Mixing
Sum of log over friables
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Let $x$ and $y$ be two positive real numbers.
What is the mean value of the function $log$ on $y$ friables integers less than $x$ i.e the value of the following sum
$$sum_substackn leq x \ P(n)leq y log(n),$$
where $P(n)$ is the greatest prime factor of $n.$
Thanks in advance.
nt.number-theory analytic-number-theory
asked Aug 9 at 9:44
Khadija Mbarki
674312
add a comment |Â
up vote
1
down vote
favorite
Let $x$ and $y$ be two positive real numbers.
What is the mean value of the function $log$ on $y$ friables integers less than $x$ i.e the value of the following sum
$$sum_substackn leq x \ P(n)leq y log(n),$$
where $P(n)$ is the greatest prime factor of $n.$
Thanks in advance.
nt.number-theory analytic-number-theory
asked Aug 9 at 9:44
Khadija Mbarki
674312
2
It should be close to $K(y)xlog(x/e)$ with $K(y)$ a version of Dickman's constant. (Maybe $K(log y/log x)$ is more standard.) You might get some improvement by a clever pairing of smooth numbers, say log(m)d(m)/2 for m a certain smooth number m larger than x. This might get you most of the sum, and then you can approximate the rest with a small multiple of log x. Gerhard "Or Try Two Large Smooths" Paseman, 2018.08.09.
– Gerhard Paseman
Aug 9 at 10:25
2
I'd never heard the term 'friable' for integers before. Is 'smooth' used interchangeably with 'friable'? (Perhaps it's a matter of linguistic background, since the paper you cite below is in French?)
– LSpice
Aug 9 at 13:24
3
@LSpice: Some people prefer friable, as the word smooth already carries too many different meanings. For others friable sounds too much like fryable.
– Jan-Christoph Schlage-Puchta
Aug 9 at 15:08
1
@Jan-ChristophSchlage-Puchta, well, there’s no denying that ‘friable’ is a much mathematically rarer term (as well as being quite apposite). In my field, we have the very inventive term ‘good’. :-)
– LSpice
Aug 9 at 18:18
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $x$ and $y$ be two positive real numbers.
What is the mean value of the function $log$ on $y$ friables integers less than $x$ i.e the value of the following sum
$$sum_substackn leq x \ P(n)leq y log(n),$$
where $P(n)$ is the greatest prime factor of $n.$
Thanks in advance.
nt.number-theory analytic-number-theory
asked Aug 9 at 9:44
Khadija Mbarki
674312
Let $x$ and $y$ be two positive real numbers.
What is the mean value of the function $log$ on $y$ friables integers less than $x$ i.e the value of the following sum
$$sum_substackn leq x \ P(n)leq y log(n),$$
where $P(n)$ is the greatest prime factor of $n.$
Thanks in advance.
nt.number-theory analytic-number-theory
asked Aug 9 at 9:44
Khadija Mbarki
674312
asked Aug 9 at 9:44
Khadija Mbarki
674312
asked Aug 9 at 9:44
Khadija Mbarki
674312
asked Aug 9 at 9:44
Khadija Mbarki
674312
674312
2
It should be close to $K(y)xlog(x/e)$ with $K(y)$ a version of Dickman's constant. (Maybe $K(log y/log x)$ is more standard.) You might get some improvement by a clever pairing of smooth numbers, say log(m)d(m)/2 for m a certain smooth number m larger than x. This might get you most of the sum, and then you can approximate the rest with a small multiple of log x. Gerhard "Or Try Two Large Smooths" Paseman, 2018.08.09.
– Gerhard Paseman
Aug 9 at 10:25
2
I'd never heard the term 'friable' for integers before. Is 'smooth' used interchangeably with 'friable'? (Perhaps it's a matter of linguistic background, since the paper you cite below is in French?)
– LSpice
Aug 9 at 13:24
3
@LSpice: Some people prefer friable, as the word smooth already carries too many different meanings. For others friable sounds too much like fryable.
