Questions tagged [nt.number-theory]

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

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4 votes
0 answers
193 views

Geometric interpretation of the rationality of the $j$-invariant

Consider the modular curve $X_0(N)$. Let $\Phi_N(X,Y)$ be the modular equation. Then the curve $\Phi_N(X,Y) = 0$ can be interpreted as a model for $ X_0(N)$ because the function field of $X_0(N)$ is $\...
3 votes
1 answer
243 views

Fermat's cubic equation in quadratic extension of $\mathbb{Q}$

Is still relevant or interesting be capable to bring a criteria in order to classifly quadratic extensions of $\mathbb{Q}$ based on the existence or not existence of non-trivial solutions of Fermat's ...
15 votes
1 answer
2k views

What did Euler do with multiple zeta values?

When reading about multiple zeta values, I often find the claim that the case of length two $$ \zeta(s_1, s_2)=\sum_{n>m \geq 1} \frac{1}{n^{s_1}m^{s_2}}, \qquad s_1 \geq 2, \quad s_2 \geq 1 $$ ...
5 votes
1 answer
460 views

Explaining patterns in modular multiplication graphs

Let the multiplication graph $n/m$ be the graph with $m$ points distributed evenly on a circle and a line between two points $a$, $b$ when $an \equiv b\operatorname{mod} m$. These graphs often look ...
4 votes
1 answer
296 views

The product of $\frac{b^i-a}{b^i-1}$ lies in a special ring (conjecture)

Consider any three positive integers $a, b, n$. Is it true that $$\frac{(b-a)(b^2-a)\cdot\dotsb\cdot(b^n-a)}{(b-1)(b^2-1)\cdot\dotsb\cdot(b^n-1)}\in\mathbb{Z}\left[\frac{a-1}{b-1},\frac{a^2-1}{b^2-1},\...
5 votes
2 answers
249 views

What integer value can be the conductor of a $g$-dimensional abelian variety over $\mathbb Q$?

Fix a positive integer $g$. What positive integer $N$ can be the conductor of a $g$-dimensional abelian variety over $\mathbb Q$ ? For example, as there is no abelian varieties over $\mathbb Z$, $N$ ...
11 votes
1 answer
590 views

Riemann zeta function: pair correlations vs. neighbor spacings

Montgomery's pair correlation conjecture states that the distribution of the pair correlations of the zeroes of the Riemann zeta function (normalized to have average spacing 1) is given by the ...
3 votes
1 answer
76 views

Dimension of fixed vectors of a semi-linear operator

Let $L$ be a field with a field embedding $\sigma:L \rightarrow L$, and $K=L^{\sigma}$ be the fixed field of $\sigma$. For $A \in M_n(L)$ a matrix, consider the set $X=\{x \in L^n|Ax=\sigma(x) \}$ ...
5 votes
0 answers
161 views

Consecutive integers each of which has a large prime factor

There are many results about consecutive integers all having small prime factors. But what about consecutive integers each of which has a large prime factor? More precisely, let $P(n)$ be the ...
9 votes
1 answer
558 views

XOR-free sets: Maximum density?

It is known that sum-free subsets of $\mathbb{N}$ can have natural density at most $\frac{1}{2}$. This density is achieved by the odd numbers: the sum of two odd numbers is even. I ask now a similar ...
6 votes
2 answers
1k views

An Expectation of Cohen-Lenstra Measure

The Cohen-Lenstra measure on the set of abelian p-groups assigns $\mathbb{P}(G) = \prod_{i \geq 1} \left( 1 - \frac{1}{p^i}\right) \cdot |\mathrm{Aut}(G)|^{-1} $. Apparently, this is equivalent to ...
8 votes
1 answer
481 views

What is the minimum $k$ such that $A^k \equiv I$ mod p for invertible matrices?

Let $F$ be a finite field of order $p$, where $p$ is prime. For any $n\times n$ matrix $A$ that is invertible over $F$, then there would appear to exist integers $k$ such that $A^{k} = I$. My question ...
2 votes
3 answers
503 views

Does this deceptively simple nonlinear recurrence relation have a closed form solution?

