All Questions
Tagged with reference-request nt.number-theory
388 questions with no upvoted or accepted answers
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
0
answers
100
views
Function equation over general number fields
Let $\chi$ be a Hecke character on a number field $k$, where could I find a precise reference for the function equation of the $GL(1)$ L-functions
$$L(s, \chi)?$$
I only find references for the case ...
- 927
2
votes
0
answers
154
views
Categorical representations of absolute Galois groups
I am looking for interesting examples of categorical representations of absolute Galois groups of arithmetic fields. Pointers to the literature would be appreciated.
user127738
2
votes
0
answers
217
views
A high dimensional generation of Dirichlet approximation theorem, linear case and nonlinear case
I am working with something on Diophantine approximation, and I found a high dimensional generation of Dirichlet approximation theorem which may be true; I will be very happy if this is true. The ...
- 697
2
votes
0
answers
519
views
Good place to learn about arithmetic schemes?
Where is a good place to learn about arithmetic schemes? There is discussion in Eisenbud-Harris's book The Geometry of Schemes (and also Mumford's red book) and I hear that there is discussion in Liu'...
- 21
2
votes
0
answers
325
views
Reference/PDF request for paper by Sathe and related note
I am looking for a PDF version of the following articles:
Sathe, L. G. - On a problem of Hardy on the distribution of
integers having a given number of prime factors, (I. - IV.) J. Indian Math. Soc. ...
- 593
2
votes
0
answers
154
views
Equi-distribution of the parity of partitions
The integer partition function $p(n)$ has a generating function given by
$$\frac1{(q)_{\infty}}=\sum_{n=0}^{\infty}p(n)q^n$$
with $(q)_{\infty}=\prod_{m=1}^{\infty}(1-q^m)$. The long-standing problem ...
- 43.2k
2
votes
0
answers
146
views
English proof for decomposition of modular polynomials
Let $\Phi_n(X,Y)$ be the modular polynomial that are the canonical equations for the modular curve $X_0(n)$. They parameterise pairs of elliptic curves related by a cyclic isogeny of degree $n$.
I am ...
- 521
2
votes
0
answers
100
views
Quasi-algebraically closed fields reference request
I am looking for a road to understanding what quasi-algebraically closed fields are with the ultimate goal of understanding the paper by Lang 1952.
My current background is the first 6 chapters from ...
- 87
2
votes
0
answers
104
views
A question on exponential increasing sequences of natural numbers
Let $(a_i)$ be an exponential increasing sequences of natural numbers; there are constants $a\in(1,2)$, $b>0$ such that $|a_{i}-ba^i|$ is exponentially decreasing. Let $(s_{k})$ be a sequences in $\...
- 1,648
2
votes
0
answers
149
views
$f(x)$-th largest number of prime factors
Given a sufficiently well-behaved function $1\le f(x)\le x$ and a multiset $S=\{\omega(n): 1\le n\le x\}$, what can be said about the asymptotics of the $f(x)$-th largest member of $S$? In other words,...
- 9,114
2
votes
0
answers
236
views
On the cardinality of the set of right-truncatable primes
We say that the (base ten) prime number $p=a_{n}a_{n-1}a_{n-2}\cdots a_{1}a_{0}$ is right-truncatable if all of the following numbers are prime:
\begin{eqnarray*}a_{n},\\a_{n}a_{n-1},\\ a_{n}a_{n-1}...
- 8,842
2
votes
0
answers
276
views
Identity with Ramanujan's generalized continued fraction
Let $F(x,q)=\sum_{n\geq 0}x^n\dfrac{q^{n^2}}{(q)_n}$, where $(q)_n=(1-q)(1-q^2)\dots(1-q^n)$. Then:
$$H(x,q)=\frac{F(-xq,q)}{F(-x,q)}=\dfrac{1}{1-\dfrac{qx}{1-\dfrac{q^2x}{1-\dots}}}$$ is the ...
