A bridge between the sum of the divisors and the Totient function
$begingroup$
Let's consider: $$tau(x,a,b)=sum_{1 le d le x \ (d,x)=d^a \} d^b$$
Where $(q,r)$ denotes the gcd of $q$ and $r$.
I think this could be interesting thing to look at because it's somehow a type of bridge between the sum of the divisor function $sigma_k(x)=tau(x,1,k)$ and Euler's totient function $phi(x)=tau(x,0,0)$.
Now, the average order of these functions is fairly well understood.
For example,
$$sum_{n=1}^x tau(n,1,1) approx frac{pi^2}{12}x^2$$
$$sum_{n=1}^x tau(n,0,1) approx frac{1}{pi^2}x^3$$
$$sum_{n=1}^x tau(n,1,0) approx xlog(x)+(2gamma+1)x$$
$$sum_{n=1}^x tau(n,0,0) approx frac{3}{pi^2}x^2$$
And these can be argued using the standard techniques which are Abel Summation formula and Dirichlet convolutions. Where $gamma$ is the Euler Macheroni constant.
Is it possible to achieve similar results for non integer values $a$?
For example, what is the average order of $tau(x,frac{1}{2},1)$? What is the average order $tau(x,frac{1}{2},0)$?
number-theory asymptotics totient-function divisor-sum
asked Dec 8 '18 at 18:14
MasonMason
1,9551530
$endgroup$
|
show 8 more comments
$begingroup$
Let's consider: $$tau(x,a,b)=sum_{1 le d le x \ (d,x)=d^a \} d^b$$
Where $(q,r)$ denotes the gcd of $q$ and $r$.
I think this could be interesting thing to look at because it's somehow a type of bridge between the sum of the divisor function $sigma_k(x)=tau(x,1,k)$ and Euler's totient function $phi(x)=tau(x,0,0)$.
Now, the average order of these functions is fairly well understood.
For example,
$$sum_{n=1}^x tau(n,1,1) approx frac{pi^2}{12}x^2$$
$$sum_{n=1}^x tau(n,0,1) approx frac{1}{pi^2}x^3$$
$$sum_{n=1}^x tau(n,1,0) approx xlog(x)+(2gamma+1)x$$
$$sum_{n=1}^x tau(n,0,0) approx frac{3}{pi^2}x^2$$
And these can be argued using the standard techniques which are Abel Summation formula and Dirichlet convolutions. Where $gamma$ is the Euler Macheroni constant.
Is it possible to achieve similar results for non integer values $a$?
For example, what is the average order of $tau(x,frac{1}{2},1)$? What is the average order $tau(x,frac{1}{2},0)$?
number-theory asymptotics totient-function divisor-sum
asked Dec 8 '18 at 18:14
MasonMason
1,9551530
$endgroup$
$begingroup$
Here's the first 100 values of the functions.
$endgroup$
– Mason
Dec 8 '18 at 19:01
$begingroup$
I just chose $tau$ because it's in between $sigma$ and $phi$ in the Greek alphabet. It shouldn't be confused with $tau$ as $sigma_0$ which is sometimes how it used in number theory. If this is confusing to anyone I guess I could change it to an upsilon.
$endgroup$
– Mason
Dec 8 '18 at 19:10
$begingroup$
Why asking for the asymptotic of many weird arithmetic functions instead of studying the famous ones (in particular the prime number theorem) ? Here and to all your questions the methods of the PNT and the Dirichlet divisor problem (poles and bounds for the Dirichlet series, Mellin inversion, tauberian theorem) apply.
$endgroup$
– reuns
Dec 8 '18 at 20:43
$begingroup$
@reuns. Undoubtedly, I should study the classics. But I think $tau$ is interesting in that is a type of bridge between important functions.
$endgroup$
– Mason
Dec 8 '18 at 20:53
1
$begingroup$
Since $sum_{d | n}phi(d)=n $ then $sum_{n=1}^infty phi(n) n^{-s}= frac{zeta(s-1)}{zeta(s)}$. The Riemann zeta function has a lot of nice properties from which we can say a lot on $sum_{n=1}^infty phi(n) n^{-s}$ and $sum_{n=1}^N phi(n)$. Can you do the same with your above functions ?
$endgroup$
– reuns
Dec 9 '18 at 2:57
|
show 8 more comments
$begingroup$
Let's consider: $$tau(x,a,b)=sum_{1 le d le x \ (d,x)=d^a \} d^b$$
Where $(q,r)$ denotes the gcd of $q$ and $r$.
I think this could be interesting thing to look at because it's somehow a type of bridge between the sum of the divisor function $sigma_k(x)=tau(x,1,k)$ and Euler's totient function $phi(x)=tau(x,0,0)$.
Now, the average order of these functions is fairly well understood.
