All Questions
Tagged with reference-request nt.number-theory
1,409 questions
10
votes
0
answers
373
views
Local Langlands Correspondence for unramified principal series representations
Let $G$ be a connected, reductive group over a $p$-adic field $k$. Assume $G$ is quasi-split with Borel subgroup $B = TU$. Consider those irreducible admissible representations $\pi$ of $G(k)$ which ...
- 6,180
2
votes
3
answers
753
views
Reference Request for Integer factorization with LP/ILP
Have there been attempts to factor integers with Linear Programming?
Searching the internet suggests that for Integer Factorization only Number Theoretic algorithms, like sieves, are taken into ...
- 13.2k
8
votes
1
answer
1k
views
Generalization of Hilbert 94 and capitulation
Let $L/K$ be a finite, cyclic extension of number fields, say with $\mathrm{Gal}(L/K)=G$. In my context $G$ is actually of order $p$, an odd prime number, but let me state my question for every cyclic ...
- 5,670
2
votes
0
answers
160
views
Where can I find a copy of this paper of Chowla and Vijayaraghavan?
Does anyone know where I can find a copy of Chowla and Vijayaraghavan's paper, ''On the largest prime divisors of numbers''?
The relevant literature say it was published in the Journal of the Indian ...
- 1,019
2
votes
3
answers
515
views
Asymptotic number of zeros for Dirichlet series with functional equation
I think the usual proof for the asymptotic number of zeros of the Riemann zeta function
$$N(T) = \#\left\{\rho : \ \zeta(\rho)=0, \begin{array}{l}\scriptstyle Im(\rho)\ \in\ [0,T]\\ \scriptstyle Re(\...
- 3,403
6
votes
1
answer
419
views
independence of $\ell$ of characteristic polynomial of Frobenius on $\ell$-adic Tate module of Abelian varieties over number fields
I am looking for a reference for the independence of $\ell$ of the characteristic polynomial of the Frobenius $\mathrm{det}(1-|\kappa(v)|^{-s}\mathrm{Frob}_v \mid (V_\ell A)^{I_v})$ acting on the $\...
user19475
2
votes
1
answer
334
views
Intuition behind the proof of key step in Minkowski's second inequality on successive minima
I recently knew of this note in which Prof. M. Henk presents a proof of Minkowski's second inequality on successive minima which is (purportedly) based on ideas in Minkowski's original proof. Let me ...
- 8,842
19
votes
1
answer
1k
views
Ehresmann's theorem over the $p$-adics
I am looking for a version of Ehresmann's theorem for analytic manifolds over the $p$-adic numbers $\mathbb{Q}_p$ or, more generally, local fields. I follow the conventions from Serre's book "Lie ...
- 21.4k
6
votes
1
answer
480
views
Twisted modular forms of half-integral weight
I am looking for references (or explainations) about the twist of modular forms of half-integral weight. I try to mimic the proof of the "integral weight case" to prove that the twist of
$$ \theta(\...
- 1,479
10
votes
1
answer
554
views
Who was/were the first to note that if $\sum_{x \in X} \frac{1}{x} < \infty$ then the natural density of $X$ is zero?
It is a result of folklore that the natural density of a set $X$ of positive integers such that $\sum_{x \in X} \frac{1}{x} < \infty$ is zero. This is reproved, e.g., in T. Šalát's paper: ...
- 10.5k
5
votes
1
answer
2k
views
Generalizing Dedekind's Factorization Theorem
A classical theorem due to Dedekind states the following:
Let $O_{K}$ be the ring of integers of a number field $K$, and assume
$K$ is generated by adjoining the algebraic integer $\alpha$ to
$...
- 14.6k
1
vote
0
answers
174
views
Reference to a particular result of Scholl and Faltings
Let $f=\sum_{n\geq 1} a_n q^n$ be a normalized eigenform which is supersingular and crystalline at a prime $p$ and let $V_f$ be the associated crystalline representation, then it follows from the work ...
user130124
14
votes
2
answers
858
views
References for particular topics related to Langlands
I have never really concentrated on Langlands, which explains my poor level of understanding of it. But I have read quite a few introductory papers related to Langlands, and to the circle of ideas ...
