All Questions
150 questions
3
votes
2
answers
363
views
Largest prime factors of integer polynomials
I have a question in analytic number theory which is closely related to the open problem (Bunyakovsky conjecture and more generally, Schinzel's hypothesis H) that asks you if, any irreducible ...
- 302
8
votes
1
answer
868
views
Brauer–Siegel's Theorem and application
$\newcommand\underto[1]{\xrightarrow[#1]{}}$Brauer–Siegel's Theorem says that if we consider an infinite sequence of number fields $K_i/\mathbb{Q}$ such that $$\frac{[K_i:\mathbb{Q}]}{\log{|D_i|}}\...
- 12.9k
3
votes
1
answer
203
views
Chowla's theorem on class number of real quadratic field
Let $p\equiv1\bmod 4$ be a prime number and $h$
the class number of real quadratic field $\mathbb Q(\sqrt{p})$, $\epsilon=\frac{t+u\sqrt{p}}{2}$ its fundamental unit. In this paper https://www.pnas....
- 7,193
9
votes
2
answers
902
views
How to compute with the Stark conjectures?
I would like a convenient basis for the elements of a fixed abelian extension $E$ of a real quadratic field $\mathbb{Q}(\sqrt{d})$. The accepted answer to this MO question suggests that the Stark ...
- 2,753
11
votes
1
answer
1k
views
Upper bounds for regulators of real quadratic fields
We have an elementary sharp lower bound for the regulator of a real quadratic field as a function of the discriminant
$$R\geq \tfrac{1}{2}(\sqrt{d-4}+\sqrt{d})$$
It is sharp because the equality ...
3
votes
2
answers
257
views
On the arithmetic of powers of subseries of the exponential series
Let $p$ be a prime number and $q=p-1$. I’m trying to prove that the nonzero coefficients $a_{qk}$ ($k\ge1$) of the power series
$$ \sum_{k\ge1} a_{qk} z^{qk} := \left( \sum_{k\ge0} \frac{z^{qk+1}}{(qk+...
- 17k
2
votes
0
answers
107
views
Record for determining complete list of imaginary quadratic fields with small class number
In 2003, Mark Watkins (Class numbers of imaginary quadratic fields) determined all imaginary quadratic fields having class number at most 100.
Has this list been improved? That is, what is the largest ...
- 869
1
vote
0
answers
133
views
Automorphy of the twisted representation
The Artin reciprocity says that if
$$
\chi: \operatorname{Gal}(K/\mathbb Q) \to \mathbb C
$$
is a 1-dimensional representation of a finite Galois extension $K/ \mathbb Q$, then it corresponds to a ...
2
votes
0
answers
110
views
Gaussian primes in translations of lattices in $\mathbb{Z}[i]$
I am considering undertaking some independent research in my summer break studying Gaussian primes in translations of lattices in $\mathbb{Z}[i]$, i.e. sets of the form $ \{ a+sx+tw:s,t \in \mathbb{Z} ...
- 21
1
vote
0
answers
59
views
A question on generalized bases
I just came to know that it is possible to define a generalized base as an infinite sequence of natural numbers $\mathbf b=(b_1,b_2,\dots)$ where $b_i\ge 2$ for all $i$. With this definition, any $m\...
4
votes
1
answer
224
views
Generators of the ideal class group
Theorem 4 of Eric Bach's "Explicit bounds for primality testing and related problems" states the following:
Let $K$ be a number field of degree greater than 1. Let $d$ be the absolute value ...
- 105k
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 ...
- 141
2
votes
1
answer
264
views
'$\times$' or '$\otimes$' when writing $L$-functions?
Recently, I came across the Langlands correspondence theorem, there is the following line:
$$L(s,\pi(\sigma) \times \pi(\tau)) = L(s,\sigma \otimes \tau), $$
where $\sigma$ and $\tau$ are ...
- 105k
0
votes
0
answers
44
views
Asymptotic counts for imaginary quadratic discriminants with fixed splitting conditions
Let $p$ be prime and $r$ be a positive integer. I am interested in asymptotics for the number of imaginary quadratic discriminants $d$ such that
$p$ does not divide the conductor of $d$,
$p$ splits ...