For example,
$$sum_{n=1}^x tau(n,1,1) approx frac{pi^2}{12}x^2$$
$$sum_{n=1}^x tau(n,0,1) approx frac{1}{pi^2}x^3$$
$$sum_{n=1}^x tau(n,1,0) approx xlog(x)+(2gamma+1)x$$
$$sum_{n=1}^x tau(n,0,0) approx frac{3}{pi^2}x^2$$
And these can be argued using the standard techniques which are Abel Summation formula and Dirichlet convolutions. Where $gamma$ is the Euler Macheroni constant.
Is it possible to achieve similar results for non integer values $a$?
For example, what is the average order of $tau(x,frac{1}{2},1)$? What is the average order $tau(x,frac{1}{2},0)$?
number-theory asymptotics totient-function divisor-sum
asked Dec 8 '18 at 18:14
MasonMason
1,9551530
$endgroup$
Let's consider: $$tau(x,a,b)=sum_{1 le d le x \ (d,x)=d^a \} d^b$$
Where $(q,r)$ denotes the gcd of $q$ and $r$.
I think this could be interesting thing to look at because it's somehow a type of bridge between the sum of the divisor function $sigma_k(x)=tau(x,1,k)$ and Euler's totient function $phi(x)=tau(x,0,0)$.
Now, the average order of these functions is fairly well understood.
For example,
$$sum_{n=1}^x tau(n,1,1) approx frac{pi^2}{12}x^2$$
$$sum_{n=1}^x tau(n,0,1) approx frac{1}{pi^2}x^3$$
$$sum_{n=1}^x tau(n,1,0) approx xlog(x)+(2gamma+1)x$$
$$sum_{n=1}^x tau(n,0,0) approx frac{3}{pi^2}x^2$$
And these can be argued using the standard techniques which are Abel Summation formula and Dirichlet convolutions. Where $gamma$ is the Euler Macheroni constant.
Is it possible to achieve similar results for non integer values $a$?
For example, what is the average order of $tau(x,frac{1}{2},1)$? What is the average order $tau(x,frac{1}{2},0)$?
number-theory asymptotics totient-function divisor-sum
number-theory asymptotics totient-function divisor-sum
asked Dec 8 '18 at 18:14
MasonMason
1,9551530
asked Dec 8 '18 at 18:14
MasonMason
1,9551530
edited Dec 8 '18 at 18:47
Mason
asked Dec 8 '18 at 18:14
MasonMason
1,9551530
asked Dec 8 '18 at 18:14
MasonMason
1,9551530
asked Dec 8 '18 at 18:14
MasonMason
1,9551530
1,9551530
$begingroup$
Here's the first 100 values of the functions.
$endgroup$
– Mason
Dec 8 '18 at 19:01
$begingroup$
I just chose $tau$ because it's in between $sigma$ and $phi$ in the Greek alphabet. It shouldn't be confused with $tau$ as $sigma_0$ which is sometimes how it used in number theory. If this is confusing to anyone I guess I could change it to an upsilon.
$endgroup$
– Mason
Dec 8 '18 at 19:10
$begingroup$
Why asking for the asymptotic of many weird arithmetic functions instead of studying the famous ones (in particular the prime number theorem) ? Here and to all your questions the methods of the PNT and the Dirichlet divisor problem (poles and bounds for the Dirichlet series, Mellin inversion, tauberian theorem) apply.
$endgroup$
– reuns
Dec 8 '18 at 20:43
$begingroup$
@reuns. Undoubtedly, I should study the classics. But I think $tau$ is interesting in that is a type of bridge between important functions.
$endgroup$
– Mason
Dec 8 '18 at 20:53
1
$begingroup$
Since $sum_{d | n}phi(d)=n $ then $sum_{n=1}^infty phi(n) n^{-s}= frac{zeta(s-1)}{zeta(s)}$. The Riemann zeta function has a lot of nice properties from which we can say a lot on $sum_{n=1}^infty phi(n) n^{-s}$ and $sum_{n=1}^N phi(n)$. Can you do the same with your above functions ?
$endgroup$
– reuns
Dec 9 '18 at 2:57
|
show 8 more comments
$begingroup$
Here's the first 100 values of the functions.
$endgroup$
– Mason
Dec 8 '18 at 19:01
$begingroup$
I just chose $tau$ because it's in between $sigma$ and $phi$ in the Greek alphabet. It shouldn't be confused with $tau$ as $sigma_0$ which is sometimes how it used in number theory. If this is confusing to anyone I guess I could change it to an upsilon.
$endgroup$
– Mason
Dec 8 '18 at 19:10
$begingroup$
Why asking for the asymptotic of many weird arithmetic functions instead of studying the famous ones (in particular the prime number theorem) ? Here and to all your questions the methods of the PNT and the Dirichlet divisor problem (poles and bounds for the Dirichlet series, Mellin inversion, tauberian theorem) apply.
$endgroup$
– reuns
Dec 8 '18 at 20:43
$begingroup$
@reuns. Undoubtedly, I should study the classics. But I think $tau$ is interesting in that is a type of bridge between important functions.
