero as its defined measure of primness decreases then the mean value of $f$ must tend to zero because as we go higher up the number line, for any $k ge 2$, the probability of of finding numbers with $le k$ factors should decrease
$endgroup$
– Nilotpal Kanti Sinha
yesterday
add a comment |
$begingroup$
The answer to Question 1 is "yes".
To see this, notice that $s_n$ is at most square root of the average square of divisor, i.e.
$$
s_nleq sqrt{frac{sum_{dmid n}d^2}{sum_{dmid n} 1}}=sqrt{frac{sigma_2(n)}{sigma_0(n)}},
$$
where $sigma_k(n)$ is the sum of $k$-th powers of divisors of $n$. Now,
$$
sigma_2(n)=n^2sigma_{-2}(n),
$$
so
$$
sigma_2(n)<frac{pi^2}{6}n^2
$$
for all $n$. Therefore we have
$$
f(n)leq frac{2}{n-1} sqrt{frac{pi^2}{6}n^2/sigma_0(n)}leq frac{5.14}{sqrt{sigma_0(n)}}
$$
for all $n$. Now, almost all $nleq N$ have at least $0.5lnln N$ distinct prime factors. In particular, for almost all $nleq N$ we have $sigma_0(n)geq 0.5lnln N$. Therefore, our bound for $f(n)$ together with the trivial observation that $0leq f(n)leq 1$ gives
$$
sum_{nleq N} f(n)leq sum_{nleq N, sigma_0(n)geq 0.5lnln N} frac{5.14}{sqrt{sigma_0(n)}}+sum_{nleq N, sigma_0(n)<0.5lnln N} 1= o(N),
$$
as needed.
Using contour integration method one can even prove something like
$$
sum_{nleq N} f(n)=O(N(ln N)^{1/sqrt{2}-1})
$$
$endgroup$
5
$begingroup$
That last expression reminds me an XKCD alt-text: "If you ever find yourself raising $log(text{anything})^{1/sqrt{2}}$, set down the marker and back away from the whiteboard; something has gone horribly wrong."
$endgroup$
– Michael Seifert
2 days ago
2
$begingroup$
@MichaelSeifert Bah, it was $log(anything)^e$ not $frac{1}{sqrt{2}}$. Log to the power of one over the sqrt of 2 is mundane; log of something to the power e is a sign of insanity.
$endgroup$
– Yakk
2 days ago
1
$begingroup$
@Yakk Yeah, but Randall also says that taking $pi$-th root of anything is insane. However, the paper "Mean values of multiplicative functions" by Montgomery and Vaughan, Theorem 5, contains $(log x)^{1/pi-1}$ and is totally fine! (P.S. Is inequality $n_p<p^{frac{1}{4sqrt{e}}+o(1)}$ ok?..)
$endgroup$
– Asymptotiac K
2 days ago
1
$begingroup$
@AsymptotiacK: Thanks for the good answer to Question 1. I think that in general if $f(n)$ is any function who value decreases from 1 to zero as its defined measure of primness decreases then the mean value of $f$ must tend to zero because as we go higher up the number line, for any $k ge 2$, the probability of of finding numbers with $le k$ factors should decrease
$endgroup$
– Nilotpal Kanti Sinha
yesterday
add a comment |
$begingroup$
The answer to Question 1 is "yes".
To see this, notice that $s_n$ is at most square root of the average square of divisor, i.e.
$$
s_nleq sqrt{frac{sum_{dmid n}d^2}{sum_{dmid n} 1}}=sqrt{frac{sigma_2(n)}{sigma_0(n)}},
$$
where $sigma_k(n)$ is the sum of $k$-th powers of divisors of $n$. Now,
$$
sigma_2(n)=n^2sigma_{-2}(n),
$$
so
$$
sigma_2(n)<frac{pi^2}{6}n^2
$$
for all $n$. Therefore we have
$$
f(n)leq frac{2}{n-1} sqrt{frac{pi^2}{6}n^2/sigma_0(n)}leq frac{5.14}{sqrt{sigma_0(n)}}
$$
for all $n$. Now, almost all $nleq N$ have at least $0.5lnln N$ distinct prime factors. In particular, for almost all $nleq N$ we have $sigma_0(n)geq 0.5lnln N$. Therefore, our bound for $f(n)$ together with the trivial observation that $0leq f(n)leq 1$ gives
$$
sum_{nleq N} f(n)leq sum_{nleq N, sigma_0(n)geq 0.5lnln N} frac{5.14}{sqrt{sigma_0(n)}}+sum_{nleq N, sigma_0(n)<0.5lnln N} 1= o(N),
$$
as needed.
