Prime numbers, diophantine equations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

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3
votes
1answer
189 views

Self-containing trees

Suppose that $r^2-r-1=0$ and that $T$ is the tree with root $1$ such that the children of each node $x$ are $rx$ and $x+1$. Remove all duplicates as they occur, and let $T(r)$ denote the remaining ...
6
votes
2answers
650 views

Generalizing Ramanujan's “1729 story”

Whenever I read the anecdote about Hardy, Ramanujan and the taxi number 1729 I'm amazed that it could have occurred to anyone just off the top of their head that 1729 can be written as the sum of two ...
1
vote
0answers
65 views

For which primes $p$ is the field $\mathbb{Q}(\Gamma(1/p^{j}))$ a strict subfield of $\mathbb{Q}(\Gamma(1/p^{i}))$ whenever $0<i<j$?

I already asked this question on a French math forum but eventually came to think that as silly it may turn out to be, perhaps something interesting could finally emerge from it, so I decided to take ...
18
votes
0answers
374 views

Checking Mertens and the like in less than linear time or less than $\sqrt{x}$ space

Say you want to check that $|\sum_{n\leq x} \mu(n)|\leq \sqrt{x}$ for all $x\leq X$. (I am actually interested in checking that $\sum_{n\leq x} \mu(n)/n|\leq c/\sqrt{x}$, where $c$ is a constant, and ...
0
votes
2answers
148 views

Function that gives 1 only when an integer is divisible by another integer [closed]

I need a function that takes two inputs, a and b, and returns 1 only when a is divisible by b and 0 otherwise. Can this be written in a nice mathematical way (other than using indicator functions)?
0
votes
0answers
117 views

A number theory question [duplicate]

Let $m,n$ be integers and $P$ be a prime and $\lambda$ be an integer such that $0 < \lambda <\sqrt p$, Can the following equation have any answer for any $m,n,p,\lambda$ except $\lambda=+-1$? $$...
0
votes
0answers
217 views

On the coherence of formal power series ring

Let $A = {\Bbb F}_p[[X_1,X_2,...]]$ be the ring of formal power series with infinitely many variables over the finite field ${\Bbb F}_p.$ $A$ consists of such formal sum elements as $\sum c_{e_1,.....
4
votes
1answer
243 views

Does anyone recognize this exponential sum?

For $a$, $b$ two integers, let $(a,b)$ denotes their gcd. We define the following exponential sum : $$G_q(n):=\sum_{d|q,~(d,q/d)=1}{e^{2i\pi n\frac{dd'}{q}}}$$ for $n$ a non-negative integer and $q$ ...
2
votes
1answer
74 views

On cluster points of a particular sequence

This is the sequel of a previous question. Let us consider the sequence $$ \xi_n = 2n \{n\xi\}-n, $$ where $\xi>0$ is a given real irrational number and $\{\cdot\}$ is the fractional part. Do ...
2
votes
1answer
190 views

Number of rational points in a non-smooth variety

Let $X$ be an algebraic variety over $\mathbb{F}_q$ with dimensional $n$. We know that if $X$ is smooth than $X$ has about $q^{nk}$ rational points over $\mathbb{F}_{q^k}$ (Weil hypothesis). Is there ...
-2
votes
2answers
167 views

Precise asymptotic of diophantine approximation

I know that every irrational number $\xi$ can be approximated by rational numbers in such a way that $$ \left| \xi - \frac{p}{q} \right| \leq \frac{c}{q^2} $$ for infinitely many choices of $p$ and $q$...
3
votes
1answer
427 views

origin of analogy “primes as the atoms of number theory/ arithmetic”

a math student recently challenged me on the old comparison/ analogy of prime numbers to "the atoms of number theory or arithmetic" and then was wondering the origin of the phrase. where does this ...
4
votes
1answer
111 views

Nearly Be Bruijn sequences constructed from De Bruijn sequences

Let $w$ be a De Bruijn $01$-sequence of the type $B(2,n)$; i.e., a cyclic $01$-sequence that contains every $n$-digit $01$-sequence exactly once. Let $x$ be a $01$-sequence of length $n$. When and ...
1
vote
1answer
113 views

is there any more odd near-perfect number?

