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

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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
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0answers
114 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
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1answer
125 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 ...
7
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0answers
133 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
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1answer
255 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
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2answers
153 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
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0answers
74 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 ...
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0answers
149 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. ...
2
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1answer
149 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 ...
2
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0answers
172 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 ...
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1answer
349 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
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1answer
204 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
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1answer
206 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?
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111 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 ...
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1answer
125 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 ...
3
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1answer
108 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 ...
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0answers
108 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
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88 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
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1answer
106 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 ...
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0answers
132 views

Mahler's theorem in the primes

Let $F(x,y)$ be an irreducible binary form with integer coefficients and degree $d \geq 3$. Denote by $A_F$ the area of the region $$\displaystyle \{(x,y) \in \mathbb{R}^2 : |F(x,y)| \leq 1\}.$$ It ...
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3answers
542 views

Philosophy behind cohomological representations

For a given real reductive Lie group $G$, we have the notion of a representation being cohomological using the Lie algebra cohomology. In particular we know that the discrete series representations of ...
26
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3answers
694 views

Does the category of (algebraically closed) fields of characteristic $p$ change when $p$ changes?

Let $\mathrm{ACF}_p$ denote the category of algebraically closed fields of characteristic $p$, with all homomorphisms as morphisms. The question is: when is there an equivalence of categories between ...
12
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5answers
1k views

How much do I need to learn algebraic geometry to understand arithmetics over number fields

I am at the stage of learning. Mostly, I am attracted by algebraic number theory. Roughly speaking, I am interested in the rational points of algebraic varieties. I am little bit afraid to start to ...
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1answer
132 views

On even almost perfect numbers other than powers of two

(Note: This question is an improved version of and has been cross-posted from this MSE post.) Let $\sigma(x)$ denote the sum of the divisors of $x$. If $\sigma(x) = 2x - 1$, then we call $x$ an ...
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0answers
178 views

Zero balancing fractions, number theory

Motivation: The main property was found in a paper analyzing networks of the human brain but was likely rooted in the realm of social networks. In a social network the influence of a person/vertex can ...
6
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2answers
274 views

Integer solutions of (x+1)(xy+1)=z^3

Consider the equation $$(x+1)(xy+1)=z^3,$$ where $x,y$ and $z$ are positive integers with $x$ and $y$ both at least $2$ (and so $z$ is necessarily at least $3$). For every $z\geq 3$, there exists ...
3
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0answers
94 views

Minimization module $p$

Let $p$ be a prime number, let $m$ be a fixed number (for example $2^{20}$), and $i=k^{-1}\cdot (j-t) \pmod{p}$ where $j \leq m$ and $m<k$. In the general case we have $t=0$, and $k,j$ are ...
2
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1answer
148 views

Applications of Level Lowering

What are some applications/consequences of level lowering of Galois representations? I understand the application of Ribet's theorem in the proof of Fermat's last theorem but I am wondering what other ...
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2answers
375 views

Famous results about the value of a given limit assuming it exists

Chebyshev got famous showing that if the limit $l:=\lim_{x\to\infty}\frac{\pi(x)}{x/\log x}$ exists, then necessarily $l=1$, constituting a major breakthrough towards a proof of the famous prime ...
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51 views

Log-concavity of difference of theta functions

My knowledge on theta functions is limited, but I suspect that this is a quite challenging question. The 3rd Jacobian Theta function is given by \begin{equation} ...
2
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1answer
136 views

How many monoids with $n$ arrows exist?

How many monoids with strictly $n$ arrows exist? Is this known? I ask this only out of curiosity. Looking at $n=1,2,3,4$, this number doesn't appear to be very large relative to $n$.
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0answers
96 views

The Galois side of the Norm map

Let $K$ be an abelian extension of $\mathbb{Q}$. We know that $[x, K]|_{\mathrm{Gal}{\mathbb{Q}^{ab}}}=[\mathrm{N}^{K}_{\mathbb{Q}} x, \mathbb{Q}]$ where $[x, F]$ is the Artin reciprocity map. Given a ...
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1answer
190 views

A question regarding lines on a cubic surface

Let $X$ be a smooth cubic surface in $\mathbb{P}^3$. It is a classical theorem of Cayley and Salmon that $X$ contains exactly 27 lines over an algebraically closed field. In 2002, Heath-Brown proved ...
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1answer
1k views

How many primes can there be in a short interval?