4
votes
0
answers
200
views
When does a continuous function's "Fourier series" converge pointwise almost everywhere to the function?
Let $G$ be a compact topological group. By the Peter-Weyl theorem, the complex Hilbert space $L^2(G)$ is the Hilbert space direct sum of the spaces of matrix coefficients of all the irreducible ...
- 6,180
5
votes
1
answer
224
views
Locating a certain result on primes represented by a certain polynomial
In Theorem 2 of the paper "A polynomial divisor problem" by Friedlander and Iwaniec, Theorem 2 states that $$\sum_{a^6 + b^2\le x} \Lambda(a^6 + b^2)\sim cx^{2/3}$$ for some constant $c > 0$ (in ...
- 1,890
14
votes
4
answers
3k
views
Fourier decay rate of Cantor measures
For $0<\theta<\frac{1}{2}$, denote by $C_\theta$ the Cantor set with dissection ratio $\theta$, i.e. the Cantor set obtained from dissection parttern $(\theta, 1-2\theta,\theta)$. It is known ...
- 981
7
votes
1
answer
443
views
Density of numbers whose prime factors belong to given arithmetic progressions
By a theorem of Landau, the number of integers $n\leq x$ whose prime divisors belong to only arithmetic progressions $a_1,\dots,a_r$ mod $q$, with $r\leq\varphi(q)$ and $a_i$ coprime to $q$ for each $...
- 3,799
8
votes
2
answers
5k
views
Inverse of the Riemann zeta function [closed]
I'm wondering if there is any information on the inverse of the Riemann zeta function (not it's reciprocal, but its functional inverse). This would obviously be a multi-valued function.
- 757
-1
votes
3
answers
301
views
Reference for Siegel-Walfisz Theorem under GRH
Let $\Lambda$ be the von Mangoldt function.
I think the following is known to hold under GRH: Given any $q \geq 1$, $(a,q)=1$, and $X \geq 1$, we have
$$
\sum_{\substack{1 \leq n \leq X \\ n \equiv ...
- 3,625
1
vote
1
answer
443
views
Distribution of Mobius function
Let $\mu(n)$ be the Mobius function.
Let us define $\mu^+(n)$ to be $\mu(n)$ if $\mu(n)>0$ and $0$ otherwise. Is there a known asymptotic formula for
$$
\sum_{n \leq N} \mu^+(n),
$$
and ...
- 3,625
10
votes
4
answers
1k
views
Introductory reading on the Scholz reflection principle?
The Scholz reflection principle says, among other things, that if $D < 0$ is a negative fundamental discriminant, not $-3$, then the 3-ranks of the class group of $\mathbb{Q}(\sqrt{D})$ is either ...
3
votes
1
answer
188
views
Reference request: denominators of lonely runner numbers
The lonely runner conjecture states that for $M=\{m_1,...,m_n\}$ a set of distinct positive integers the quantity
$$
\kappa(M):=\sup_{t \in \mathbb{R}} \min_i ||tm_i||
$$
satisfies $\kappa(M) \geq \...
- 2,753
7
votes
1
answer
281
views
Sato-Tate conjecture when Fourier coefficients are complex numbers
Let $k\geq 1$ and let $f=\sum_{n\geq 1}a(n)q^{n}$, $a(n)\in\mathbb{R}$, be a normalised cuspidal Hecke eigenform of weight $2k$ for $\Gamma_{0}(N)$ without complex multiplication. So the result of ...
- 433
5
votes
1
answer
1k
views
Reference to "bounds of Weil and Deligne"
In the this paper by Friedlander and Iwaniec, it is said that they are "able to avoid much of the high-powered technology frequently used in modern analytic number theory such as the bounds of Weil ...
- 1,890
2
votes
0
answers
136
views
numbers independent over $\mathbb{Q}$ but not BA? numbers that aren't a basis for a number field but are BA?