we know the first odd near-perfect number is $3^4*7^2*11^2*19^2$(of course the number must be square).and from one paper ,which is studying the odd perfect number and giving some estimate on it, we ...
0
votes
0answers
115 views

A reflection on the Fermat equation

$xy(x+y)^n=1$ with $n\in\mathbb{N}$ without any solution for $x,y\in\mathbb{Q}^+$ is equivalent to (and therefore a trivial generalization of) the Fermat Theorem $a^{n+2}+b^{n+2}=c^{n+2}$ without any ...
4
votes
2answers
275 views

The best possible density in Hilbert's Irreducibility Theorem

Let $f(X,t_1,\dots,t_s)$ be an irreducible polynomial with coefficients in $\mathcal{O}_K$, the ring of integers of a number field $K$. By work of S. D. Cohen (http://plms.oxfordjournals.org/content/...
1
vote
1answer
149 views

Powers of two with coefficients {1,−1}

Given a vector $(n_0, n_1, \dots, n_l)$ where $n_i \in \{-1, 1\}$, $i = \overline{0, l-1}, n_l = 1$ and $l \in \mathbb{N}$. Prove that for all $a$ such that $$0 < a \leq 2^0\cdot n_0 + 2^1 \cdot ...
2
votes
0answers
261 views

An explicit formula for $\zeta(2m+1)$ with good convergence

The question: Is the following formula known? $$\zeta(2m+1)=\frac{(-1)^m 2^{4m+2}\pi^{2m}}{2^{2m}-1} \sum\limits_{k=1}^m \frac{(2^{2k}-1)b_{2k}}{2^{2k}(2k)!}\cdot \sum\limits_{v=k}^m \frac{(2^{2v-2k+...
3
votes
0answers
371 views

Finding the number of rational points effectively

Consider $\# P$ and $\oplus P$. There is a $\# P$-hard problem: to find number of rational solutions of a system of polynomial equations over $\mathbb{F}_2$. The corresponding $\oplus P$-hard ...
11
votes
1answer
278 views

Computing endomorphism rings of supersingular elliptic curves

I would like to know what algorithms there are to compute the linearly independent generators $(1,i,j,k)$ for quaternion algebra containing the endomorphism ring of a supersingular curve. The curve in ...
7
votes
1answer
209 views

If two Hecke characters cut out the same field, are they Galois conjugates?

First question on MathOverflow, I hope it is appropriate for this site. There are two related questions. Let $K$ be a number field, $G_K = Gal(\overline{K}/K)$, $p$ a prime, and $$\chi_1,\chi_2:G_K\...
5
votes
2answers
196 views

Mahler measure of a totally positive, expanding algebraic integer

Consider a degree-$d$ algebraic integer $\alpha$ all of whose conjugates (including itself) are real numbers greater than 1. Its Mahler measure $M(\alpha)$ is simply equal to the norm $N(\alpha)$. ...
2
votes
0answers
77 views

How to estimate $\prod_{t=1}^{N}\frac{1}{2-z^t}$ for large $N$?

Based on the top answer to How to estimate of $\prod_{k=a}^N \frac{1}{e^{k\kappa}-1}$ for large $N$? Can anyone find an approximate closed form for $$ \frac{\mathrm{d}^k}{\mathrm{d}z^k}\prod_{t=1}^{N}...
1
vote
0answers
78 views

Are there only finitely many fixed degree nontrivial polynomial parametrizations of the surface $x^4+y^4=z^4+t^4$?

Consider the surface over the rationals $$ x^4+y^4=z^4+t^4 \qquad (1)$$ Consider parametrizations of form: $$ f_1(u)^4+f_2(u)^4=f_3(u)^4+f_4(u)^4 \qquad(2) $$ where $f_i$ are polynomials with integer ...
15
votes
1answer
453 views

What do Hecke eigensheaves actually look like?

