What is sum of squared elements of Farey sequence?
1
$begingroup$
Let's consider a Farey sequence:
$$F_{n}={a_{1},...,a_{k}}$$;
Where given elements satisfy definition of $n$-th Farey sequence.
My problem: Find the formula for the following sum:
$$sum_{l=1}^{k}a_{l}^{2}=?$$
Alternatively find asymptotic growth in dependence of $n$.
Only thing i know is that denominator of this sum is $(n!)^{2}$.
Regards.
number-theory analytic-number-theory diophantine-approximation farey-sequences
asked Jan 19 at 14:22
mkultramkultra
658
$endgroup$
add a comment |
1
$begingroup$
Let's consider a Farey sequence:
$$F_{n}={a_{1},...,a_{k}}$$;
Where given elements satisfy definition of $n$-th Farey sequence.
My problem: Find the formula for the following sum:
$$sum_{l=1}^{k}a_{l}^{2}=?$$
Alternatively find asymptotic growth in dependence of $n$.
Only thing i know is that denominator of this sum is $(n!)^{2}$.
Regards.
number-theory analytic-number-theory diophantine-approximation farey-sequences
asked Jan 19 at 14:22
mkultramkultra
658
$endgroup$
1
$begingroup$
Solution in iosrjournals.org/iosr-jm/papers/Vol12-issue2/Version-6/…
$endgroup$
– Jean Marie
Jan 19 at 15:13
$begingroup$
Sorry, but i asked for sum of squares Farey elements not squared sum of Farey elements.
$endgroup$
– mkultra
Jan 19 at 15:26
$begingroup$
I don't understand : the article gives a method for the sum of the squares of a Farey sequence...
$endgroup$
– Jean Marie
Jan 19 at 15:29
$begingroup$
Yes, it basically says: If you want to calculate the sum of squares of the elements of a Farey sequence, you have to add the squares of the elements of the Farey sequence.
$endgroup$
– Reinhard Meier
Jan 19 at 15:31
$begingroup$
By the way, confirming the currently assumed asymptotic growth would prove the Riemann hypothesis. Farey sequence
$endgroup$
– Reinhard Meier
Jan 19 at 15:34
add a comment |
1
1
1
0
$begingroup$
Let's consider a Farey sequence:
$$F_{n}={a_{1},...,a_{k}}$$;
Where given elements satisfy definition of $n$-th Farey sequence.
My problem: Find the formula for the following sum:
$$sum_{l=1}^{k}a_{l}^{2}=?$$
Alternatively find asymptotic growth in dependence of $n$.
Only thing i know is that denominator of this sum is $(n!)^{2}$.
Regards.
number-theory analytic-number-theory diophantine-approximation farey-sequences
asked Jan 19 at 14:22
mkultramkultra
658
$endgroup$
Let's consider a Farey sequence:
$$F_{n}={a_{1},...,a_{k}}$$;
Where given elements satisfy definition of $n$-th Farey sequence.
My problem: Find the formula for the following sum:
$$sum_{l=1}^{k}a_{l}^{2}=?$$
Alternatively find asymptotic growth in dependence of $n$.
Only thing i know is that denominator of this sum is $(n!)^{2}$.
Regards.
number-theory analytic-number-theory diophantine-approximation farey-sequences
number-theory analytic-number-theory diophantine-approximation farey-sequences
asked Jan 19 at 14:22
mkultramkultra
658
asked Jan 19 at 14:22
mkultramkultra
658
asked Jan 19 at 14:22
mkultramkultra
658
asked Jan 19 at 14:22
mkultramkultra
658
asked Jan 19 at 14:22
mkultramkultra
658
658
1
$begingroup$
Solution in iosrjournals.org/iosr-jm/papers/Vol12-issue2/Version-6/…
$endgroup$
– Jean Marie
Jan 19 at 15:13
$begingroup$
Sorry, but i asked for sum of squares Farey elements not squared sum of Farey elements.
$endgroup$
– mkultra
Jan 19 at 15:26
$begingroup$
I don't understand : the article gives a method for the sum of the squares of a Farey sequence...
