Trying to understand a certain form of zeta function
$begingroup$
A month ago I have asked a question about a certain form of the zeta function. I will now try to be more accurate.
facts:
Let $N_s$ be the number of points on the projective hypersurface $bar{H}_f(F_s)$.
Under the condition $N_s = sum_j beta_j - sum_i alpha_i$ for $beta_j, alpha_i in mathbb{C}$ we can say that the zeta function is rational with $$ Z(u) = frac{prod_i 1-alpha_iu}{prod_j 1-beta_ju}. $$
First Question:
Now I have $$ N_s = sum_{k=0}^{n-1} q^{ks} -(-1)^n sum_{chi_0,...,chi_n} [ frac{(-1)^{n+1}}{q} chi_0(a_o^{-1}) cdots chi_n(a_n^{-1})g(chi_0) cdots g(chi_n)]^s .$$ the $chi_i$ are characters of $F$ with the following condition: $chi_i^m = varepsilon, chi_i neq varepsilon$ and $chi_0chi_1 cdots chi_n = varepsilon$. And here comes my question: why does the zeta function has the following form?
$$ Z(u) = frac{P(u)^{(-1)^n}}{(1-u)(1-qu) cdots (1-q^{n-1}u)}, $$ where $ P(u) = prod_{chi_0,...,chi_n} (1-(-1)^{n+1}frac{1}{q}chi_0(a_o^{-1}) cdots chi_n(a_n^{-1})g(chi_0) cdots g(chi_n)u) $.
Thoughts
It is clear to me that
$sum_j beta_j^s = sum_{k=0}^{n-1} q^{ks} $ and $sum_i alpha_i^s =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 $ but the term $(-1)^n$ disturbs me. Without this term I would have $Z(u) = frac{P(u)}{(1-u)(1-qu) cdots (1-q^{n-1}u)} $. That's totally fine. Just using the fact above. But like I said I don't get it why I get $(-1)^n$ in the exponent of $P(u)$ if I have the term $(-1)^n$.
Second question:
Do you see why $deg (P(u)) = m^{-1}[(m-1)^{n+1}+(-1)^{n+1}(m-1)]$ ? Sorry that I am clueless, here.
number-theory riemann-zeta characters
asked Jan 5 at 14:53
RukiaKuchikiRukiaKuchiki
337211
$endgroup$
add a comment |
$begingroup$
A month ago I have asked a question about a certain form of the zeta function. I will now try to be more accurate.
facts:
Let $N_s$ be the number of points on the projective hypersurface $bar{H}_f(F_s)$.
Under the condition $N_s = sum_j beta_j - sum_i alpha_i$ for $beta_j, alpha_i in mathbb{C}$ we can say that the zeta function is rational with $$ Z(u) = frac{prod_i 1-alpha_iu}{prod_j 1-beta_ju}. $$
First Question:
Now I have $$ N_s = sum_{k=0}^{n-1} q^{ks} -(-1)^n sum_{chi_0,...,chi_n} [ frac{(-1)^{n+1}}{q} chi_0(a_o^{-1}) cdots chi_n(a_n^{-1})g(chi_0) cdots g(chi_n)]^s .$$ the $chi_i$ are characters of $F$ with the following condition: $chi_i^m = varepsilon, chi_i neq varepsilon$ and $chi_0chi_1 cdots chi_n = varepsilon$. And here comes my question: why does the zeta function has the following form?
$$ Z(u) = frac{P(u)^{(-1)^n}}{(1-u)(1-qu) cdots (1-q^{n-1}u)}, $$ where $ P(u) = prod_{chi_0,...,chi_n} (1-(-1)^{n+1}frac{1}{q}chi_0(a_o^{-1}) cdots chi_n(a_n^{-1})g(chi_0) cdots g(chi_n)u) $.
Thoughts
It is clear to me that
$sum_j beta_j^s = sum_{k=0}^{n-1} q^{ks} $ and $sum_i alpha_i^s =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 $ but the term $(-1)^n$ disturbs me. Without this term I would have $Z(u) = frac{P(u)}{(1-u)(1-qu) cdots (1-q^{n-1}u)} $. That's totally fine. Just using the fact above. But like I said I don't get it why I get $(-1)^n$ in the exponent of $P(u)$ if I have the term $(-1)^n$.
Second question:
Do you see why $deg (P(u)) = m^{-1}[(m-1)^{n+1}+(-1)^{n+1}(m-1)]$ ? Sorry that I am clueless, here.
number-theory riemann-zeta characters
asked Jan 5 at 14:53
RukiaKuchikiRukiaKuchiki
337211
$endgroup$
add a comment |
$begingroup$
A month ago I have asked a question about a certain form of the zeta function. I will now try to be more accurate.
facts:
Let $N_s$ be the number of points on the projective hypersurface $bar{H}_f(F_s)$.
