Density of twin square-free numbersare there infinitely many triples of consecutive square-free...
Density of twin square-free numbers
are there infinitely many triples of consecutive square-free integers?Squarefree numbers $n$ such that $432n+1$ is also squarefreeorthogonality relation for quadratic Dirichlet charactersInfinite sets of primes of density 0A pair of subset of natural numbers having density, but whose intersection has no densityDensity of numbers whose prime factors all come from a fixed congruence classThe density of square-free integers represented by a cubic polynomialIf the natural density (relative to the primes) exists, then the Dirichlet density also exists, and the two are equalExistence of relative Dirichlet density of primes starting with 1Density of integers with many prime factorsGrowth Rate of the Square-Free PartNumber of $k$-free integers of bounded radical
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It is well-known how to compute the density of square-free numbers, to get
$$ lim_{Ntoinfty} frac{#{ n leq N : n text{ square-free}}}{N} = frac{6}{pi^2}.$$
What is the density of numbers such that both $n$ and $n+1$ are square-free?
In other words, what is
$$lim_{Ntoinfty} frac{#{ n leq N : n(n+1) text{ square-free}}}{N} $$
(if the limit exists)?
I'm guessing this has been studied before. Does anyone have a textbook or paper reference?
nt.number-theory reference-request analytic-number-theory
asked 8 hours ago
Harry RichmanHarry Richman
9886 silver badges18 bronze badges
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add a comment |
$begingroup$
It is well-known how to compute the density of square-free numbers, to get
$$ lim_{Ntoinfty} frac{#{ n leq N : n text{ square-free}}}{N} = frac{6}{pi^2}.$$
What is the density of numbers such that both $n$ and $n+1$ are square-free?
In other words, what is
$$lim_{Ntoinfty} frac{#{ n leq N : n(n+1) text{ square-free}}}{N} $$
(if the limit exists)?
I'm guessing this has been studied before. Does anyone have a textbook or paper reference?
nt.number-theory reference-request analytic-number-theory
asked 8 hours ago
Harry RichmanHarry Richman
9886 silver badges18 bronze badges
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1
$begingroup$
This question is essentially the same as mathoverflow.net/questions/177849/… See my answer there.
$endgroup$
– GH from MO
7 hours ago
1
$begingroup$
Also essentially a duplicate of MO 59741 <mathoverflow.net/questions/59741> which asked the same question about squarefree triples $(4a+1,4a+2,4a+3)$.
$endgroup$
– Noam D. Elkies
6 hours ago
add a comment |
$begingroup$
It is well-known how to compute the density of square-free numbers, to get
$$ lim_{Ntoinfty} frac{#{ n leq N : n text{ square-free}}}{N} = frac{6}{pi^2}.$$
What is the density of numbers such that both $n$ and $n+1$ are square-free?
In other words, what is
$$lim_{Ntoinfty} frac{#{ n leq N : n(n+1) text{ square-free}}}{N} $$
(if the limit exists)?
I'm guessing this has been studied before. Does anyone have a textbook or paper reference?
nt.number-theory reference-request analytic-number-theory
asked 8 hours ago
Harry RichmanHarry Richman
9886 silver badges18 bronze badges
$endgroup$
It is well-known how to compute the density of square-free numbers, to get
$$ lim_{Ntoinfty} frac{#{ n leq N : n text{ square-free}}}{N} = frac{6}{pi^2}.$$
What is the density of numbers such that both $n$ and $n+1$ are square-free?
In other words, what is
$$lim_{Ntoinfty} frac{#{ n leq N : n(n+1) text{ square-free}}}{N} $$
(if the limit exists)?
I'm guessing this has been studied before. Does anyone have a textbook or paper reference?
nt.number-theory reference-request analytic-number-theory
nt.number-theory reference-request analytic-number-theory
asked 8 hours ago
Harry RichmanHarry Richman
9886 silver badges18 bronze badges
asked 8 hours ago
Harry RichmanHarry Richman
9886 silver badges18 bronze badges
asked 8 hours ago
Harry RichmanHarry Richman
9886 silver badges18 bronze badges
asked 8 hours ago
Harry RichmanHarry Richman
9886 silver badges18 bronze badges
asked 8 hours ago
Harry RichmanHarry Richman
9886 silver badges18 bronze badges
9886 silver badges18 bronze badges
1
$begingroup$
This question is essentially the same as mathoverflow.net/questions/177849/… See my answer there.
$endgroup$
– GH from MO
7 hours ago
1
$begingroup$
Also essentially a duplicate of MO 59741 <mathoverflow.net/questions/59741> which asked the same question about squarefree triples $(4a+1,4a+2,4a+3)$.
$endgroup$
– Noam D. Elkies
6 hours ago
add a comment |
1
$begingroup$
This question is essentially the same as mathoverflow.net/questions/177849/… See my answer there.
$endgroup$
– GH from MO
7 hours ago
1
$begingroup$
Also essentially a duplicate of MO 59741 <mathoverflow.net/questions/59741> which asked the same question about squarefree triples $(4a+1,4a+2,4a+3)$.
$endgroup$
– Noam D. Elkies
6 hours ago
1
1
$begingroup$
This question is essentially the same as mathoverflow.net/questions/177849/… See my answer there.
$endgroup$
– GH from MO
7 hours ago
$begingroup$
This question is essentially the same as mathoverflow.net/questions/177849/… See my answer there.