Has anyone discovered a vector of algebraic real numbers $(a_1,...,a_k)$ such that $1,a_1,...,a_k$ are linearly independent over $\mathbb{Q}$ and such that $(a_1,...,a_k)$ is not "badly approximable"?
...
- 21
2
votes
1
answer
516
views
On comparing two almost injective divisor maps
Edit 2018.08.08 This answer https://mathoverflow.net/a/307881 will be updated to give recent information about S, especially a forthcoming preprint. End Edit 2018.08.08
In an introductory post on ...
- 13k
22
votes
1
answer
2k
views
Reference request: The first cohomology of SL(2,Z) with coefficients in homogeneous polynomials
Let $H_k$ be the vector space of degree $k$ homogeneous polynomials in two variables.I'm looking for a reference for the fact that $H^1(SL(2,\mathbb Z);H_k)=M^0(k+2)\oplus\overline{M^0(k+2)}\oplus E_{...
- 4,898
5
votes
2
answers
369
views
Links between tight closure and deformation theory
I am looking for links between tight closure and deformation theory. As a sample question:
Question 1. Are there geometric interpretations in terms of deformation theory of
Frobenius rationality?
...
- 32.2k
1
vote
2
answers
287
views
On the notion of primary representation of a natural number by a quadratic form
This "discussion" has to do with some of the material we can find in pages 183-186 of the translation into English of the first part of E. Landau's Vorlesungen über Zahlentheorie (published by the ...
- 8,842
8
votes
1
answer
476
views
How does Jacquet's "Generic Representations" classify tempered representations?
Let $L$ be a $p$-adic field $G = GL_n(L)$. Let $P$ be a standard parabolic subgroup with Levi decomposition $P = MU$, where $M \cong G_1\times \ldots \times G_r$, for $G_i \cong GL_{n_i}(L)$.
The ...
- 1,453
9
votes
2
answers
647
views
On bounds for idoneal integer
What is the best known lower bound and upper bound known for such a number if it exists and have there been any attempts (computational including) to eliminate the existence of such a number in known ...
user76479
3
votes
0
answers
540
views
Questions about the exceptional zeros of Dirichlet $L$-functions
I have couple questions regarding the exceptional zeros of Dirichlet $L$-functions. We have the following result:
There is a constant $c_1 > 0$ such that $L(\sigma, \chi) \not = 0$ whenever
$$
\...
- 3,625
3
votes
2
answers
506
views
Indefinite quadratic form universal over negative integers
Here's a question that (I hope) may seem very trivial for you, and I hope one of you may provide me with a reference answering it (unless it's a trivial colloquial knowledge).
Let $f$ be an ...
- 1,268
0
votes
0
answers
132
views
Final step in Coppersmith?
In the final step in Coppersmith technique we have $n$ polynomials (possibly non-homogeneous) in $\mathbb Z[x_1,\dots,x_m]$ where $m\leq n$ and using elimination theory we extract the common integer ...
- 13.9k
5
votes
0
answers
772
views
The Grimm Machine(s): A Collatz Conjecture Rival?
Edit 2018.08.08 This answer https://mathoverflow.net/a/307881 will be updated to give recent information about S, especially a forthcoming preprint. End Edit 2018.08.08
Just as the Collatz ...
- 13k
5
votes
4
answers
4k
views
Values of Dirichlet L-funcions at natural numbers
I want to know about reference of formulas for
$$
L(s,D)=\sum_{n=1}^\infty \left(\frac{D}{n}\right)\,n^{-s}
$$
for $s$ a positive integer number and $D$ a fundamental discriminant. For $s=1$ we have ...
- 2,446
34
votes
2
answers
3k
views
Shimura-Taniyama-Weil VS Grothendieck's dessins
When listening to the beautiful lectures by Gilles Schaeffer at
the SLC68, the following (perhaps crazy) question occurred to me:
did anyone attempt (succeed?) to combinatorially prove modularity of ...
19
votes
1
answer
2k
views
"The Galois group of $\pi$ is $\mathbb{Z}$."