$endgroup$
– Mason
Dec 8 '18 at 20:53
1
$begingroup$
Since $sum_{d | n}phi(d)=n $ then $sum_{n=1}^infty phi(n) n^{-s}= frac{zeta(s-1)}{zeta(s)}$. The Riemann zeta function has a lot of nice properties from which we can say a lot on $sum_{n=1}^infty phi(n) n^{-s}$ and $sum_{n=1}^N phi(n)$. Can you do the same with your above functions ?
$endgroup$
– reuns
Dec 9 '18 at 2:57
$begingroup$
Here's the first 100 values of the functions.
$endgroup$
– Mason
Dec 8 '18 at 19:01
$begingroup$
Here's the first 100 values of the functions.
$endgroup$
– Mason
Dec 8 '18 at 19:01
$begingroup$
I just chose $tau$ because it's in between $sigma$ and $phi$ in the Greek alphabet. It shouldn't be confused with $tau$ as $sigma_0$ which is sometimes how it used in number theory. If this is confusing to anyone I guess I could change it to an upsilon.
$endgroup$
– Mason
Dec 8 '18 at 19:10
$begingroup$
I just chose $tau$ because it's in between $sigma$ and $phi$ in the Greek alphabet. It shouldn't be confused with $tau$ as $sigma_0$ which is sometimes how it used in number theory. If this is confusing to anyone I guess I could change it to an upsilon.
$endgroup$
– Mason
Dec 8 '18 at 19:10
$begingroup$
Why asking for the asymptotic of many weird arithmetic functions instead of studying the famous ones (in particular the prime number theorem) ? Here and to all your questions the methods of the PNT and the Dirichlet divisor problem (poles and bounds for the Dirichlet series, Mellin inversion, tauberian theorem) apply.
$endgroup$
– reuns
Dec 8 '18 at 20:43
$begingroup$
Why asking for the asymptotic of many weird arithmetic functions instead of studying the famous ones (in particular the prime number theorem) ? Here and to all your questions the methods of the PNT and the Dirichlet divisor problem (poles and bounds for the Dirichlet series, Mellin inversion, tauberian theorem) apply.
$endgroup$
– reuns
Dec 8 '18 at 20:43
$begingroup$
@reuns. Undoubtedly, I should study the classics. But I think $tau$ is interesting in that is a type of bridge between important functions.
$endgroup$
– Mason
Dec 8 '18 at 20:53
$begingroup$
@reuns. Undoubtedly, I should study the classics. But I think $tau$ is interesting in that is a type of bridge between important functions.
$endgroup$
– Mason
Dec 8 '18 at 20:53
1
1
$begingroup$
Since $sum_{d | n}phi(d)=n $ then $sum_{n=1}^infty phi(n) n^{-s}= frac{zeta(s-1)}{zeta(s)}$. The Riemann zeta function has a lot of nice properties from which we can say a lot on $sum_{n=1}^infty phi(n) n^{-s}$ and $sum_{n=1}^N phi(n)$. Can you do the same with your above functions ?
$endgroup$
– reuns
Dec 9 '18 at 2:57
$begingroup$
Since $sum_{d | n}phi(d)=n $ then $sum_{n=1}^infty phi(n) n^{-s}= frac{zeta(s-1)}{zeta(s)}$. The Riemann zeta function has a lot of nice properties from which we can say a lot on $sum_{n=1}^infty phi(n) n^{-s}$ and $sum_{n=1}^N phi(n)$. Can you do the same with your above functions ?
$endgroup$
– reuns
Dec 9 '18 at 2:57
|
show 8 more comments
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$begingroup$
Here's the first 100 values of the functions.
$endgroup$
– Mason
Dec 8 '18 at 19:01
$begingroup$
I just chose $tau$ because it's in between $sigma$ and $phi$ in the Greek alphabet. It shouldn't be confused with $tau$ as $sigma_0$ which is sometimes how it used in number theory. If this is confusing to anyone I guess I could change it to an upsilon.
$endgroup$
– Mason
Dec 8 '18 at 19:10
$begingroup$
Why asking for the asymptotic of many weird arithmetic functions instead of studying the famous ones (in particular the prime number theorem) ? Here and to all your questions the methods of the PNT and the Dirichlet divisor problem (poles and bounds for the Dirichlet series, Mellin inversion, tauberian theorem) apply.
$endgroup$
– reuns
Dec 8 '18 at 20:43
$begingroup$
@reuns. Undoubtedly, I should study the classics. But I think $tau$ is interesting in that is a type of bridge between important functions.
$endgroup$
– Mason
Dec 8 '18 at 20:53
1
$begingroup$
Since $sum_{d | n}phi(d)=n $ then $sum_{n=1}^infty phi(n) n^{-s}= frac{zeta(s-1)}{zeta(s)}$. The Riemann zeta function has a lot of nice properties from which we can say a lot on $sum_{n=1}^infty phi(n) n^{-s}$ and $sum_{n=1}^N phi(n)$. Can you do the same with your above functions ?
$endgroup$
– reuns
Dec 9 '18 at 2:57