$endgroup$
– GH from MO
7 hours ago
1
1
$begingroup$
Also essentially a duplicate of MO 59741 <mathoverflow.net/questions/59741> which asked the same question about squarefree triples $(4a+1,4a+2,4a+3)$.
$endgroup$
– Noam D. Elkies
6 hours ago
$begingroup$
Also essentially a duplicate of MO 59741 <mathoverflow.net/questions/59741> which asked the same question about squarefree triples $(4a+1,4a+2,4a+3)$.
$endgroup$
– Noam D. Elkies
6 hours ago
add a comment |
1 Answer
1
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See NOTE ON AN ASYMPTOTIC FORMULA CONNECTED WITH r-FREE INTEGERS by
L. MIRSKY, The Quarterly Journal of Mathematics, Volume os-18, Issue 1, 1947, Pages 178–182, https://doi.org/10.1093/qmath/os-18.1.178
This paper is more general, i.e., for $r$ tuples of square free numbers with fixed gap sizes. The number of such integer pairs $leq x$ is given by
$$
Ax+O( x^{frac{2}{3}+epsilon}(log x)^{frac{4}{3}}),
$$
where $A$ is a constant. See also here where the constant $A$ is evaluated in terms of Euler products.
answered 7 hours ago
kodlukodlu
4,5582 gold badges21 silver badges32 bronze badges
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add a comment |
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1 Answer
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$begingroup$
See NOTE ON AN ASYMPTOTIC FORMULA CONNECTED WITH r-FREE INTEGERS by
L. MIRSKY, The Quarterly Journal of Mathematics, Volume os-18, Issue 1, 1947, Pages 178–182, https://doi.org/10.1093/qmath/os-18.1.178
This paper is more general, i.e., for $r$ tuples of square free numbers with fixed gap sizes. The number of such integer pairs $leq x$ is given by
$$
Ax+O( x^{frac{2}{3}+epsilon}(log x)^{frac{4}{3}}),
$$
where $A$ is a constant. See also here where the constant $A$ is evaluated in terms of Euler products.
answered 7 hours ago
kodlukodlu
4,5582 gold badges21 silver badges32 bronze badges
$endgroup$
add a comment |
$begingroup$
See NOTE ON AN ASYMPTOTIC FORMULA CONNECTED WITH r-FREE INTEGERS by
L. MIRSKY, The Quarterly Journal of Mathematics, Volume os-18, Issue 1, 1947, Pages 178–182, https://doi.org/10.1093/qmath/os-18.1.178
This paper is more general, i.e., for $r$ tuples of square free numbers with fixed gap sizes. The number of such integer pairs $leq x$ is given by
$$
Ax+O( x^{frac{2}{3}+epsilon}(log x)^{frac{4}{3}}),
$$
where $A$ is a constant. See also here where the constant $A$ is evaluated in terms of Euler products.
answered 7 hours ago
kodlukodlu
4,5582 gold badges21 silver badges32 bronze badges
$endgroup$
add a comment |
$begingroup$
See NOTE ON AN ASYMPTOTIC FORMULA CONNECTED WITH r-FREE INTEGERS by
L. MIRSKY, The Quarterly Journal of Mathematics, Volume os-18, Issue 1, 1947, Pages 178–182, https://doi.org/10.1093/qmath/os-18.1.178
This paper is more general, i.e., for $r$ tuples of square free numbers with fixed gap sizes. The number of such integer pairs $leq x$ is given by
$$
Ax+O( x^{frac{2}{3}+epsilon}(log x)^{frac{4}{3}}),
$$
where $A$ is a constant. See also here where the constant $A$ is evaluated in terms of Euler products.
answered 7 hours ago
kodlukodlu
4,5582 gold badges21 silver badges32 bronze badges
$endgroup$
See NOTE ON AN ASYMPTOTIC FORMULA CONNECTED WITH r-FREE INTEGERS by
L. MIRSKY, The Quarterly Journal of Mathematics, Volume os-18, Issue 1, 1947, Pages 178–182, https://doi.org/10.1093/qmath/os-18.1.178
This paper is more general, i.e., for $r$ tuples of square free numbers with fixed gap sizes. The number of such integer pairs $leq x$ is given by
$$
Ax+O( x^{frac{2}{3}+epsilon}(log x)^{frac{4}{3}}),
$$
where $A$ is a constant. See also here where the constant $A$ is evaluated in terms of Euler products.
answered 7 hours ago
kodlukodlu
4,5582 gold badges21 silver badges32 bronze badges
answered 7 hours ago
kodlukodlu
4,5582 gold badges21 silver badges32 bronze badges
answered 7 hours ago
kodlukodlu
4,5582 gold badges21 silver badges32 bronze badges
answered 7 hours ago
kodlukodlu
4,5582 gold badges21 silver badges32 bronze badges
4,5582 gold badges21 silver badges32 bronze badges
add a comment |
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1
$begingroup$
This question is essentially the same as mathoverflow.net/questions/177849/… See my answer there.
$endgroup$
– GH from MO
7 hours ago
1
$begingroup$
Also essentially a duplicate of MO 59741 <mathoverflow.net/questions/59741> which asked the same question about squarefree triples $(4a+1,4a+2,4a+3)$.
$endgroup$
– Noam D. Elkies
6 hours ago