Given $n \in \mathbb{N}$, let $\pi(n)$ denote the number of prime numbers $\leq n$. What is $$ \limsup_{m \rightarrow \infty} \left( \limsup_{n \rightarrow \infty} \frac{\pi(n+m) - \pi(n)}{\pi(m)} ...
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0answers
73 views

Reference request: Structure of $H^1({{\mathbf{Q}}_{q}},{{{{E}}_{{p}^{\infty}}}})$

I need reference on the structure of ${H}^{1}({{\mathbf{Q}}_{q}},{{E}_{{{p}^{\infty}}}})$, in particular when: (1.) $q=p$ and/or (2.) $E$ has multiplicative reduction at $q$. Here, $E$ is an ...
3
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0answers
103 views

Complex multiplication and ray class fields

This question is mainly referring to the proof of Theorem 5.6, Chapter 2 of Silverman's "Advanced Topics in the AEC". Basically, let $K$ be an imaginary quadratic field, and $E$ be an elliptic curve ...
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0answers
156 views

Any heuristics explaining why one seems to have $2n=p+q\Rightarrow\pi(p)+\pi(q)=Li(2n)+o(1)$?

The title explains almost everything: suppose $2n$ is a Goldbach number, i.e. an even positive integer such that there exist two primes $p$ and $q$ fulfilling $2n=p+q$. It seems that "most of the ...
2
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1answer
144 views

theta functions and Brownian motion

I did some plots of the theta function $\theta(z) = \sum q^{n^2}$ near the real axis, so $q = e^{2\pi i \, n z}$ and $z = 0.001 + i \mathbb{R}$. At first it looks like some random sine curve and then ...
13
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2answers
483 views

What is the smallest x such that [x^n] has the same parity as n?

A previous MO question asked for information about a number $x$ such that $[x^n]$ has the same parity as $n$ for all positive $n$. Answers to this post included two values of x which meet the ...
1
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0answers
93 views

Expressing Numbers with a Minimal Sum in Powers of 2 [closed]

The first 64 bits of pi are: 11.00100100001111110110101010001000100001011010001100001000110100 Computer multiplication can be sped up by looking for patterns and ...
4
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1answer
237 views

Do there exist infinitely many $n$ such that $n^3+an+b$ is squarefree?

Question: Assume that $a,b\in Z$, and $4a^3+27b^2\neq 0$. Prove that there exist infinitely many positive integers $n$ such $n^3+an+b$ is square-free. I have following There exist infinitely ...
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253 views

Controlling residue class sizes

$\forall k>10^2$ is there $m_k$ such that at infinite primes $q>m_k$ $\exists$ pairwise coprime $a,b,c$ such that $$(1)\quad q^{\frac14+\frac1k}<a,b,|a-b|<q^{\frac14+\frac2k} < ...
18
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1answer
701 views

Sphere packings : what next after the recent breakthrough of Viazovska (et al.)?

Given the march 2016 breakthrough concerning sphere packings by Viazovska for the case of dimension 8, and by Cohn, Kumar, Miller, Radchenko and Viazovska for the case of dimension 24, it follows that ...
15
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1answer
625 views

An equality involving roots of unity which holds most of the times, but not always

Let $m$ and $n$ be distinct odd positive integers. The equality $$ \prod_{k=0}^{mn-1} \left( e^{\frac{2\pi i k}{m}} + e^{\frac{2\pi i k}{n}} \right) \ = \ 2^{\gcd(m,n)} ...
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2answers
102 views

Sturmian subword whose reverse is not a subword

Let ${\cal L}_n$ be the set of all subwords of length $n$ of a biinfinite Sturmian sequence, induced by a rotation coding with irrational angle $\theta$. Take a word $w \in {\cal L}_{2^n}$ and write ...
7
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0answers
322 views

Field of definition of a point in $[p]^{-1}E(K)$

Let $E$ be an ordinary elliptic curve defined over a non-perfect field $K$ of characteristic $p$. If $P \in E(K)$ satisfies $P \not\in [p]E(K)$, is it true that its $p^m$-division points of $P$ are ...
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102 views

One of the coefficients of $P(x)$ has absolute value $\geq |\alpha(\alpha-1)|$

Let $P \in \mathbb{Z}[x]$ be a polynomial with non-negative coefficients and degree $k$. Let $\alpha \leq -2$ be an integer such that: $P(\alpha) \equiv 0 (mod (\alpha^{k+1} -1))$ and ...
2
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0answers
58 views

Carlitz factorials and Euler-like series

Let $q$ be a power of a prime $p$. For every $i\in\mathbb N$, one denotes $D_i=\prod_{\substack{h\in\mathbb F_q[T]\text{ monic}\\\deg h=i}}\limits h$. For $n\in\mathbb N$, write ...
5
votes
2answers
160 views

How to construct particular De Bruijn sequences

For $n \ge 2$, there is at least one binary DeBruijn sequence beginning with $n$ zeros followed by $n$ ones. Is there a straightforward way to construct such a sequence for each $n \ge 2$? Examples: ...
2
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0answers
42 views

Functional equations about Conway's box function

Conway's box function is the inverse of Minkowski's question mark function. It maps the dyadic rationals on the unit interval to the rationals using the Stern-Brocot tree (Farey sequence). The ...