- 301
2
votes
0
answers
245
views
Help for reference of moduli stack of fake elliptic curves
I see everywhere the following:
Let $B$ be an indefinite quaternion algebra over $\mathbb{Q}$ of discriminant $D$, $\mathcal{O}_B$ be a maximal order, $N$ be an positive integer coprime to $D$.
...
- 21
2
votes
0
answers
114
views
Distribution of residue classes of totients of (univariate) polynomials
Let $\phi$ denote Euler's totient function and $f$ a non-constant univariate polynomial with integer coefficients such that $f(u) \in \mathbf N^+$ for all $u \in \mathbf N^+$ (assume $f$ is ...
- 10.5k
2
votes
0
answers
99
views
Does there exist $k\ge2$ s.t. $X \subseteq \mathbf N^+$ has positive upper Banach density if the counting function of $X$ is $\gg n/\log^{[k]}(n)$?
Does there exist an integer $k \ge 1$ such that ${\sf bd}^\ast(X) > 0$ whenever $X \subseteq \mathbf N^+$ and $\pi_X(n) \gg \frac{n}{\log^{[k]}(n)}$ as $n \to \infty$? Here, ${\sf bd}^\ast$ is the ...
- 10.5k
2
votes
0
answers
127
views
Nonnegative integers represented by $\prod_{i=1}^m \sum_{j=1}^n a_{i,j} x_j $, where the $a_{i,j}$ are positive integers
Fix $m, n \in \mathbf N^+$ with $m+n \ge 3$, and let $A = (a_{i,j})_{1 \le i \le m, 1 \le j \le n}$ be an $m$-by-$n$ matrix of positive integers. What is known about the asymptotic behavior of the ...
- 10.5k
2
votes
0
answers
159
views
The dimension of the space of automorphic forms with multiplier system
Let $\Gamma$ be a discrete subgroup of $SL_{2}(\mathbb{Z})$ and $\vartheta$ a multiplier system of weight $k$ for $\Gamma$, by which we mean a function $\vartheta:\Gamma \rightarrow \mathbb{C}$ ...
- 433
2
votes
0
answers
487
views
On Descartes / spoof odd perfect numbers
Descartes, Frenicle, and subsequently Sorli, conjectured that $k = 1$, if $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form (i.e., $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and $\...
2
votes
0
answers
564
views
Sets of coprime numbers
Consider the set $\{0, 3, 7, 15\}$ of four integers. If you add each of these numbers to a fixed power of 2, then the resulting four numbers are pairwise coprime. For example, $\{4, 7, 11, 19\}$ are ...
- 595
2
votes
0
answers
179
views
Questions about transformation or integral transformation
I have asked several mathematicians about the following questions,but all of them think they are good questions,but can not give a complete answer.Now I have to come here to ask mathematicians all ...
- 3,094
2
votes
0
answers
116
views
Name of a difference of continuants
I am getting ready to publish the manuscript
http://arxiv.org/pdf/1408.4631v2.pdf
and I am trying to do due diligence on a quantity I study before it gets published. (This is cross-posted from Name/...
- 1,521
2
votes
0
answers
412
views
Conjectures on fractions where each digit appears once in numerator and denominator
This is a highly redacted version of a question that was asked before. Please see Criteria of considering relevance of the question to the domain of research topics for details.
Some numerical ...
- 335
2
votes
0
answers
227
views
Kloosterman-like sum with inverse to different moduli
In some recent work, the following strange-looking exponential sum arose:
$$
\sum_k \sum_r \sum_s e\bigg( \frac{r \bar s^{(r)} \bar k^{(r^2+s^2)}}{r^2+s^2} \bigg).
$$
Here $e(x) = e^{2\pi i x}$ as ...
- 12.8k
2
votes
0
answers
207
views
n-ary quadratic forms with $S$-integer values
Let $Q(x_1,\ldots,x_n):=x_1^2+\cdots+x_n^2$ be an $n$-ary quadratic form.
