All Questions
Tagged with analytic-number-theory algebraic-number-theory
184
questions
5
votes
1
answer
82
views
Given nonzero $p(x)\in\mathbb Z[x]$. Are there infinitely many integers $n$ such that $p(n)\mid n!$ is satisfied?
This problem comes from a famous exercise in elementary number theory:
Prove that there are infinitely many $n\in\mathbb Z_+$ such that $n^2+1\mid n!$.
I know a lot of ways to do this. A fairly easy ...
- 786
2
votes
0
answers
38
views
Prime ideals dividing the Artin conductor
Let $L/K$ be a Galois extension of number fields, and let $(\phi,V)$ be a representation of $\operatorname{Gal}(L/K)$. Let $\mathfrak{f}_\phi$ be the Artin conductor of this representation, which is ...
- 787
2
votes
0
answers
65
views
Dedekind zeta function of abelian number fields is a product of Dirichlet L-functions
Can somebody give an explicit (and self-contained) proof of the fact that the Dedekind zeta function of an abelian number field $K$ is a product of Dirichlet L-functions? I.e.,
$$
\zeta_K(s)=\prod_{\...
- 787
2
votes
1
answer
108
views
Degree of extension of the field of coefficients of modular forms
I am beginning to study modular forms and I came across an inequality defining the bound for the degree of extension of the field of coefficients of a modular form $f\in S_k(\Gamma_1(N))$. This goes ...
- 720
3
votes
0
answers
49
views
Are Hecke L-functions associated to Artin L-functions primitive?
I'm reading a famous paper by Lagarias & Odlyzko on effective versions of the Chebotarev density theorem. There is one thing about Artin L-functions that is a little perplexing to me.
Let $L/K$ be ...
- 787
0
votes
1
answer
90
views
Understanding a key definition in Lagarias Odlyzko's paper on Chebotarev density theorem
Lagarias & Odlyzko has a 1977 paper where they prove effective versions of the Chebotarev density theorem. I am having trouble understanding equation (3.1).
Here, $L/K$ is a Galois extension of ...
- 787
3
votes
1
answer
143
views
$1^\alpha+2^\alpha+3^\alpha+\cdots+n^\alpha$
Let $\alpha>0$ and $m$ be a positive integers, use Euler's summation formula we can prove that there exists a constant $C$ such that
$$ \sum_{k=1}^nn^\alpha=\frac{n^{\alpha+1}}{\alpha+1}+\frac{n^...
- 915
2
votes
0
answers
53
views
Reconciling different ideal-theoretic definitions of Hecke Characters
I'm reading Chapter 7 of Neukirch's book on algebraic number theory, where the author defines a Größencharakter (6.1) as:
Let $\mathfrak{m}$ be an integral ideal of the number field $K$, and let $J^{\...
- 787
1
vote
2
answers
58
views
Additive characters over a Number Field
Given any non-zero
integral ideal $\mathfrak{b}$ of $K$, an additive character modulo $\mathfrak{b}$
is defined to be a non-zero function $\sigma$ on $\mathfrak{o}/\mathfrak{b}$ which satisfies
$$
\...
- 304
-1
votes
1
answer
79
views
What is known about the equation $x^2+ay^2=b^2$, where $a$ is a fixed square free positive integer and $b$ is a fixed positive integer. [closed]
$(b,0)$ and $(-b,0)$ are two trivial solutions. What do we know about the nontrivial solutions of the equation $x^2+ay^2=b^2$.
- 396
1
vote
0
answers
46
views
Is there a useful/meaningful notion of a multi-variable L-function in number theory?
I recently encountered multi-variable generalizations of various classical zeta functions. For example, the multi-variable Riemann zeta function
$$
\zeta(s_1, \ldots , s_r) := \sum_{0 < n_1 < \...
- 470
2
votes
1
answer
61
views
Asymptotic Approximation towards Sum of the Composite Number's Smallest Prime Factor
I wonder if there is any asymptotic approximation towards the sum of the smallest prime factor of the composite numbers which are less than $n$. This is also the sum of terms whose index is not prime ...
- 21
1
vote
1
answer
103
views
How to show $\sum_{n=0}^{q-1} ne^{2\pi i\frac{p}{q} n^2 } = -\frac{q}{2} $ a connection between Quadratic Gauss Sums and Cesaro Summation
It can be shown that if $q \equiv 2 \mod 4 $ and $(p,q) = 1$ then
$$ \sum_{n=0}^{q-1} e^{2\pi i \frac{p}{q} n^2} = 0 $$
By the theory of Quadratic Gauss Sums which is intimately tied to Quadratic ...
- 17.5k
6
votes
1
answer
75
views
What is the adelic Schwartz function that gives the xi function?
Let $f_p = 1_{\mathbb{Z}p}$ for $p$ finite and $f_\infty(x) = e^{-\pi x^2}$ for $p$ at infinity, and $f = \otimes_v f_v$ where the product is over the places of $\mathbb{Q}$.
Then the zeta integral of ...
- 121
1
vote
0
answers
72
views
Is $ f$ bounded as $\Im z\to \infty$
Let $k$ be an integer and $\mathcal{H}$ denotes the upper half plane. Let $f: \mathcal{H}\to \mathbb{C}$ be a function satisfying the following conditions:
$1.$ $f$ is holomorphic on $\mathcal{H}$.
$2....
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