Let $\mathbb F_q$ be a finite field, $C$ a curve over $\mathbb F_q$ of genus $g\geq 2$, $\rho: \pi_1(C) \to GL_2(\overline{\mathbb Q}_\ell)$ an irreducible local system. The geometric Langlands ...
1
vote
0answers
124 views

how to solve this symmetrical equation in number theory

i just have no idea about this equation, i would thank you to you to give me some suggestions on this. $$m_{1}m_{2}m_{3}+2^{\alpha-s-t}m_{1}+2^{\alpha-\gamma-t}m_{2}+2^{\alpha-\gamma -s}m_{3}\\-2^{\...
4
votes
0answers
203 views

Equations for Elliptic Curves

An elliptic curve $C$ over a field $k$ is a smooth, genus 1 curve defined over $k$ with an associated $k$-rational point. If char$(k) \ne 2$, we can show that $C$ has a model of the form $y^2 = f(x)$ ...
0
votes
0answers
76 views

A cyclic subgroup as a decomposition group

Let $G$ a finite group appeared as galois group of an extension of $\mathbb{Q}$. Is it true that any cyclic subgroup $C \subset G$ can be realized as a decomposition group of an ideal $\mathfrak{P}$ ...
2
votes
0answers
83 views

what is the structure of the group of isogenies between two ordinary elliptic curve?

Let $E_1$ and $E_2$ be two ordinary elliptic curves. It is well known that the group $Home(E_1,E_2)$ is a $\mathbb{Z}$ module of rank at most four. In the case where $E_1=E_2$, this module has rank ...
2
votes
0answers
148 views

Comparison of algebraic and analytic q-expansion

I would like to check that algebraic and analytic q-expansion of a modular form coincide. I'm thinking about modular forms as global sections of some sheaf on modular curves. If $X$ is a modular ...
10
votes
1answer
255 views

About positive upper density

For $S\subset \mathbb{N}$ define the upper density as $D^{\ast }(S)=\limsup_{n\rightarrow \infty }\frac{\left\vert S\cap \{1,2,\ldots,n\} \right\vert }{% \left\vert n\right\vert }.$ Question: ...
0
votes
0answers
79 views

applications of ergodic theory to periodicity of regular continued-fractions

The usual application one sees of ergodic theory to the regular continued-fractions is the Gauss-Kuzmin Theorem on the frequency of positive integers in the continued fraction expansion for almost all ...
4
votes
0answers
115 views

Tamagawa numbers of elliptic curves and quadratic base change

Let $E/Q$ be an elliptic curve of conductor $N$, $F$ an imaginary quadratic number field of discriminant $d$ with $d$ coprime to $N$, and $E^d/Q$ the quadratic twist of $E$ by $d$. Let $p$ be prime ...
4
votes
1answer
135 views

Congruence Primes and Modular Degrees

Let $\mathcal{S}=S_2(\Gamma_0(N) \cap \mathbf{Z} [[ q ]]$ be the set of cusp forms of weight $2$ on $\Gamma_0(N)$ with integral coefficients. Let $f \in \mathcal{S}$ be a normalized newform, so it ...
8
votes
0answers
142 views

Sets of natural numbers which are almost closed under addition

I am interested in a classification of sets $A \subseteq \mathbb{N}$ such that for all $k \in A$, $d( A+k \cap \mathbb{N} \setminus A) = 0$ where $d$ is the asymptotic density and $A+k = \{n \in \...
4
votes
1answer
261 views

Time-efficient way of calculating the least number of 1s in a representation of $n$ using only the operations $+,!$

This was inspired by the following paper: J. Arias de Reyna, J. van de Lune, "How many $1$s are needed?" revisited, arXiv link. It might help explain my question better, because my question is ...
4
votes
2answers
166 views

Asymptotic of $Lcm(\binom{2k}k)_{1\le k\le n}$

Everything is in the title. I wonder if there is known good asymptotic (when $n\to+\infty$) for the quantity $LCM(\binom{2k}k)_{1\le k\le n}$ Thanks in advance
5
votes
0answers
81 views