Given the base case $a_0 = 1$, does $a_n = a_{n-1} + \frac{1}{\left\lfloor{a_{n-1}}\right \rfloor}$ have a closed form solution? The sequence itself is divergent and simply goes {$1, 2, 2+\frac{1}{2}, ...
28 votes
0 answers
561 views

A sequence potentially consisting of only integers

I will first ask the question which can be stated very simply. Afterwards I will explain some motivation and give references to related sequences. Consider the sequence defined by $$b_n = \frac{(...
1 vote
0 answers
156 views

Absolute convergence of the Fourier series of a smooth adelic function

Let $f: \mathbb A/\mathbb Q \rightarrow \mathbb C$ be a smooth function. Smooth means that $f$ is continuous, smooth in the archimedean argument, for every $(x_0,y_0) \in \mathbb A = \mathbb R \...
5 votes
0 answers
409 views

Pisot conjugates

An informal version of my question is "If we have a Pisot number between 1 and 2 of a very large degree, is it true that all its other conjugates are very close to 1 in modulus?" A more formal ...
1 vote
0 answers
254 views

On Primes in Arithmetic Progressions

I was wondering if the following approach is being attempted to prove the twin-prime conjecture. Tao and Green proved in their paper (2006), that there are arbitrarily long arithmetic progressions ...
1 vote
1 answer
110 views

Density of a set of numbers dividing a fixed number with polynomial exponent

Fix a positive integer $a>1$ and let $f\in\mathbb{Z}[x]$ be a polynomial with positive leading coefficient. We define a set $S$ of positive integers, $$ S=\{n\in\mathbb{Z}^+:n\mid a^{f(n)}-1\}. $$ ...
1 vote
0 answers
69 views

On equality of two quotients of a congruence subgroup

Related question: Non-torsion part of the abelianisation of congruence subgroups Let $A = \mathbb{F}_q[T]$ be the ring of polynomials with coefficients in a finite field, with $N$ a nonconstant ...
1 vote
0 answers
118 views

On the exponent of a certain matrix $A$ in characteristic $p > 0$

Let $A$ be a square matrix in characteristic $p > 0$ with both column and row having length $(1 + p^0 + p + \cdots + p^i)$, where $i \geq 0$. Suppose that further the $(m,n)$-component $a_{m,n}$ ...
1 vote
1 answer
102 views

Size of a multi-segment of a representation of $GL_n(F)$

Let $F$ be a p-adic field and $GL_n(F)$ the general linear group over $F$. The irreducible complex finite length smooth representations are parametrized by multi-segements in the paper. A multi-...
9 votes
0 answers
207 views

Kronecker's theorem in higher dimension

Recall the following classical theorem of Kronecker: if $P(x) \in \mathbb{Z}[x]$ is a monic irreducible polynomial with all roots on the unit circle $S^1$, then $P(x)$ is a cyclotomic polynomial (and ...
28 votes
1 answer
2k views

Can we use the Rogers-Ramanujan cfrac to parameterize the Fermat quintic $x^5+y^5=1$?

Define $\color{blue}{q=e^{2\pi i \tau}}$ and Dedekind eta function $\eta(\tau)$. Note: I found these relations empirically, but their consistent forms suggest they can be rigorously proven. I. $p=2$...
6 votes
1 answer
258 views

Differential equations satisfied by quasi modular forms?

It is known that modular forms are solutions of differential equations. More precisely, let me cite the statement from the following question. Differential Equations Satisfied by Modular Forms ...
2 votes
2 answers
1k views

What is the natural density of hyper prime numbers?