Using contour integration method one can even prove something like
$$
sum_{nleq N} f(n)=O(N(ln N)^{1/sqrt{2}-1})
$$
$endgroup$
The answer to Question 1 is "yes".
To see this, notice that $s_n$ is at most square root of the average square of divisor, i.e.
$$
s_nleq sqrt{frac{sum_{dmid n}d^2}{sum_{dmid n} 1}}=sqrt{frac{sigma_2(n)}{sigma_0(n)}},
$$
where $sigma_k(n)$ is the sum of $k$-th powers of divisors of $n$. Now,
$$
sigma_2(n)=n^2sigma_{-2}(n),
$$
so
$$
sigma_2(n)<frac{pi^2}{6}n^2
$$
for all $n$. Therefore we have
$$
f(n)leq frac{2}{n-1} sqrt{frac{pi^2}{6}n^2/sigma_0(n)}leq frac{5.14}{sqrt{sigma_0(n)}}
$$
for all $n$. Now, almost all $nleq N$ have at least $0.5lnln N$ distinct prime factors. In particular, for almost all $nleq N$ we have $sigma_0(n)geq 0.5lnln N$. Therefore, our bound for $f(n)$ together with the trivial observation that $0leq f(n)leq 1$ gives
$$
sum_{nleq N} f(n)leq sum_{nleq N, sigma_0(n)geq 0.5lnln N} frac{5.14}{sqrt{sigma_0(n)}}+sum_{nleq N, sigma_0(n)<0.5lnln N} 1= o(N),
$$
as needed.
Using contour integration method one can even prove something like
$$
sum_{nleq N} f(n)=O(N(ln N)^{1/sqrt{2}-1})
$$
edited 2 days ago
answered 2 days ago
Asymptotiac KAsymptotiac K
1,5941314
1,5941314
5
$begingroup$
That last expression reminds me an XKCD alt-text: "If you ever find yourself raising $log(text{anything})^{1/sqrt{2}}$, set down the marker and back away from the whiteboard; something has gone horribly wrong."
$endgroup$
– Michael Seifert
2 days ago
2
$begingroup$
@MichaelSeifert Bah, it was $log(anything)^e$ not $frac{1}{sqrt{2}}$. Log to the power of one over the sqrt of 2 is mundane; log of something to the power e is a sign of insanity.
$endgroup$
– Yakk
2 days ago
1
$begingroup$
@Yakk Yeah, but Randall also says that taking $pi$-th root of anything is insane. However, the paper "Mean values of multiplicative functions" by Montgomery and Vaughan, Theorem 5, contains $(log x)^{1/pi-1}$ and is totally fine! (P.S. Is inequality $n_p<p^{frac{1}{4sqrt{e}}+o(1)}$ ok?..)
$endgroup$
– Asymptotiac K
2 days ago
1
$begingroup$
@AsymptotiacK: Thanks for the good answer to Question 1. I think that in general if $f(n)$ is any function who value decreases from 1 to zero as its defined measure of primness decreases then the mean value of $f$ must tend to zero because as we go higher up the number line, for any $k ge 2$, the probability of of finding numbers with $le k$ factors should decrease
$endgroup$
– Nilotpal Kanti Sinha
yesterday
add a comment |
5
$begingroup$
That last expression reminds me an XKCD alt-text: "If you ever find yourself raising $log(text{anything})^{1/sqrt{2}}$, set down the marker and back away from the whiteboard; something has gone horribly wrong."
$endgroup$
– Michael Seifert
2 days ago
2
$begingroup$
@MichaelSeifert Bah, it was $log(anything)^e$ not $frac{1}{sqrt{2}}$. Log to the power of one over the sqrt of 2 is mundane; log of something to the power e is a sign of insanity.
$endgroup$
– Yakk
2 days ago
1
$begingroup$
@Yakk Yeah, but Randall also says that taking $pi$-th root of anything is insane. However, the paper "Mean values of multiplicative functions" by Montgomery and Vaughan, Theorem 5, contains $(log x)^{1/pi-1}$ and is totally fine! (P.S. Is inequality $n_p<p^{frac{1}{4sqrt{e}}+o(1)}$ ok?..)