– Jan-Christoph Schlage-Puchta
Aug 9 at 15:08
1
@Jan-ChristophSchlage-Puchta, well, there’s no denying that ‘friable’ is a much mathematically rarer term (as well as being quite apposite). In my field, we have the very inventive term ‘good’. :-)
– LSpice
Aug 9 at 18:18
add a comment |Â
2
It should be close to $K(y)xlog(x/e)$ with $K(y)$ a version of Dickman's constant. (Maybe $K(log y/log x)$ is more standard.) You might get some improvement by a clever pairing of smooth numbers, say log(m)d(m)/2 for m a certain smooth number m larger than x. This might get you most of the sum, and then you can approximate the rest with a small multiple of log x. Gerhard "Or Try Two Large Smooths" Paseman, 2018.08.09.
– Gerhard Paseman
Aug 9 at 10:25
2
I'd never heard the term 'friable' for integers before. Is 'smooth' used interchangeably with 'friable'? (Perhaps it's a matter of linguistic background, since the paper you cite below is in French?)
– LSpice
Aug 9 at 13:24
3
@LSpice: Some people prefer friable, as the word smooth already carries too many different meanings. For others friable sounds too much like fryable.
– Jan-Christoph Schlage-Puchta
Aug 9 at 15:08
1
@Jan-ChristophSchlage-Puchta, well, there’s no denying that ‘friable’ is a much mathematically rarer term (as well as being quite apposite). In my field, we have the very inventive term ‘good’. :-)
– LSpice
Aug 9 at 18:18
2
2
It should be close to $K(y)xlog(x/e)$ with $K(y)$ a version of Dickman's constant. (Maybe $K(log y/log x)$ is more standard.) You might get some improvement by a clever pairing of smooth numbers, say log(m)d(m)/2 for m a certain smooth number m larger than x. This might get you most of the sum, and then you can approximate the rest with a small multiple of log x. Gerhard "Or Try Two Large Smooths" Paseman, 2018.08.09.
– Gerhard Paseman
Aug 9 at 10:25
It should be close to $K(y)xlog(x/e)$ with $K(y)$ a version of Dickman's constant. (Maybe $K(log y/log x)$ is more standard.) You might get some improvement by a clever pairing of smooth numbers, say log(m)d(m)/2 for m a certain smooth number m larger than x. This might get you most of the sum, and then you can approximate the rest with a small multiple of log x. Gerhard "Or Try Two Large Smooths" Paseman, 2018.08.09.
– Gerhard Paseman
Aug 9 at 10:25
2
2
I'd never heard the term 'friable' for integers before. Is 'smooth' used interchangeably with 'friable'? (Perhaps it's a matter of linguistic background, since the paper you cite below is in French?)
– LSpice
Aug 9 at 13:24
I'd never heard the term 'friable' for integers before. Is 'smooth' used interchangeably with 'friable'? (Perhaps it's a matter of linguistic background, since the paper you cite below is in French?)
– LSpice
Aug 9 at 13:24
3
3
@LSpice: Some people prefer friable, as the word smooth already carries too many different meanings. For others friable sounds too much like fryable.
– Jan-Christoph Schlage-Puchta
Aug 9 at 15:08
@LSpice: Some people prefer friable, as the word smooth already carries too many different meanings. For others friable sounds too much like fryable.