Given a finite set of (rational) primes $S$ is there an algorithm or theorem that describes all solutions to $Q(x_1,\ldots,...
- 1,639
2
votes
0
answers
152
views
Reference Request: Properties of the Integer Factorization Polytope
The complexity of Integer Factorization is to my knowledge still an open problem, whereas deciding, whether a given integer is a prime number is known to be in $P$ and a proof is available online here:...
- 13.2k
2
votes
0
answers
224
views
Abel-Jacobi map isomorphism galois representations
Let $X/\mathbb{Q}$ be an irreducible smooth projective curve with a $\mathbb{Q}$-rational point $p$. Then there is a map $\phi: X \rightarrow \textrm{Pic}^{0}(X)$ with the property that $q \mapsto [q]-...
- 3,555
2
votes
0
answers
236
views
Is there a generalization of Granville-Langevin conjecture for number fields?
According to Wikipedia and other sources the Granville-Langevin conjecture
states:
If $f$ is a square-free binary form of degree $n > 2$, then for every real $\beta > 2$ there is a constant $...
- 25.4k
2
votes
0
answers
398
views
Counting factors: is this approach in the literature on multiperfect numbers?
Does the following approach (or something near it) exist in the number theory
literature?
I will provide some motivation for $\omega(p^n - 1)$ as $n \rightarrow \infty$
and for this question. First, ...
- 2,158
2
votes
0
answers
172
views
Bost's slopes method for Lindemann theorem
Hello,
I am looking for a proof of Lindemann theorem (transcendence of $e^a$ for $a$ algebraic) that uses Bost's slopes method. Does anyone know where I could find that?
- 3,998
2
votes
0
answers
293
views
Zeta function of abelian variety with CM
In his book, "Introduction to the arithmetic theory ..." (Ch 7, section 8) Shimura constructs the Zeta function of an abelian variety with CM and expresses it as a product of L-functions.
Since I ...
- 21
2
votes
0
answers
228
views
Progress towards Selberg's conjecture A
Hello,
Selberg's conjecture A states that whenever $F$ is an element of the Selberg class, there exists a non negative integer $n_F$ depending only on $F$ such that $\sum_{p\leq x}\dfrac{\vert a_{p}(...
- 6,984
2
votes
0
answers
243
views
Hurwitz integers and $F_4$
The Hurwitz integers are
$$
\mathcal H=
\{a+bi+cj+dk:a,b,c,d\in\mathbb Z\;\text{ or } \;a,b,c,d\in \tfrac12+\mathbb Z\}.
$$
I want to know if there is a formula, for $m\in\mathbb Z$, for the number ...
- 2,446
2
votes
0
answers
214
views
structure of $T_\ell A$ for $A/\mathbf{F}_q$ an abelian variety
Can someone give me references for the structure of the $G_{\mathbf{F}_q}$-module $T_\ell A$, $A/\mathbf{F}_q$ an abelian variety?
- 417
2
votes
0
answers
687
views
Strong Bezout's Identity?
Let $\{ a_i \}_{i=1}^N $ be a set of elements of the ring of integers, $\mathbb{Z}_D$ and define $g = \text{gcd}(a_1, a_2,\ldots, a_N, D)$. Then Bezout's Identity states that there exists another set $...
- 133
1
vote
0
answers
265
views
Are there connections between Calabi-Yau manifolds and number theory?
I am interested in understanding whether there are any significant connections between Calabi-Yau manifolds and number theory. Calabi-Yau manifolds are central objects in algebraic geometry and string ...
1
vote
0
answers
45
views
Proper Pisot n-tuples
Recall that x is a Pisot number if it is real and x>1, while all of its conjugates have magnitude less than 1. Then $\{(x)^k\}$ (where $\{\cdot\}$ is the fractional part of x) approaches 0 ...
- 680
1
vote
0
answers
150
views
What are alternative mathematical definitions of observers beyond Bennett and Hoffman's framework?
