Finishing Dirichlets Prime Number Theorem

Question 1

What is the dedekind ζ function for a number field K?

What is the respective ruler product formula?

Answer 1

ζκ(s)=Σ1/(Na)^s
where we sum over all ideals in the ring if integers.

ζκ(s)=Π(1-1/(Np)^s)^-1

where now we sum over all prime ideals in Οκ.

Question 2

What is the analytic class number formula?

Answer 2

K a number field of degree n=r1+2•r2. Then:
limζκ(s)/s^(r1+r2-1)=-hK•RK/ωκ

Where the limit is as s tends to 0, and:
hK =class number formula
ωκ=number of roots of unity in K
RK=regulator of K

Question 3

Show that:
ζκ(s)=ζ(s)•L(s,(D/•))
Use this to show that limit as s tends to 1 of L(s,(D/•)) is positive

Answer 3

First part follows once we see that each prime ideal of the ring of integers of K divides a prime p in Z and we know how these factor in our quadratic extension.
Second part, split into case K is imaginary and real.
IMAG:
Then (D/•) is odd, and Rκ is 1. Now use the above observation, the analytic class number formula, and the functional equation for L-functions.
REAL:
L-function is even, the regulator is logε for some fundamental unit ε. Then use the same info as above.