Under the condition $N_s = sum_j beta_j - sum_i alpha_i$ for $beta_j, alpha_i in mathbb{C}$ we can say that the zeta function is rational with $$ Z(u) = frac{prod_i 1-alpha_iu}{prod_j 1-beta_ju}. $$
First Question:
Now I have $$ N_s = sum_{k=0}^{n-1} q^{ks} -(-1)^n sum_{chi_0,...,chi_n} [ frac{(-1)^{n+1}}{q} chi_0(a_o^{-1}) cdots chi_n(a_n^{-1})g(chi_0) cdots g(chi_n)]^s .$$ the $chi_i$ are characters of $F$ with the following condition: $chi_i^m = varepsilon, chi_i neq varepsilon$ and $chi_0chi_1 cdots chi_n = varepsilon$. And here comes my question: why does the zeta function has the following form?
$$ Z(u) = frac{P(u)^{(-1)^n}}{(1-u)(1-qu) cdots (1-q^{n-1}u)}, $$ where $ P(u) = prod_{chi_0,...,chi_n} (1-(-1)^{n+1}frac{1}{q}chi_0(a_o^{-1}) cdots chi_n(a_n^{-1})g(chi_0) cdots g(chi_n)u) $.
Thoughts
It is clear to me that
$sum_j beta_j^s = sum_{k=0}^{n-1} q^{ks} $ and $sum_i alpha_i^s =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 $ but the term $(-1)^n$ disturbs me. Without this term I would have $Z(u) = frac{P(u)}{(1-u)(1-qu) cdots (1-q^{n-1}u)} $. That's totally fine. Just using the fact above. But like I said I don't get it why I get $(-1)^n$ in the exponent of $P(u)$ if I have the term $(-1)^n$.
Second question:
Do you see why $deg (P(u)) = m^{-1}[(m-1)^{n+1}+(-1)^{n+1}(m-1)]$ ? Sorry that I am clueless, here.
number-theory riemann-zeta characters
asked Jan 5 at 14:53
RukiaKuchikiRukiaKuchiki
337211
$endgroup$
A month ago I have asked a question about a certain form of the zeta function. I will now try to be more accurate.
facts:
Let $N_s$ be the number of points on the projective hypersurface $bar{H}_f(F_s)$.
Under the condition $N_s = sum_j beta_j - sum_i alpha_i$ for $beta_j, alpha_i in mathbb{C}$ we can say that the zeta function is rational with $$ Z(u) = frac{prod_i 1-alpha_iu}{prod_j 1-beta_ju}. $$
First Question:
Now I have $$ N_s = sum_{k=0}^{n-1} q^{ks} -(-1)^n sum_{chi_0,...,chi_n} [ frac{(-1)^{n+1}}{q} chi_0(a_o^{-1}) cdots chi_n(a_n^{-1})g(chi_0) cdots g(chi_n)]^s .$$ the $chi_i$ are characters of $F$ with the following condition: $chi_i^m = varepsilon, chi_i neq varepsilon$ and $chi_0chi_1 cdots chi_n = varepsilon$. And here comes my question: why does the zeta function has the following form?
$$ Z(u) = frac{P(u)^{(-1)^n}}{(1-u)(1-qu) cdots (1-q^{n-1}u)}, $$ where $ P(u) = prod_{chi_0,...,chi_n} (1-(-1)^{n+1}frac{1}{q}chi_0(a_o^{-1}) cdots chi_n(a_n^{-1})g(chi_0) cdots g(chi_n)u) $.
Thoughts
It is clear to me that
$sum_j beta_j^s = sum_{k=0}^{n-1} q^{ks} $ and $sum_i alpha_i^s =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 $ but the term $(-1)^n$ disturbs me. Without this term I would have $Z(u) = frac{P(u)}{(1-u)(1-qu) cdots (1-q^{n-1}u)} $. That's totally fine. Just using the fact above. But like I said I don't get it why I get $(-1)^n$ in the exponent of $P(u)$ if I have the term $(-1)^n$.
Second question:
Do you see why $deg (P(u)) = m^{-1}[(m-1)^{n+1}+(-1)^{n+1}(m-1)]$ ? Sorry that I am clueless, here.
number-theory riemann-zeta characters
number-theory riemann-zeta characters
asked Jan 5 at 14:53
RukiaKuchikiRukiaKuchiki
337211
asked Jan 5 at 14:53
RukiaKuchikiRukiaKuchiki
337211
edited Jan 5 at 15:00
RukiaKuchiki
asked Jan 5 at 14:53
RukiaKuchikiRukiaKuchiki
337211
asked Jan 5 at 14:53
RukiaKuchikiRukiaKuchiki
337211
asked Jan 5 at 14:53
RukiaKuchikiRukiaKuchiki
337211
337211
add a comment |
add a comment |
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