- 1,354
2
votes
2
answers
283
views
Expressions for binomial residue sum $\sum_{k=0}^n {n \choose k} x^k \left( \frac{k}{q} \right)$
I'm interested in the sum:
$$\sum_{k=0}^n {n \choose k} x^k \left( \frac{k}{q} \right)$$
where $q$ is a prime number. This is just the binomial expansion with an extra weight on quadratic residues ...
- 14.6k
2
votes
0
answers
103
views
On equidistribution of primes in positive characteristic
In S. Lang's book "Algebraic Number Theory" (1986), page 317, Theorem 6 states essentially that given $P$ a set of primes, let $\tau:P\longrightarrow J$ be the typical idèle map taking ...
- 869
3
votes
1
answer
538
views
Density of prime ideals of a given degree
Let $K$ be a number field. For each ideal $I$ of the ring of integers $\mathcal{O}_K$ let $N_K(I)$ denote the norm of $I$. For a prime $\mathfrak{p}\subset \mathcal{O}_K$ above the rational prime $p\...
- 28.2k
0
votes
1
answer
112
views
Statistics of action of Galois group of number field on primes over unramified rational primes
Let $p \in \mathbb{Z}$ be prime and $K / \mathbb{Q}$ be a finite Galois extension. The Galois group $G$ of $K$ acts on the primes of $\mathcal{O}_K$ over $p$. Do we know any statistical information ...
- 105k
7
votes
0
answers
162
views
Relation between the additive Haar measure on $(K,+)$ and the multiplicative Haar measure on $K^{*}$ for a global field $K$
The following question comes from my studying of Alain Connes's paper Trace Formula in Noncommutative Geometry and the Zeros of the Riemann Zeta Function. In it, on p. 11, Connes notes that if $K$ is ...
- 50.6k
-2
votes
2
answers
149
views
Calculate the great common factor between $2^{2n+1}-1$ and $2^{4m+2}+1$ [closed]
How to calculate the great common factor between $2^{2n+1}-1$ and $2^{4m+2}+1$, where $n$ and $m$ are positive numbers.
We guess that: the great common factor is $1$.
- 34.3k
10
votes
1
answer
398
views
Vinogradov-Korobov prime number theorem for number fields
Without assuming the Riemann hypothesis, the traditional error bound of the prime-counting function $\pi(x)$ is $O(x\exp(-c(\log(x))^{1/2}))$. As shown by the Wikipedia page for the Landau prime ...
- 8,374
14
votes
0
answers
446
views
Is every prime $q$ of the form $x^2 + py^2$ for some prime $p<q$?
For every odd prime $q \geq 3$, does there exist a prime $p < q$ and integers $x,y$ such that
$$\displaystyle x^2 + py^2 = q?$$
One can easily show that all primes $q \not \equiv -1 \pmod{3}$ can ...
- 63.9k
1
vote
0
answers
115
views
Integral points in smooth cubic curves
Let $X$ be a smooth affine cubic curve in $\mathbb A^2$ defined by $f(T_1,T_2)\in\mathbb Z[T_1,T_2]$ (of course $\deg(f)=3$ by definition), and
$$n(f, B)=\{(x_1,x_2)\in\mathbb Z^2| |x_1|\leq B, |x_2|\...
- 403
4
votes
1
answer
520
views
Proving $\zeta_K\left(\frac12\right)\neq 0 \implies \zeta_K'\left(\frac12\right)\neq0?$
For an algebraic number field $K$, let $\zeta_K(s)$ be the Dedekind zeta function associated to $K$, and let $\zeta_K'(s)$ be its derivative.
I believe that the following statement is true:
$$\zeta_K\...
- 79.9k
3
votes
1
answer
758
views
Looking for a paper of Lagarias and Odlyzko
I have been studying about the Chebotarev Density Theorem and have been hunting for the following paper of Lagarias and Odlyzko for quite a while:
Effective versions of the Chebotarev density theorem, ...
- 190
2
votes
0
answers
356
views
Classifying solutions of a certain Diophantine Equation
The following question arose from a problem I am working on.
