Trying to understand why the zeta function is a rational function under certain conditions. Questions about...
Information: I linked the pages below, which relate to my questions.
I am currently reading " A Classic Introduction to Modern Number Theory " by Kenneth Ireland and Michael Rosen. In the 11th chapter they consider the zeta function. In the third section of this chapter they want to show that the Zeta Function is associated to $a_0x_0^m+a_1x_1^m+...+a_nx_n^m$.
They start with:
$$N_s = q^{s(n-1)}+q^{s(n-2)}+ ... + q + 1 + frac{1}{q^s} sum_{chi_0^{(s)},...,chi_n^{(s)}} chi_0^{(s)}(a_o^{-1}) cdots chi_n^{(s)}(a_n^{-1})g(chi_0^{(s)}) cdots g(chi_n^{(s)}) $$
That's ok, but I don't see why $$ q^{s(n-1)}+q^{s(n-2)}+ ... + q + 1 + frac{1}{q^s} sum_{chi_0^{(s)},...,chi_n^{(s)}} chi_0^{(s)}(a_o^{-1}) cdots chi_n^{(s)}(a_n^{-1})g(chi_0^{(s)}) cdots g(chi_n^{(s)}) = $$
$$q^{s(n-1)}+q^{s(n-2)}+ ... + q + 1 + frac{1}{q^s} sum_{chi_0,...,chi_n} chi_0(a_o^{-1})^s cdots chi_n(a_n^{-1})^sg(chi_0) cdots g(chi_n) $$
That was my first question. Next:
$$ q^{s(n-1)}+q^{s(n-2)}+ ... + q + 1 + frac{1}{q^s} sum_{chi_0,...,chi_n} chi_0(a_o^{-1})^s cdots chi_n(a_n^{-1})^sg(chi_0) cdots g(chi_n) = $$
$$sum_{k=0}^{n-1} q^{ks} + (-1)^{n+1} sum_{chi_0,...,chi_n} [ frac{(-1)^{n+1}}{q} chi_0(a_o^{-1}) cdots chi_n(a_n^{-1})g(chi_0) cdots g(chi_n)]^s $$
in this equation I don't understand where the first $(-1)^{n+1} $ came from (second question). and here comes my third question:
Now they use Proposition 11.1.1 to get:
$$Z_f(u) = frac{P(u)^{(-1)^n}}{(1-u)(1-qu)(1-q^{n-1}u)} $$ Here I don't see where the $(-1)^n$ came from.
I'm aware that you need context to answer my questions. So here are the pages:
And here is Proposition 11.1.1:
If you need something more, let me know.
Thank you for your help.
number-theory finite-fields characters zeta-functions
asked Nov 24 at 12:15
RukiaKuchiki
299211
add a comment |
Information: I linked the pages below, which relate to my questions.
I am currently reading " A Classic Introduction to Modern Number Theory " by Kenneth Ireland and Michael Rosen. In the 11th chapter they consider the zeta function. In the third section of this chapter they want to show that the Zeta Function is associated to $a_0x_0^m+a_1x_1^m+...+a_nx_n^m$.
