Mixing
The definition of polar density of a set of primes in a number field
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In Chapter 7 of Marcus' Number Fields, he defines the polar density of a set $A$ of primes in a number field $K$, as follows: if for some positive integer $n$, the $n$-th power of the Dedekind zeta function $zeta_{K, A}(s) = prod_{P in A} (1-||P||^{-s})^{-1}$ can be extended to a meromorphic function on some neighborhood (in the complex plane) of $s=1$ , with the order of pole at $s=1$ equal to $m$, then the polar density is defined to be $m/n$.
My question regarding this definition is: why do we expect the existence of a meromorphic extension when RAISING TO A POWER, in the first place?
Many thanks in advance!
complex-analysis number-theory algebraic-number-theory analytic-number-theory
asked Jan 16 at 23:59
mathidiotmathidiot
93
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add a comment |
$begingroup$
In Chapter 7 of Marcus' Number Fields, he defines the polar density of a set $A$ of primes in a number field $K$, as follows: if for some positive integer $n$, the $n$-th power of the Dedekind zeta function $zeta_{K, A}(s) = prod_{P in A} (1-||P||^{-s})^{-1}$ can be extended to a meromorphic function on some neighborhood (in the complex plane) of $s=1$ , with the order of pole at $s=1$ equal to $m$, then the polar density is defined to be $m/n$.
My question regarding this definition is: why do we expect the existence of a meromorphic extension when RAISING TO A POWER, in the first place?
Many thanks in advance!
complex-analysis number-theory algebraic-number-theory analytic-number-theory
asked Jan 16 at 23:59
mathidiotmathidiot
93
$endgroup$
$begingroup$
In general $zeta_{K,A}^{, m}$ is not meromorphic for any $m$. At first the polar density should be $limsup_{s to 1,s > 1} frac{log zeta_{K,A}(s)}{log (s-1)}$ comprised between $[0,1]$. If $K/F$ is Galois and $A$ is (Artin map) a conjugacy class $C$ in the Galois group then $zeta_{K,A}^{, |G|}$ is meromorphic with a pole of order $|C|$ and no poles nor zeros on $Re(s) ge 1,s ne 1$ (Chebotarev theorem)
$endgroup$
– reuns
Jan 17 at 11:59
add a comment |
$begingroup$
In Chapter 7 of Marcus' Number Fields, he defines the polar density of a set $A$ of primes in a number field $K$, as follows: if for some positive integer $n$, the $n$-th power of the Dedekind zeta function $zeta_{K, A}(s) = prod_{P in A} (1-||P||^{-s})^{-1}$ can be extended to a meromorphic function on some neighborhood (in the complex plane) of $s=1$ , with the order of pole at $s=1$ equal to $m$, then the polar density is defined to be $m/n$.
My question regarding this definition is: why do we expect the existence of a meromorphic extension when RAISING TO A POWER, in the first place?
Many thanks in advance!
complex-analysis number-theory algebraic-number-theory analytic-number-theory
asked Jan 16 at 23:59
mathidiotmathidiot
93
$endgroup$
In Chapter 7 of Marcus' Number Fields, he defines the polar density of a set $A$ of primes in a number field $K$, as follows: if for some positive integer $n$, the $n$-th power of the Dedekind zeta function $zeta_{K, A}(s) = prod_{P in A} (1-||P||^{-s})^{-1}$ can be extended to a meromorphic function on some neighborhood (in the complex plane) of $s=1$ , with the order of pole at $s=1$ equal to $m$, then the polar density is defined to be $m/n$.
My question regarding this definition is: why do we expect the existence of a meromorphic extension when RAISING TO A POWER, in the first place?
Many thanks in advance!
complex-analysis number-theory algebraic-number-theory analytic-number-theory
complex-analysis number-theory algebraic-number-theory analytic-number-theory
asked Jan 16 at 23:59
mathidiotmathidiot
93
asked Jan 16 at 23:59
mathidiotmathidiot
93
asked Jan 16 at 23:59
mathidiotmathidiot
93
asked Jan 16 at 23:59
mathidiotmathidiot
93
asked Jan 16 at 23:59
mathidiotmathidiot
93
93
$begingroup$
In general $zeta_{K,A}^{, m}$ is not meromorphic for any $m$. At first the polar density should be $limsup_{s to 1,s > 1} frac{log zeta_{K,A}(s)}{log (s-1)}$ comprised between $[0,1]$. If $K/F$ is Galois and $A$ is (Artin map) a conjugacy class $C$ in the Galois group then $zeta_{K,A}^{, |G|}$ is meromorphic with a pole of order $|C|$ and no poles nor zeros on $Re(s) ge 1,s ne 1$ (Chebotarev theorem)
$endgroup$
– reuns
Jan 17 at 11:59
add a comment |
$begingroup$
In general $zeta_{K,A}^{, m}$ is not meromorphic for any $m$. At first the polar density should be $limsup_{s to 1,s > 1} frac{log zeta_{K,A}(s)}{log (s-1)}$ comprised between $[0,1]$. If $K/F$ is Galois and $A$ is (Artin map) a conjugacy class $C$ in the Galois group then $zeta_{K,A}^{, |G|}$ is meromorphic with a pole of order $|C|$ and no poles nor zeros on $Re(s) ge 1,s ne 1$ (Chebotarev theorem)
$endgroup$
– reuns
Jan 17 at 11:59
$begingroup$
In general $zeta_{K,A}^{, m}$ is not meromorphic for any $m$. At first the polar density should be $limsup_{s to 1,s > 1} frac{log zeta_{K,A}(s)}{log (s-1)}$ comprised between $[0,1]$. If $K/F$ is Galois and $A$ is (Artin map) a conjugacy class $C$ in the Galois group then $zeta_{K,A}^{, |G|}$ is meromorphic with a pole of order $|C|$ and no poles nor zeros on $Re(s) ge 1,s ne 1$ (Chebotarev theorem)
$endgroup$
– reuns
Jan 17 at 11:59
$begingroup$
In general $zeta_{K,A}^{, m}$ is not meromorphic for any $m$. At first the polar density should be $limsup_{s to 1,s > 1} frac{log zeta_{K,A}(s)}{log (s-1)}$ comprised between $[0,1]$. If $K/F$ is Galois and $A$ is (Artin map) a conjugacy class $C$ in the Galois group then $zeta_{K,A}^{, |G|}$ is meromorphic with a pole of order $|C|$ and no poles nor zeros on $Re(s) ge 1,s ne 1$ (Chebotarev theorem)
$endgroup$
– reuns
Jan 17 at 11:59
add a comment |
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$begingroup$
In general $zeta_{K,A}^{, m}$ is not meromorphic for any $m$. At first the polar density should be $limsup_{s to 1,s > 1} frac{log zeta_{K,A}(s)}{log (s-1)}$ comprised between $[0,1]$. If $K/F$ is Galois and $A$ is (Artin map) a conjugacy class $C$ in the Galois group then $zeta_{K,A}^{, |G|}$ is meromorphic with a pole of order $|C|$ and no poles nor zeros on $Re(s) ge 1,s ne 1$ (Chebotarev theorem)
$endgroup$
– reuns
Jan 17 at 11:59