Let $N, k$ be positive integers. Consider the Diophantine equation in $a, b, c$:
$$
\frac{1}{a} + \frac{N - 1}{b} = \frac{N^k}{c}
$$
with ...
- 791
7
votes
0
answers
118
views
Upper bound on $\#\{p \leq B :\#E(\mathbb{F}_p) \equiv a \mod b\}$ for large $b$
Let $E$ be a fixed elliptic curve over $\mathbb{Q}$. Is there a good upper bound on $\#\{p \leq B :\#E(\mathbb{F}_p) \equiv a \mod b\}$ when $b$ is large (maybe around $\sqrt{B}$)? I don't mind ...
- 270
0
votes
1
answer
305
views
Generalization of Gauss's class number one problem
Gauss's class number one problem for imaginary quadratic fields, now a theorem due to Heegner, Stark, and Baker (independently), asserts that the set of imaginary quadratic fields having class number ...
- 11
10
votes
1
answer
841
views
Infinitely many primes that split completely in an arithmetic progression
Let $d \geq 1$ be an integer. Dirichlet's theorem on arithmetic progression implies that the arithmetic progression $a, a+d, a+2d, \ldots$ contains infinitely many primes if and only if $\gcd(a,d)=1$.
...
- 50.6k
4
votes
0
answers
197
views
Bailey's lemma in number theory
A pair of sequences $(α_n,β_n)$ is called a Bailey pair if they are related by
$$\beta_n=\sum_{r=0}^n\frac{\alpha_r}{(q;q)_{n-r}(aq;q)_{n+r}}$$
or equivalently
$$\alpha_n = (1-aq^{2n})\sum_{j=0}^n\...
- 41
2
votes
0
answers
245
views
Ambiguity about the exact definition of coefficients of modular forms
You can see the parts after my questions in the boxes. I received the answer to my first question in the comments.
I am confused about the definition of $a_n$ and $b_n$ in Part II below. I know the ...
- 2,784
2
votes
0
answers
1k
views
Advanced texts on analytic number theory?
So a friend of mine is very interested in analytic number theory, and is looking for resources past the basic level.
He has studied analytic number theory from several books, among them are Hardy’s ...
- 6,215
12
votes
1
answer
2k
views
Artin conjecture on L-functions
Artin conjecture on Artin $L$-functions asserts that the Artin $L$-function $L(\rho,s)$ of a non-trivial irreducible representation $\rho$ of the Galois group $\Gamma$ of a number field admits ...
- 105k
1
vote
0
answers
84
views
How common are semiprimes with equally bitsized factors among semiprimes with equal bitsize?
I am curious about the following after having looked at the paper "Almost primes in almost all short intervals", theorem 3 says:
Almost all intervals $[x, x + \log^{3.51}{(x)}]$ with $x ≤ X$...
- 11
2
votes
2
answers
263
views
Sign of the special value at s=0 of Hecke L-functions
Let $L/K$ be an abelian extension of number fields with Galois group $G$ and let $\chi : G \to \{\pm 1\}$ denote a real linear character of $G$. Denote $L(\chi,s)$ the Artin L-function associated to $\...
- 617
3
votes
1
answer
261
views
Consecutive integers that are coprime to a given number
Let $n \in \mathbb{N}$. Is there a general formula for $|\{1 \leq k \leq n \mid (k(k+1),n）= 1\}|$?
Or even more generally, for $1 \leq r < n$, is there a formula for $|\{1 \leq k \leq n \mid (k(k+r)...
- 1,381
6
votes
0
answers
456
views
Conditions under which an $\eta$-quotient becomes a **weak** modular form (reference request for theorems similar to Ligozat's theorem)
For any $z \in \mathcal{H}$, let $q = e^{2\pi iz}$; and the eta function is defined as
${\displaystyle \eta (q)
=q^{\frac {1}{24}}\prod _{n=1}^{\infty }\left(1-q^{n}\right).}$
By an $\eta$-quotient ...
2
votes
1
answer
159
views
On some claims on cyclic modules over Hecke algebra used in Serre's "Quelques applications du théorème de densité de Chebotarev"
I have been reading section 7 of Serre's "Quelques applications du théorème de densité de Chebotarev" (http://www.numdam.org/item/PMIHES_1981__54__123_0/), and in particular have been trying ...