$endgroup$
– Jean Marie
Jan 19 at 15:29
$begingroup$
Yes, it basically says: If you want to calculate the sum of squares of the elements of a Farey sequence, you have to add the squares of the elements of the Farey sequence.
$endgroup$
– Reinhard Meier
Jan 19 at 15:31
$begingroup$
By the way, confirming the currently assumed asymptotic growth would prove the Riemann hypothesis. Farey sequence
$endgroup$
– Reinhard Meier
Jan 19 at 15:34
add a comment |
1
$begingroup$
Solution in iosrjournals.org/iosr-jm/papers/Vol12-issue2/Version-6/…
$endgroup$
– Jean Marie
Jan 19 at 15:13
$begingroup$
Sorry, but i asked for sum of squares Farey elements not squared sum of Farey elements.
$endgroup$
– mkultra
Jan 19 at 15:26
$begingroup$
I don't understand : the article gives a method for the sum of the squares of a Farey sequence...
$endgroup$
– Jean Marie
Jan 19 at 15:29
$begingroup$
Yes, it basically says: If you want to calculate the sum of squares of the elements of a Farey sequence, you have to add the squares of the elements of the Farey sequence.
$endgroup$
– Reinhard Meier
Jan 19 at 15:31
$begingroup$
By the way, confirming the currently assumed asymptotic growth would prove the Riemann hypothesis. Farey sequence
$endgroup$
– Reinhard Meier
Jan 19 at 15:34
1
1
$begingroup$
Solution in iosrjournals.org/iosr-jm/papers/Vol12-issue2/Version-6/…
$endgroup$
– Jean Marie
Jan 19 at 15:13
$begingroup$
Solution in iosrjournals.org/iosr-jm/papers/Vol12-issue2/Version-6/…
$endgroup$
– Jean Marie
Jan 19 at 15:13
$begingroup$
Sorry, but i asked for sum of squares Farey elements not squared sum of Farey elements.
$endgroup$
– mkultra
Jan 19 at 15:26
$begingroup$
Sorry, but i asked for sum of squares Farey elements not squared sum of Farey elements.
$endgroup$
– mkultra
Jan 19 at 15:26
$begingroup$
I don't understand : the article gives a method for the sum of the squares of a Farey sequence...
$endgroup$
– Jean Marie
Jan 19 at 15:29
$begingroup$
I don't understand : the article gives a method for the sum of the squares of a Farey sequence...
$endgroup$
– Jean Marie
Jan 19 at 15:29
$begingroup$
Yes, it basically says: If you want to calculate the sum of squares of the elements of a Farey sequence, you have to add the squares of the elements of the Farey sequence.
$endgroup$
– Reinhard Meier
Jan 19 at 15:31
$begingroup$
Yes, it basically says: If you want to calculate the sum of squares of the elements of a Farey sequence, you have to add the squares of the elements of the Farey sequence.
$endgroup$
– Reinhard Meier
Jan 19 at 15:31
$begingroup$
By the way, confirming the currently assumed asymptotic growth would prove the Riemann hypothesis. Farey sequence
$endgroup$
– Reinhard Meier
Jan 19 at 15:34
$begingroup$
By the way, confirming the currently assumed asymptotic growth would prove the Riemann hypothesis. Farey sequence
$endgroup$
– Reinhard Meier
Jan 19 at 15:34
add a comment |
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1
$begingroup$
Solution in iosrjournals.org/iosr-jm/papers/Vol12-issue2/Version-6/…
$endgroup$
– Jean Marie
Jan 19 at 15:13
$begingroup$
Sorry, but i asked for sum of squares Farey elements not squared sum of Farey elements.
$endgroup$
– mkultra
Jan 19 at 15:26
$begingroup$
I don't understand : the article gives a method for the sum of the squares of a Farey sequence...
$endgroup$
– Jean Marie
Jan 19 at 15:29
$begingroup$
Yes, it basically says: If you want to calculate the sum of squares of the elements of a Farey sequence, you have to add the squares of the elements of the Farey sequence.
$endgroup$
– Reinhard Meier
Jan 19 at 15:31
$begingroup$
By the way, confirming the currently assumed asymptotic growth would prove the Riemann hypothesis. Farey sequence
$endgroup$
– Reinhard Meier
Jan 19 at 15:34