What do we mean by hyper prime numbers? Well, roughly speaking they are natural numbers which are prime with respect to hyperoperators in arithmetic such as exponentiation, tetration, pentation, et ...
4 votes
1 answer
194 views

Bound of size $X\subset \mathbb{Z}/N\mathbb{Z}$ which satisfies $X+X=\mathbb{Z}/N\mathbb{Z}$

(Sorry for my poor english skill..) Let $N$ be a large integer and the set $X$ be the subset of $\mathbb{Z}/N\mathbb{Z}$. For two sets $A$ and $B$, we define \begin{equation} A+B:=\{a+b : a\in A, b\...
21 votes
0 answers
764 views

Class field theory and the class group

Let $k$ be a finite abelian extension of $\mathbb{Q}$. Class field theory states that $k$ corresponds to some open subgroup of finite index $U_k \subset \mathbb{A}_{\mathbb{Q}}^*/ \mathbb{Q}^*$ where $...
2 votes
0 answers
193 views

Shattering with sinusoids

Let $d \geq 2$ and $K$ some positive integer. Consider distinct points $\theta_1, \ldots, \theta_K\in \mathbb{T}^d$ and (not necessarily distinct) $z_1, \ldots, z_K \in \{-1,1\}$ such that $\sum\...
4 votes
1 answer
716 views

Is the integral closure of a valuation ring in a finite separable extension of its fraction field étale?

Let $K$ be a field endowed with a rank (height) one valuation with completion $\hat{K}$, which is not discrete. Let $R$ be the valuation ring of $K$. Let $L \subset \hat{K}$ be a separable finite ...
8 votes
0 answers
164 views

Why is the set of lifts of a p-divisible group canonically the same as the set of lines that span $M(G)/FM(G)$?

Let $M$ be the Dieudonne module of a p-divisible group $G_0$ over $k$, and let a lift of $G_0$ to $A$ be a p-divisible group $G$ over $A$ such that $G \otimes_A k \simeq G_0$. Let $\omega_G$ be the ...
1 vote
0 answers
150 views

Do we believe that the distribution of spacings of successive critical zeros of zeta is log-symmetric?

Let $\gamma^{+}(T)$ be the imaginary part of the critical zero of $\zeta$ closest to $1/2+iT$ with $\gamma^{+}(T)\ge T$ and define similarly $\gamma^{-}(T)$ with a reversed inequality. Let $g(T)$ be ...
6 votes
2 answers
318 views

Number of integer partitions modulo 3

Motivated by Parity of number of partitions of $n!/6$ and $n!/2$, I asked my computer (and my FriCAS package for guessing) for an algebraic differential equation for the number of integer partitions ...
1 vote
0 answers
205 views

Parity of number of partitions of $n!/6$ and $n!/2$

The parities of the number of partitions of $n!/6$ and $n!/2$ appear to be non-random initially, as follows — is there an explanation for this other than chance? With $p$ being the partition ...
31 votes
2 answers
3k views

Motivation behind Analytic Number Theory

I am an undergraduate student of mathematics and recently took an introductory course in analytic number theory, where the instructor roughly followed Apostol's first text on the subject. I have now ...
9 votes
2 answers
660 views

Examples of automorphic forms over $\mathbb{H}^3/\text{PSL}_2(\mathbb{Z}[i])$

If I understood my automorphic forms correctly, at least cusp forms can be thought of as elements of $L^2(G/\Gamma)$ for a $G = \text{SL}_2(\mathbb{R})$ and $\Gamma = \text{SL}_2(\mathbb{Z})$ or a ...
17 votes
1 answer
2k views

Rational points à la Chabauty-Coleman

I have been trying to learn the method of Chabauty and Coleman to find rational points on curves; I have been reading an exposition by McCallum and Poonen which was pointed out to me by Emerton in ...
5 votes
1 answer
627 views

rank of Jacobian of Fermat curve and Chabauty-Coleman method

Consider the fermat curve $F(p)$ over $\mathbb Q$ which is the projective closure of $X^p+Y^p=1$ inside projective plane, where $p$ is a prime number and without loss of generality we assume $p>2$. ...
6 votes
0 answers
191 views

A recursion with a number-theoretic function

For a positive integer $Q$, let $$ s(Q) := Q\,\sum_{p\mid Q} \frac1p, $$ where the sum extends over all prime divisors of $Q$; also, let $s(0)=0$. Thus, we have, for instance, $s(1)=0$, while $s(p^\...
9 votes
0 answers
253 views

How small may the discriminant of an $S_d$-field be?

