All Questions
10 questions
22
votes
3
answers
2k
views
Hecke equidistribution
For a prime $p\equiv 1\pmod{4}$, we can write $p=a^2+b^2=N(a+bi)$. Therefore
$$
a+bi=p^{1/2}e^{i\varphi}
$$
where $\varphi\in [0,2\pi]$. I know that Hecke proved that $\varphi$ is equidistributed. I ...
- 2,508
23
votes
3
answers
2k
views
Why are values of Eisenstein $E_2^*$ algebraic integers?
I'm looking for a proof that the following term is an algebraic integer whenever $\tau_N=\frac{N+\sqrt{-N}}{2}$ is a quadratic irrationality with class number $1$:
$$A_N:=\sqrt{-N}\cdot\frac{E_2(\...
- 598
4
votes
1
answer
504
views
How does this calculation of Siegel make sense?
I am reading Siegel's paper Zum Beweise des Starkschen Satzes. Let $K$ be an imaginary quadratic field with $d_K=-p$, $p=4k+3$ a prime, and such that $h_K=1$.
Let $f=4m+1$ be a prime inert in $K$, ...
- 2,375
3
votes
1
answer
363
views
How many non principal prime ideals does a number field contain?
Let $K$ be a number field with ring of integers $O_K$ is not PID. Can we estimate the cardinality of the following sets
$$\mathcal{A}= \{\mathcal{P}\subset O_K \ |\ Nm(\mathcal{P})\leq x, \mathcal{P}\...
- 743
47
votes
3
answers
5k
views
Class Numbers and 163
This is a bit fluffier of a question than I usually aim for, so apologies in advance if this doesn't pass the smell test for suitability.
Likely my favorite fun fact in all of number theory is the ...
- 8,467
13
votes
2
answers
2k
views
Why is $\frac{\sqrt{6}}{32}(29 + \sqrt{145}) \approx \pi$ ?
Apologies in advance if this is a stupid question; also, disclaimer: this is purely for fun; but:
Why is $\frac{\sqrt{6}}{32}(29 + \sqrt{145})$ such a good approximation to $\pi$?
(Correct to 8 ...
- 1,990
11
votes
1
answer
1k
views
Upper bounds for regulators of real quadratic fields
We have an elementary sharp lower bound for the regulator of a real quadratic field as a function of the discriminant
$$R\geq \tfrac{1}{2}(\sqrt{d-4}+\sqrt{d})$$
It is sharp because the equality ...
- 17.6k
8
votes
2
answers
973
views
Easiest way to see that $\zeta_{\mathbb{Z}[i]}(s) = \zeta(s) L(s, \chi)$?
As the question suggests, what is the easiest way to see that$$\zeta_{\mathbb{Z}[i]}(s) = \zeta(s)L(s, \chi)?$$Here, $\chi$ is the homomorphism $(\mathbb{Z}/4\mathbb{Z})^\times \to \mathbb{C}^\times$ ...
user75660
5
votes
0
answers
136
views
When are the Artin symbols of two primes equal?
Let $K$ be an abelian number field. Let $p$, $q$ be rational primes. Is there some condition like $p\equiv q$ modulo some integer which depends on conductor of $K$ or $\operatorname{disc}(K)$ that ...
- 343
4
votes
1
answer
545
views
class number of biquadratic fields
Can any one provide some references which treat the relation between the class number of a biquadratic field and the class numbers of its sub-fields using the analytic class number formula ?
- 347