- 105k
5
votes
0
answers
349
views
Smallest prime $p$ such that $2\mid\operatorname{ord}_p(q)$, the multiplicative order of $q$ modulo $p$
$\DeclareMathOperator\ord{ord}$Let $q$ be prime. I want to upper bound the smallest odd prime $p$ such that $2\mid\ord_p(q)$ (where $\ord_p(q)$ is the multiplicative order of $q$ modulo $p$).
Using ...
- 101
2
votes
0
answers
270
views
Generalized Siegel Weil formula
I am studying the following Poincare-like series,
\begin{equation}
F_k(\tau,\bar{\tau})=\sum_{\gamma\in\Gamma_{\infty}\backslash\Gamma}\sqrt{\text{Im}\gamma\tau}(q_{\gamma}\bar{q}_{\gamma})^k,
\end{...
- 103
1
vote
0
answers
172
views
Does Lemma 5.4 in Deligne's Ramanujan paper generalize to Shimura varieties of PEL type?
It is generally not known if a smooth variety over a perfect field embeds into a smooth proper variety.
Lemma 5.4 in Formes modulaires et représentations $\ell$-adiques provides such an embedding for ...
- 31
1
vote
0
answers
135
views
Modularity for $\mathrm{GL}_2/\mathbb{Q}(\sqrt[3]{2})$ [closed]
What is known about modularity for $\mathrm{GL}_2/\mathbb{Q}(\sqrt[3]{2})$?
- 11
4
votes
1
answer
355
views
Clarification regarding a claim in Heilbronn’s 1934 paper
I was reading Heilbronn’s 1934 paper where he proves that $H(d) \to \infty$ as $d \to -\infty$, where $H(d)$ is the ideal class number of the imaginary quadratic field with discriminant $d$. I couldn'...
- 39.9k
5
votes
0
answers
205
views
Is there a polynomial version of Wilson's theorem which can avoid Cramer flavored conjectures?
Wilson's theorem states that a natural number $n > 1$ is a prime number if and only if the product of all the positive integers less than $n$ is one less than a multiple of $n$.
Is there a version ...
- 13.9k
7
votes
1
answer
635
views
Is there a Chebotarev‘s theorem for non-Galois extension over Q？
For a Galois extension $K/\mathbb{Q}$, the Chebotarev Density Theorem predicts the density of primes with a certain splitting type.
I'm wondering if there is a similar result for non-Galois extension?
...
- 105k
5
votes
0
answers
274
views
Goldfeld resolution of the quadratic class number problem
Goldfeld proved the following result. Let $E$ be an elliptic curve (with conductor $N$) over $\mathbb{Q}$ whose Hasse-Weil L-function has a zero at $s = 1$ with multiplicity $g$ then for sufficiently ...
- 63.9k
3
votes
0
answers
186
views
Maximum value of newform from Galois representation
One can attach $\ell$-adic Galois representations to holomorphic cuspidal newforms of weight $2$ on the upper half-plane.
If a newform is $L^2$-normalized, can one extract its maximum value from the ...
- 105k
9
votes
1
answer
1k
views
Functional equation Dedekind zeta function
I'd like to know to what point is it possible to generalize
this method
for obtaining the functional equation for the Dedekind zeta function $\zeta_K(s)$ of a number field ?
Let $\mathfrak{C}$ be ...
CommunityBot
- 1
7
votes
1
answer
400
views
Splitting of small primes in number fields generated by the torsion of elliptic curves
Suppose $E/\mathbb Q$ is a non CM elliptic curve and we look at the number field $K_d$ generated by the $d$-torsion of $E$. What is known about the (complete) splitting of small primes in $K_d$?
More ...
- 3,065
12
votes
1
answer
1k
views
Where should I learn about the p-adic L-functions of elliptic curves?
Where is the best place to learn about the p-adic L-functions of Elliptic Curves? Doing a bit of research I have found books like "An Introduction to Cyclotomic Fields" by Washington, but ...
- 26k