They start with:
$$N_s = q^{s(n-1)}+q^{s(n-2)}+ ... + q + 1 + frac{1}{q^s} sum_{chi_0^{(s)},...,chi_n^{(s)}} chi_0^{(s)}(a_o^{-1}) cdots chi_n^{(s)}(a_n^{-1})g(chi_0^{(s)}) cdots g(chi_n^{(s)}) $$
That's ok, but I don't see why $$ q^{s(n-1)}+q^{s(n-2)}+ ... + q + 1 + frac{1}{q^s} sum_{chi_0^{(s)},...,chi_n^{(s)}} chi_0^{(s)}(a_o^{-1}) cdots chi_n^{(s)}(a_n^{-1})g(chi_0^{(s)}) cdots g(chi_n^{(s)}) = $$
$$q^{s(n-1)}+q^{s(n-2)}+ ... + q + 1 + frac{1}{q^s} sum_{chi_0,...,chi_n} chi_0(a_o^{-1})^s cdots chi_n(a_n^{-1})^sg(chi_0) cdots g(chi_n) $$
That was my first question. Next:
$$ q^{s(n-1)}+q^{s(n-2)}+ ... + q + 1 + frac{1}{q^s} sum_{chi_0,...,chi_n} chi_0(a_o^{-1})^s cdots chi_n(a_n^{-1})^sg(chi_0) cdots g(chi_n) = $$
$$sum_{k=0}^{n-1} q^{ks} + (-1)^{n+1} sum_{chi_0,...,chi_n} [ frac{(-1)^{n+1}}{q} chi_0(a_o^{-1}) cdots chi_n(a_n^{-1})g(chi_0) cdots g(chi_n)]^s $$
in this equation I don't understand where the first $(-1)^{n+1} $ came from (second question). and here comes my third question:
Now they use Proposition 11.1.1 to get:
$$Z_f(u) = frac{P(u)^{(-1)^n}}{(1-u)(1-qu)(1-q^{n-1}u)} $$ Here I don't see where the $(-1)^n$ came from.
I'm aware that you need context to answer my questions. So here are the pages:
And here is Proposition 11.1.1:
If you need something more, let me know.
Thank you for your help.
number-theory finite-fields characters zeta-functions
asked Nov 24 at 12:15
RukiaKuchiki
299211
$n=2$... Then express "there exists $b$ such that $c = b^m$" in term of characters of $E$ and $F$.
– reuns
Nov 25 at 1:17
add a comment |
Information: I linked the pages below, which relate to my questions.
I am currently reading " A Classic Introduction to Modern Number Theory " by Kenneth Ireland and Michael Rosen. In the 11th chapter they consider the zeta function. In the third section of this chapter they want to show that the Zeta Function is associated to $a_0x_0^m+a_1x_1^m+...+a_nx_n^m$.
They start with:
$$N_s = q^{s(n-1)}+q^{s(n-2)}+ ... + q + 1 + frac{1}{q^s} sum_{chi_0^{(s)},...,chi_n^{(s)}} chi_0^{(s)}(a_o^{-1}) cdots chi_n^{(s)}(a_n^{-1})g(chi_0^{(s)}) cdots g(chi_n^{(s)}) $$
That's ok, but I don't see why $$ q^{s(n-1)}+q^{s(n-2)}+ ... + q + 1 + frac{1}{q^s} sum_{chi_0^{(s)},...,chi_n^{(s)}} chi_0^{(s)}(a_o^{-1}) cdots chi_n^{(s)}(a_n^{-1})g(chi_0^{(s)}) cdots g(chi_n^{(s)}) = $$
$$q^{s(n-1)}+q^{s(n-2)}+ ... + q + 1 + frac{1}{q^s} sum_{chi_0,...,chi_n} chi_0(a_o^{-1})^s cdots chi_n(a_n^{-1})^sg(chi_0) cdots g(chi_n) $$
That was my first question. Next:
$$ q^{s(n-1)}+q^{s(n-2)}+ ... + q + 1 + frac{1}{q^s} sum_{chi_0,...,chi_n} chi_0(a_o^{-1})^s cdots chi_n(a_n^{-1})^sg(chi_0) cdots g(chi_n) = $$
$$sum_{k=0}^{n-1} q^{ks} + (-1)^{n+1} sum_{chi_0,...,chi_n} [ frac{(-1)^{n+1}}{q} chi_0(a_o^{-1}) cdots chi_n(a_n^{-1})g(chi_0) cdots g(chi_n)]^s $$
in this equation I don't understand where the first $(-1)^{n+1} $ came from (second question). and here comes my third question:
Now they use Proposition 11.1.1 to get:
$$Z_f(u) = frac{P(u)^{(-1)^n}}{(1-u)(1-qu)(1-q^{n-1}u)} $$ Here I don't see where the $(-1)^n$ came from.