$endgroup$
– Asymptotiac K
2 days ago
1
$begingroup$
@AsymptotiacK: Thanks for the good answer to Question 1. I think that in general if $f(n)$ is any function who value decreases from 1 to zero as its defined measure of primness decreases then the mean value of $f$ must tend to zero because as we go higher up the number line, for any $k ge 2$, the probability of of finding numbers with $le k$ factors should decrease
$endgroup$
– Nilotpal Kanti Sinha
yesterday
5
5
$begingroup$
That last expression reminds me an XKCD alt-text: "If you ever find yourself raising $log(text{anything})^{1/sqrt{2}}$, set down the marker and back away from the whiteboard; something has gone horribly wrong."
$endgroup$
– Michael Seifert
2 days ago
$begingroup$
That last expression reminds me an XKCD alt-text: "If you ever find yourself raising $log(text{anything})^{1/sqrt{2}}$, set down the marker and back away from the whiteboard; something has gone horribly wrong."
$endgroup$
– Michael Seifert
2 days ago
2
2
$begingroup$
@MichaelSeifert Bah, it was $log(anything)^e$ not $frac{1}{sqrt{2}}$. Log to the power of one over the sqrt of 2 is mundane; log of something to the power e is a sign of insanity.
$endgroup$
– Yakk
2 days ago
$begingroup$
@MichaelSeifert Bah, it was $log(anything)^e$ not $frac{1}{sqrt{2}}$. Log to the power of one over the sqrt of 2 is mundane; log of something to the power e is a sign of insanity.
$endgroup$
– Yakk
2 days ago
1
1
$begingroup$
@Yakk Yeah, but Randall also says that taking $pi$-th root of anything is insane. However, the paper "Mean values of multiplicative functions" by Montgomery and Vaughan, Theorem 5, contains $(log x)^{1/pi-1}$ and is totally fine! (P.S. Is inequality $n_p<p^{frac{1}{4sqrt{e}}+o(1)}$ ok?..)
$endgroup$
– Asymptotiac K
2 days ago
$begingroup$
@Yakk Yeah, but Randall also says that taking $pi$-th root of anything is insane. However, the paper "Mean values of multiplicative functions" by Montgomery and Vaughan, Theorem 5, contains $(log x)^{1/pi-1}$ and is totally fine! (P.S. Is inequality $n_p<p^{frac{1}{4sqrt{e}}+o(1)}$ ok?..)
$endgroup$
– Asymptotiac K
2 days ago
1
1
$begingroup$
@AsymptotiacK: Thanks for the good answer to Question 1. I think that in general if $f(n)$ is any function who value decreases from 1 to zero as its defined measure of primness decreases then the mean value of $f$ must tend to zero because as we go higher up the number line, for any $k ge 2$, the probability of of finding numbers with $le k$ factors should decrease
$endgroup$
– Nilotpal Kanti Sinha
yesterday
$begingroup$
@AsymptotiacK: Thanks for the good answer to Question 1. I think that in general if $f(n)$ is any function who value decreases from 1 to zero as its defined measure of primness decreases then the mean value of $f$ must tend to zero because as we go higher up the number line, for any $k ge 2$, the probability of of finding numbers with $le k$ factors should decrease
$endgroup$
– Nilotpal Kanti Sinha
yesterday
add a comment |
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$begingroup$
From the linked question it seems that $s_n$ grows faster than $n$ so that $f(n)$ doesn't go to zero.
$endgroup$
– lcv
2 days ago
1
$begingroup$
@lcv No $s_n$ doesn't grow faster than $n$. What are you looking at?
$endgroup$
– Nilos
2 days ago
1
$begingroup$
"...I wanted to have a continuous function...". In what topology is $f$ continuous? If you put discrete topology on natural numbers, then any function is continuous so you probably have something else in mind.
$endgroup$
– Aknazar Kazhymurat
2 days ago
$begingroup$
I have verified that $f$ is injective over composites less than 10,000,000.
$endgroup$
– Matt F.
2 days ago
$begingroup$
@AknazarKazhymurat I have reworded that line. Hope it is clearer now?
$endgroup$
– Nilos
2 days ago