Density of integers represented by norm form (cyclotomic field)
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Let $K=mathbb Q(zeta_n)$ be the $n$th cyclotomic field, and $N$ be the norm form constructed from the field norm of $K$. An interesting problem would be the number of integers $l$ less than $x$ that can be represented by the norm form $N$ (in other words, there is an integral element of $K$, $f$, such that the field norm of $f$ is $l$) (assuming x is finite).
(I) More specifically to this problem, let $n$ be a prime and $K=mathbb Q(zeta_n)$ as above. Let $L(x)$ represent the number of integers less than $x$ which can be represented by the norm form $N$ also defined above. What is a good upper and lower bound for $L(2^n)$ as $n$ goes to infinity?
My conjecture (II) $2^{(n-1)/2}<L(2^n)<2^{(3n-3)/4}$
Is anyone able to provide a better and more accurate lower bound than II for $L(2^n)$ to answer (I)?
For instance, when $n=7$, $L(2^7)=11$ because 11 integers less than or equal to $2^7$ are represented by $N: (1, 7, 8, 29, 43, 49, 56, 64, 71, 113, 127)$. It seems to fit my conjectured bound II as $2^3 < L(2^7) < 2^{4.5}$, $8 < 11 < 22.6$.
Thanks for help.
elementary-number-theory probability-distributions cyclotomic-fields norm-forms
asked Dec 29 '18 at 8:31
J. LinneJ. Linne
883415
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add a comment |
$begingroup$
Let $K=mathbb Q(zeta_n)$ be the $n$th cyclotomic field, and $N$ be the norm form constructed from the field norm of $K$. An interesting problem would be the number of integers $l$ less than $x$ that can be represented by the norm form $N$ (in other words, there is an integral element of $K$, $f$, such that the field norm of $f$ is $l$) (assuming x is finite).
(I) More specifically to this problem, let $n$ be a prime and $K=mathbb Q(zeta_n)$ as above. Let $L(x)$ represent the number of integers less than $x$ which can be represented by the norm form $N$ also defined above. What is a good upper and lower bound for $L(2^n)$ as $n$ goes to infinity?
My conjecture (II) $2^{(n-1)/2}<L(2^n)<2^{(3n-3)/4}$
Is anyone able to provide a better and more accurate lower bound than II for $L(2^n)$ to answer (I)?
For instance, when $n=7$, $L(2^7)=11$ because 11 integers less than or equal to $2^7$ are represented by $N: (1, 7, 8, 29, 43, 49, 56, 64, 71, 113, 127)$. It seems to fit my conjectured bound II as $2^3 < L(2^7) < 2^{4.5}$, $8 < 11 < 22.6$.
Thanks for help.
elementary-number-theory probability-distributions cyclotomic-fields norm-forms
asked Dec 29 '18 at 8:31
J. LinneJ. Linne
883415
$endgroup$
1
$begingroup$
I think you can use the prime number theorem for Hecke L-functions of characters of the ideal class group to obtain an asymptotic for $L(x)$. With $nu(I) = 1$ if $I$ is a product of prime ideals of distinct norms, with $r(I) = #{ J, N(J) = N(I)}$ then $sum_I frac{nu(I)psi(I)}{r(I)} N(I)^{-s}$ has an Euler product, which is close to $L(s,psi)^{1/varphi(n)}$, and summing over $psi$ yields the Dirichlet series counting the integers being the norm of a principal ideal which is a product of prime ideals of distinct norms.
$endgroup$
– reuns
Dec 29 '18 at 9:47
$begingroup$
@reuns I remember learning this a couple years ago, would you mind giving me an example of this if you can? Thanks much appreciated.
$endgroup$
– J. Linne
Dec 30 '18 at 5:08
add a comment |
$begingroup$
Let $K=mathbb Q(zeta_n)$ be the $n$th cyclotomic field, and $N$ be the norm form constructed from the field norm of $K$. An interesting problem would be the number of integers $l$ less than $x$ that can be represented by the norm form $N$ (in other words, there is an integral element of $K$, $f$, such that the field norm of $f$ is $l$) (assuming x is finite).
(I) More specifically to this problem, let $n$ be a prime and $K=mathbb Q(zeta_n)$ as above. Let $L(x)$ represent the number of integers less than $x$ which can be represented by the norm form $N$ also defined above. What is a good upper and lower bound for $L(2^n)$ as $n$ goes to infinity?
My conjecture (II) $2^{(n-1)/2}<L(2^n)<2^{(3n-3)/4}$
Is anyone able to provide a better and more accurate lower bound than II for $L(2^n)$ to answer (I)?
For instance, when $n=7$, $L(2^7)=11$ because 11 integers less than or equal to $2^7$ are represented by $N: (1, 7, 8, 29, 43, 49, 56, 64, 71, 113, 127)$. It seems to fit my conjectured bound II as $2^3 < L(2^7) < 2^{4.5}$, $8 < 11 < 22.6$.
Thanks for help.
elementary-number-theory probability-distributions cyclotomic-fields norm-forms
asked Dec 29 '18 at 8:31
J. LinneJ. Linne
883415
$endgroup$
Let $K=mathbb Q(zeta_n)$ be the $n$th cyclotomic field, and $N$ be the norm form constructed from the field norm of $K$. An interesting problem would be the number of integers $l$ less than $x$ that can be represented by the norm form $N$ (in other words, there is an integral element of $K$, $f$, such that the field norm of $f$ is $l$) (assuming x is finite).