In every degree $d$, the Galois closure of the typical number field has the maximal possible Galois group $S_d$. Denote by $f(d)$ the least absolute value of a discriminant of an $S_d$-field of degree ...
2 votes
0 answers
153 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 ...
4 votes
1 answer
565 views

Piltz Divisor Problem

Let $\tau_k(n)$ count the number of ways of representing $n$ as the product of $k$ natural numbers. It is known that: $$D_k(x) = \sum_{n \leq x} \tau_k(n) = xP_k(\log x) + O(x ^{1 - \frac{1}{k-1}}(\...
3 votes
0 answers
80 views

The analogue of difference operator in Drinfeld module theory

The theory of Drinfeld modules is, in part, inspired by the analogy between Frobenius and the operator of differentiation. My question is: what should be the analogue of a difference operator $f(x)\...
2 votes
0 answers
154 views

Classification of mod p Galois Representations for l not equal to p

Let $l\neq p$ be primes and let $\text{G}_l:=\text{G}_{\mathbb{Q}_l}$. Let $k$ be a finite field of characteristic $p$ and $\bar{\rho}:\text{G}_l\rightarrow \text{GL}_2(k)$ a local Galois ...
6 votes
0 answers
131 views

On a certain $(-1)$-Eulerian polynomials of type $B$

Let $(q)_n=(1-q)(1-q^2)\cdots(1-q^n)$ with $(q)_0:=1$. Define a $q$-exponential by $$e_q(z)=\sum_{n\geq0}\frac{z^n}{(q)_n}.$$ There is a notion of $q$-Eulerian polynomials of type $A$, see the ...
4 votes
1 answer
939 views

Arithmetic properties of a sum related to the first Hardy-Littlewood conjecture

The starting point of this post is an earlier question, where I conjectured (and GH from MO confirmed) that the von Mangoldt function is the limit at $s=1$ of a certain Dirichlet series, $$\Lambda(m)=\...
3 votes
1 answer
272 views

On the convergence of $\sum_{n=1}^{\infty} \frac{\lambda(n)}{n}$ and the Prime Number Theorem

Let $\lambda$ be the Lioville function of number theory. I've heard several times that if $L=\sum_{n=1}^{\infty} \frac{\lambda(n)}{n} =O(1)$ then $L=0$ (the Prime Number Theorem). How can this be ...
28 votes
0 answers
695 views

Does this infinite primes snake-product converge?

This re-asks a question I posed on MSE: Q. Does this infinite product converge? $$ \frac{2}{3}\cdot\frac{7}{5}\cdot\frac{11}{13}\cdot\frac{19}{17}\cdot\frac{23}{29}\cdot\frac{37}{31} \cdot \cdots \...
6 votes
0 answers
732 views

$f(x) \ne g(x)$ but $f(f(x))=g(g(x))$ - is there a name/some discussion of this property?

In the context of iteration of functions I look at the eigenvalues of the associated matrixoperator/Carleman-matrix . If a function $\small f(x)$ has a negative eigenvalue in its associated ...
4 votes
0 answers
161 views

Smoothed Weyl sum inequality

One version of Weyl's inequality states that for any $\alpha\in\mathbb{R}$ and $(a, q) = 1$ such that $|\alpha - a/q|\le 1/q^2$, we have that $$\sum_{n\le X} e(n^k\alpha)\ll X^{1 + \varepsilon}(q^{-1}...
-1 votes
1 answer
231 views

Second Differences of Primes Pattern [closed]

I am wondering why this property exists or if I have discovered a new property of prime numbers? Let's do a sample here: 2,3,5,7,11,13,17 You obtain the gaps (subtracting the left number from the ...

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