Last year, in a talk of Michel Waldschmidt's, I remember hearing a statement along the lines of the title of this question, that is, "The Galois group of $\pi$ is $\mathbb{Z}$.". In what sense/...
- 1,105
5
votes
1
answer
431
views
About Morley congruence
Let $p>3$ be an odd prime and $a$ be a positive integer, is the following congruence true?
$$\binom{p^a-1}{\frac{p^a-1}{2}}\equiv(-1)^{\frac{p^a-1}{2}}4^{p^a-1}\pmod{p^3}.$$
When $a=1$, this is ...
- 51
18
votes
1
answer
562
views
Is special value of Epstein zeta function in 3 variables a period?
Kontsevich-Zagier's article "Periods" contains the following question
Is $\displaystyle \sum_{x,y,z \in \mathbb{Z}}' \frac{1}{(x^2+y^2+z^2)^2}$ an extended period?
($\sum'$ means we do not sum ...
- 809
5
votes
0
answers
161
views
A relation concerning the "sum of squares" counting function $r_2(n)$
This is a re-post from MSE as I did not get any response there.
Let $r_2(n)$ denote the number of ways in which a positive integer $n$ can be expressed as the sum of squares of two integers. Here ...
- 2,259
5
votes
2
answers
283
views
LDPC codes construction
According to Google Scholar original Gallager's article Low-density parity-check codes is cited more than 10000 times. It looks scary for non-experts.
I suspect that the number of algorithms for ...
- 12.3k
9
votes
0
answers
275
views
Complete list of exceptions and efficient algorithm for Waring's problem
2 weeks ago, Samir Siksek https://arxiv.org/abs/1505.00647 proved the more than 150-years-old conjecture that every positive integer other than 15, 22, 23, 50, 114, 167, 175, 186, 212, 231, 238, 239, ...
- 91
10
votes
4
answers
1k
views
Number of solutions of linear homogenous Diophantine equation inside a box
Let $a_1, ..., a_d$ be positive reals and consider the linear Diophantine equation
$$
\sum_i a_in_i = 0.
$$
I am interested in estimating the number of integer solutions of this equation inside a ...
- 960
15
votes
4
answers
2k
views
Are any good strategies known for Erdos-Turan conjecture on additive bases of order two?
The following problem can become a bit of an obsession. I'm curious if there are any serious strategies for attacking it. The problem is a certain Erdos-Turan conjecture.
Let $ B \subseteq {\mathbb ...
- 7,067
10
votes
2
answers
955
views
Semisimplicity of étale cohomology representations
Let $K$ be a number field and $G=Gal(\overline{K}/K)$ the absolute Galois group of $K$. Let $\ell$ be a prime number.
Let $A/K$ be an abelian variety. Then the representation of $G$ on $V_\ell(A)$ is ...
- 1,673
5
votes
0
answers
156
views
Reference Request: Cohomology and limits of coherent topoi. (Non-abelian case)
SGA 4 VI Discusses finiteness conditions one can impose on topoi to make limits behave correctly. I am not that familiar with SGA but it is my impression that this expose only discusses abelian ...
- 435
4
votes
2
answers
928
views
Examples of Sets with Positive Upper Density
While reading the statement of Roth's theorem I started asking myself what are examples of sets of positive upper density? It's not hard to come up with a few:
Flip a coin with probability $\mathbb{...
- 22.8k
5
votes
2
answers
619
views
Are there any patterns in simple continued fraction expansions of algebraic real numbers?
As we know there are patterns in simple continued fraction expansion of quadratic algebraic numbers,are there any patterns in simple continued fraction expansions of other algebraic real numbers?...
- 3,104
2
votes
1
answer
217
views
Diagonalising a symmetric matrix with polynomial entries
Suppose I have a symmetric $2$ by $2$ matrix $M$ whose $(i,j)$-th entry $F_{i,j}(\mathbf{x})$ belongs to $\mathbb{R}[x_1, \ldots, x_n]$ for each $i,j$. I know that for each $\mathbf{x} \in \mathbb{R}^...
- 3,625