Motivation:
This question is inspired by a talk from Avi Wigderson given on Randomness, where the idea that the randomness is in the eye of the observer is suggested.
In the study of information ...
- 3,064
1
vote
0
answers
25
views
Characterising rank-$2$ lattices $\Lambda$ and conjugate-linear translate $g \sigma(\Lambda)$, given elementary divisors
Let $E/F$ be a quadratic unramified extension of local fields with $\operatorname{char} F = 0$. Let $\Lambda \subseteq E^2$ be an $O_E$-lattice of rank $2$. Let $g \sigma \in \operatorname{GL}_2(E)$ ...
- 111
1
vote
0
answers
98
views
Are there known effective bounds on the number of semisimple Galois representations?
In continuation to my question here, are there known effective bounds on the total number of semisimple $p$-adic Galois representations unramified outside a finite set of primes $S$, of dimension $d$, ...
- 2,907
1
vote
0
answers
75
views
automorphisms and mellin transforms
If a real analytic function $f$ is involutive i.e. $f(f(x))=x$ and its Mellin transform can be taken on a section of the real axis, and is analytic for $x>0$, in certain cases can this imply that $\...
- 101
1
vote
0
answers
127
views
Multiplicities of Galois representations in the semisimplification of the reduction of a Tate module
Let $C$ be a smooth proper curve, of genus $g$ over a number field $K$. Let $v$ be a prime of good reduction for $C$ above $p>2$, and let $T_pJ$ denote the $p$-adic Tate module of $J$, the Jacobian ...
- 2,907
1
vote
0
answers
158
views
A question and reference about Bombieri's article continued fraction of algebraic numbers
Above the Comments in the article continued fraction of algebraic numbers, there are some words on the unboundedness/cycle of coefficients of continued fraction of algebraic numbers "Thus, ...
- 3,094
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\...
1
vote
0
answers
64
views
$F$-structure implies regular singularities + unipotent local monodromy?
Let $(\mathcal{E},\nabla)$ be a vector bundle with an integrable connection on a smooth quasi-projective $K$ scheme $X$, with $K$ a $p$-adic number field of characteristic $0$. Let $F$ denote a semi-...
- 2,907
1
vote
0
answers
131
views
Analytic properties of $L$-functions attached to a compatible system of $\ell$-adic Galois representations
Let $F$ and $E$ be number fields, $G_F$ be the absolute Galois group of $F$, and $S$ be a finite set of primes of $F$. For $\lambda$ a prime of $E$ we denote by $\ell$ its residual characteristic. We ...
- 663
1
vote
0
answers
89
views
Finiteness of elliptic curves with trivial conductor over function fields
Let $K=\mathbb{F}_q(C)$ be the function field of a smooth projective curve $C$ over a finite field $\mathbb{F}_q$ with $\text{cha}(K)>3$ and let $E$ be an elliptic curve over $E$. To $E$ we may ...
- 121
1
vote
0
answers
174
views
Upper bound on sum of Lambda(n) over short interval
I am looking for a bound of type
$$\sum_{x<n\leq x+y} \Lambda(n) \leq \frac{\log(x+y)}{\log y} \cdot 2y$$
(or better). Of course such a bound has to exist: the idea of the proof of Brun-Titchmarsh (...
- 20.2k
1
vote
0
answers
126
views
Reference request: unfolding of Integral representation of an L-function
Are there any text or papers that thoroughly address unfolding of integral representation of an L-function such as D. Ginzburg's On Spin L-function for Orthogonal Groups page 762-763 and page 774 (or ...
- 43
1
vote
0
answers
109
views
What is known about the average growth rate of the denominators of $n$ Egyptian fractions summing to one?
Motivation
In the following question posted here on MO and over at MSE, user Noah Schweber asks about a weighted count on Egyptian fraction representations (EFRs). To that end, he defines the ...
- 4,829