(I) More specifically to this problem, let $n$ be a prime and $K=mathbb Q(zeta_n)$ as above. Let $L(x)$ represent the number of integers less than $x$ which can be represented by the norm form $N$ also defined above. What is a good upper and lower bound for $L(2^n)$ as $n$ goes to infinity?
My conjecture (II) $2^{(n-1)/2}<L(2^n)<2^{(3n-3)/4}$
Is anyone able to provide a better and more accurate lower bound than II for $L(2^n)$ to answer (I)?
For instance, when $n=7$, $L(2^7)=11$ because 11 integers less than or equal to $2^7$ are represented by $N: (1, 7, 8, 29, 43, 49, 56, 64, 71, 113, 127)$. It seems to fit my conjectured bound II as $2^3 < L(2^7) < 2^{4.5}$, $8 < 11 < 22.6$.
Thanks for help.
elementary-number-theory probability-distributions cyclotomic-fields norm-forms
elementary-number-theory probability-distributions cyclotomic-fields norm-forms
asked Dec 29 '18 at 8:31
J. LinneJ. Linne
883415
asked Dec 29 '18 at 8:31
J. LinneJ. Linne
883415
asked Dec 29 '18 at 8:31
J. LinneJ. Linne
883415
asked Dec 29 '18 at 8:31
J. LinneJ. Linne
883415
asked Dec 29 '18 at 8:31
J. LinneJ. Linne
883415
883415
1
$begingroup$
I think you can use the prime number theorem for Hecke L-functions of characters of the ideal class group to obtain an asymptotic for $L(x)$. With $nu(I) = 1$ if $I$ is a product of prime ideals of distinct norms, with $r(I) = #{ J, N(J) = N(I)}$ then $sum_I frac{nu(I)psi(I)}{r(I)} N(I)^{-s}$ has an Euler product, which is close to $L(s,psi)^{1/varphi(n)}$, and summing over $psi$ yields the Dirichlet series counting the integers being the norm of a principal ideal which is a product of prime ideals of distinct norms.
$endgroup$
– reuns
Dec 29 '18 at 9:47
$begingroup$
@reuns I remember learning this a couple years ago, would you mind giving me an example of this if you can? Thanks much appreciated.
$endgroup$
– J. Linne
Dec 30 '18 at 5:08
add a comment |
1
$begingroup$
I think you can use the prime number theorem for Hecke L-functions of characters of the ideal class group to obtain an asymptotic for $L(x)$. With $nu(I) = 1$ if $I$ is a product of prime ideals of distinct norms, with $r(I) = #{ J, N(J) = N(I)}$ then $sum_I frac{nu(I)psi(I)}{r(I)} N(I)^{-s}$ has an Euler product, which is close to $L(s,psi)^{1/varphi(n)}$, and summing over $psi$ yields the Dirichlet series counting the integers being the norm of a principal ideal which is a product of prime ideals of distinct norms.
$endgroup$
– reuns
Dec 29 '18 at 9:47
$begingroup$
@reuns I remember learning this a couple years ago, would you mind giving me an example of this if you can? Thanks much appreciated.
$endgroup$
– J. Linne
Dec 30 '18 at 5:08
1
1
$begingroup$
I think you can use the prime number theorem for Hecke L-functions of characters of the ideal class group to obtain an asymptotic for $L(x)$. With $nu(I) = 1$ if $I$ is a product of prime ideals of distinct norms, with $r(I) = #{ J, N(J) = N(I)}$ then $sum_I frac{nu(I)psi(I)}{r(I)} N(I)^{-s}$ has an Euler product, which is close to $L(s,psi)^{1/varphi(n)}$, and summing over $psi$ yields the Dirichlet series counting the integers being the norm of a principal ideal which is a product of prime ideals of distinct norms.
$endgroup$
– reuns
Dec 29 '18 at 9:47
$begingroup$
I think you can use the prime number theorem for Hecke L-functions of characters of the ideal class group to obtain an asymptotic for $L(x)$. With $nu(I) = 1$ if $I$ is a product of prime ideals of distinct norms, with $r(I) = #{ J, N(J) = N(I)}$ then $sum_I frac{nu(I)psi(I)}{r(I)} N(I)^{-s}$ has an Euler product, which is close to $L(s,psi)^{1/varphi(n)}$, and summing over $psi$ yields the Dirichlet series counting the integers being the norm of a principal ideal which is a product of prime ideals of distinct norms.
$endgroup$
– reuns
Dec 29 '18 at 9:47
$begingroup$
@reuns I remember learning this a couple years ago, would you mind giving me an example of this if you can? Thanks much appreciated.
$endgroup$
– J. Linne
Dec 30 '18 at 5:08
$begingroup$
@reuns I remember learning this a couple years ago, would you mind giving me an example of this if you can? Thanks much appreciated.
$endgroup$
– J. Linne
Dec 30 '18 at 5:08
add a comment |
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$begingroup$
I think you can use the prime number theorem for Hecke L-functions of characters of the ideal class group to obtain an asymptotic for $L(x)$. With $nu(I) = 1$ if $I$ is a product of prime ideals of distinct norms, with $r(I) = #{ J, N(J) = N(I)}$ then $sum_I frac{nu(I)psi(I)}{r(I)} N(I)^{-s}$ has an Euler product, which is close to $L(s,psi)^{1/varphi(n)}$, and summing over $psi$ yields the Dirichlet series counting the integers being the norm of a principal ideal which is a product of prime ideals of distinct norms.
$endgroup$
– reuns
Dec 29 '18 at 9:47
$begingroup$
@reuns I remember learning this a couple years ago, would you mind giving me an example of this if you can? Thanks much appreciated.
$endgroup$
– J. Linne
Dec 30 '18 at 5:08