Reference request: "effective'' semistable reduction

I am looking for the origin of the following idea: suppose $m$ and $n$ are relatively prime integers $\geq 3$. Let $E$ be an elliptic curve over a number field $K$. Let $L/K$ be a finite extension ...
0
votes
0answers
155 views

Asymptotic value of sum over Möbius function

Consider the sum $$ S(x)=\sum_{\substack{{d\mid P(\sqrt{x})}\\{d\leq x}}}|\mu(d)|, $$ where $P(x)$ is the product of all primes less than or equal to $x$, and $\mu(d)$ is the Möbius function. Q:...
3
votes
1answer
218 views

An electronic copy of Vishik's work on $p$-adic $L$-functions for modular forms

This question is very simple. Would someone be so nice as to send me an electronic copy of M. M. Vishik, Non-Archimedean measures connected with Dirichlet series, Mat. Sb. (N.S.), 1976, Volume 99(...
2
votes
0answers
188 views

Automorphism group of $\mathbf{C}$ over $\overline{\mathbf{Q}}$

Choose an embedding $\overline{\mathbf{Q}}\rightarrow \mathbf{C}$ from an algebraic closure of the field of rationals to the field of complex numbers. Question 1: Is it true that $\mathbf{C}$ is ...
0
votes
1answer
352 views

What does this proof of Fermat's little theorem mean for Euler's theorem?

The following proof of Fermat's little theorem is semi-standard: We prove that $a^p-a \equiv 0 \mod p$ by induction on $a.$ For $a = 2,$ we write $2^p = (1+1)^p = 2 + \sum_{i=1}^{p-1} \binom{p}{i},$ ...
12
votes
1answer
214 views

Are quadratic units cyclotomic norms?

Consider the fundamental unit $\varepsilon$ of a real quadratic number field $k = {\mathbb Q}(\sqrt{p})$ for primes $p \equiv 1 \bmod 4$, and let $h$ denote its class number. By Dirichlet's work on ...
2
votes
1answer
208 views

Weil Conjectures Analog for Multivariate Zeta Functions

We know that the Riemann zeta function can be generalized to multivariate zeta functions. Is there a multivariate analog of the Weil conjectures?
3
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0answers
117 views

How to show that $h(-D)\geq \displaystyle\sum_{a\in A_1\\, 1\leq a\leq{\frac{\sqrt D}{2}}} 1$?

Here $A_1=\{u;p|u\Longrightarrow \chi(p)=1\}$ with $\chi$ a real quadratic character and $h(-D)$ the class number of the imaginary quadratic field of the fundamental discriminant. This problem occurs ...
1
vote
1answer
137 views

Orbital integral for matrix coefficients

I am currently aiming at estimating orbital integrals. Maybe surprizingly, I hope for some help in the compact case (ramified places), in proving the usual formula $$O_\gamma(f) = \int_G f(x^{-1}\...
3
votes
1answer
115 views

Reference Request on the existence of $k$ satisfying $P(\Phi_d(2))^k \gt \Phi_d(2)$ for all $d$

I am working my way through the literature regarding the following conjecture: There is a positive integer $k$ such that for all positive integers $d$, $$P(\Phi_d(2))^k \gt \Phi_d(2).$$ I am ...
2
votes
0answers
112 views

Values of Bernoulli polynomials at roots of unity

I am wondering if there are any nice results on the values of Bernoulli polynomials at roots of unity, besides those at 1 or -1.
4
votes
0answers
93 views

Are there only finitely many near-perfect numbers with more than 4 distinct prime divisors?

Given a positive integer $n$, let $\sigma(n)$ denote the sum of divisors of $n$. We say that a positive integer $n$ is a near-perfect number if $\sigma(n) - 2n$ is greater than $0$ and a proper ...
2
votes
1answer
119 views

Maass form properties and their fourier coefficients

Some Maass form can be written ($K_{iR}$ is the K-Bessel function): $$f(x+iy)=\sum_{n \ne 0}^{\infty} a_n \sqrt{y} \;K_{iR}(2\pi |n| y) \; e^{2 i\pi nx}$$ with the $a_n$ multiplicative, but inversly ...