I'm aware that you need context to answer my questions. So here are the pages:
And here is Proposition 11.1.1:
If you need something more, let me know.
Thank you for your help.
number-theory finite-fields characters zeta-functions
asked Nov 24 at 12:15
RukiaKuchiki
299211
Information: I linked the pages below, which relate to my questions.
I am currently reading " A Classic Introduction to Modern Number Theory " by Kenneth Ireland and Michael Rosen. In the 11th chapter they consider the zeta function. In the third section of this chapter they want to show that the Zeta Function is associated to $a_0x_0^m+a_1x_1^m+...+a_nx_n^m$.
They start with:
$$N_s = q^{s(n-1)}+q^{s(n-2)}+ ... + q + 1 + frac{1}{q^s} sum_{chi_0^{(s)},...,chi_n^{(s)}} chi_0^{(s)}(a_o^{-1}) cdots chi_n^{(s)}(a_n^{-1})g(chi_0^{(s)}) cdots g(chi_n^{(s)}) $$
That's ok, but I don't see why $$ q^{s(n-1)}+q^{s(n-2)}+ ... + q + 1 + frac{1}{q^s} sum_{chi_0^{(s)},...,chi_n^{(s)}} chi_0^{(s)}(a_o^{-1}) cdots chi_n^{(s)}(a_n^{-1})g(chi_0^{(s)}) cdots g(chi_n^{(s)}) = $$
$$q^{s(n-1)}+q^{s(n-2)}+ ... + q + 1 + frac{1}{q^s} sum_{chi_0,...,chi_n} chi_0(a_o^{-1})^s cdots chi_n(a_n^{-1})^sg(chi_0) cdots g(chi_n) $$
That was my first question. Next:
$$ q^{s(n-1)}+q^{s(n-2)}+ ... + q + 1 + frac{1}{q^s} sum_{chi_0,...,chi_n} chi_0(a_o^{-1})^s cdots chi_n(a_n^{-1})^sg(chi_0) cdots g(chi_n) = $$
$$sum_{k=0}^{n-1} q^{ks} + (-1)^{n+1} sum_{chi_0,...,chi_n} [ frac{(-1)^{n+1}}{q} chi_0(a_o^{-1}) cdots chi_n(a_n^{-1})g(chi_0) cdots g(chi_n)]^s $$
in this equation I don't understand where the first $(-1)^{n+1} $ came from (second question). and here comes my third question:
Now they use Proposition 11.1.1 to get:
$$Z_f(u) = frac{P(u)^{(-1)^n}}{(1-u)(1-qu)(1-q^{n-1}u)} $$ Here I don't see where the $(-1)^n$ came from.
I'm aware that you need context to answer my questions. So here are the pages:
And here is Proposition 11.1.1:
If you need something more, let me know.
Thank you for your help.
number-theory finite-fields characters zeta-functions
number-theory finite-fields characters zeta-functions
asked Nov 24 at 12:15
RukiaKuchiki
299211
asked Nov 24 at 12:15
RukiaKuchiki
299211
asked Nov 24 at 12:15
RukiaKuchiki
299211
asked Nov 24 at 12:15
RukiaKuchiki
299211
asked Nov 24 at 12:15
RukiaKuchiki
299211
299211
$n=2$... Then express "there exists $b$ such that $c = b^m$" in term of characters of $E$ and $F$.
– reuns
Nov 25 at 1:17
add a comment |
$n=2$... Then express "there exists $b$ such that $c = b^m$" in term of characters of $E$ and $F$.
– reuns
Nov 25 at 1:17
$n=2$... Then express "there exists $b$ such that $c = b^m$" in term of characters of $E$ and $F$.
– reuns
Nov 25 at 1:17
$n=2$... Then express "there exists $b$ such that $c = b^m$" in term of characters of $E$ and $F$.
– reuns
Nov 25 at 1:17
add a comment |
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$n=2$... Then express "there exists $b$ such that $c = b^m$" in term of characters of $E$ and $F$.
– reuns
Nov 25 at 1:17