– Jan-Christoph Schlage-Puchta
Aug 9 at 15:08
1
1
@Jan-ChristophSchlage-Puchta, well, there’s no denying that ‘friable’ is a much mathematically rarer term (as well as being quite apposite). In my field, we have the very inventive term ‘good’. :-)
– LSpice
Aug 9 at 18:18
@Jan-ChristophSchlage-Puchta, well, there’s no denying that ‘friable’ is a much mathematically rarer term (as well as being quite apposite). In my field, we have the very inventive term ‘good’. :-)
– LSpice
Aug 9 at 18:18
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
7
down vote
De La Breteche and Tenenbaum established in their paper 'Propriétés statistiques des entiers friables' the asymptotic formula for the above sum. They find it as a corollary; Uniformly for $2 leq y leq x,$ we have
$$sum_substackn leq x \ P(n)leq y log(n)=leftlog(x)-fraclog(y)+O(loglog(y))logleft(1+fracylog(x)right) rightPsi(x,y),$$
where $Psi(x,y)$ is the number of friables integers less than $x$ whose greatest prime factors are less than $y.$
answered Aug 9 at 13:17
Khadija Mbarki
674312
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
7
down vote
De La Breteche and Tenenbaum established in their paper 'Propriétés statistiques des entiers friables' the asymptotic formula for the above sum. They find it as a corollary; Uniformly for $2 leq y leq x,$ we have
$$sum_substackn leq x \ P(n)leq y log(n)=leftlog(x)-fraclog(y)+O(loglog(y))logleft(1+fracylog(x)right) rightPsi(x,y),$$
where $Psi(x,y)$ is the number of friables integers less than $x$ whose greatest prime factors are less than $y.$
answered Aug 9 at 13:17
Khadija Mbarki
674312
add a comment |Â
up vote
7
down vote
De La Breteche and Tenenbaum established in their paper 'Propriétés statistiques des entiers friables' the asymptotic formula for the above sum. They find it as a corollary; Uniformly for $2 leq y leq x,$ we have
$$sum_substackn leq x \ P(n)leq y log(n)=leftlog(x)-fraclog(y)+O(loglog(y))logleft(1+fracylog(x)right) rightPsi(x,y),$$
where $Psi(x,y)$ is the number of friables integers less than $x$ whose greatest prime factors are less than $y.$
answered Aug 9 at 13:17
Khadija Mbarki
674312
add a comment |Â
up vote
7
down vote
up vote
7
down vote
De La Breteche and Tenenbaum established in their paper 'Propriétés statistiques des entiers friables' the asymptotic formula for the above sum. They find it as a corollary; Uniformly for $2 leq y leq x,$ we have
$$sum_substackn leq x \ P(n)leq y log(n)=leftlog(x)-fraclog(y)+O(loglog(y))logleft(1+fracylog(x)right) rightPsi(x,y),$$
where $Psi(x,y)$ is the number of friables integers less than $x$ whose greatest prime factors are less than $y.$
answered Aug 9 at 13:17
Khadija Mbarki
674312
De La Breteche and Tenenbaum established in their paper 'Propriétés statistiques des entiers friables' the asymptotic formula for the above sum. They find it as a corollary; Uniformly for $2 leq y leq x,$ we have
$$sum_substackn leq x \ P(n)leq y log(n)=leftlog(x)-fraclog(y)+O(loglog(y))logleft(1+fracylog(x)right) rightPsi(x,y),$$
where $Psi(x,y)$ is the number of friables integers less than $x$ whose greatest prime factors are less than $y.$
answered Aug 9 at 13:17
Khadija Mbarki
674312
answered Aug 9 at 13:17
Khadija Mbarki
674312
answered Aug 9 at 13:17
Khadija Mbarki
674312
answered Aug 9 at 13:17
Khadija Mbarki
674312
674312
add a comment |Â
add a comment |Â
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2
It should be close to $K(y)xlog(x/e)$ with $K(y)$ a version of Dickman's constant. (Maybe $K(log y/log x)$ is more standard.) You might get some improvement by a clever pairing of smooth numbers, say log(m)d(m)/2 for m a certain smooth number m larger than x. This might get you most of the sum, and then you can approximate the rest with a small multiple of log x. Gerhard "Or Try Two Large Smooths" Paseman, 2018.08.09.
– Gerhard Paseman
Aug 9 at 10:25
2
I'd never heard the term 'friable' for integers before. Is 'smooth' used interchangeably with 'friable'? (Perhaps it's a matter of linguistic background, since the paper you cite below is in French?)
– LSpice
Aug 9 at 13:24
3
@LSpice: Some people prefer friable, as the word smooth already carries too many different meanings. For others friable sounds too much like fryable.
– Jan-Christoph Schlage-Puchta
Aug 9 at 15:08
1
@Jan-ChristophSchlage-Puchta, well, there’s no denying that ‘friable’ is a much mathematically rarer term (as well as being quite apposite). In my field, we have the very inventive term ‘good’. :-)
– LSpice
Aug 9 at 18:18