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

Filter by
Sorted by
Tagged with
1
vote
0answers
152 views
+200

“proper” base change for good schemes

If one has a scheme $X$ smooth over a discrete valued ring $R$, when the $\mathbb Q_l $-coefficient compactly supported etale cohomology of the special fiber and generic fiber of $X$ ($l$ is ...
11
votes
1answer
189 views

Greatest common divisor in $\mathbb{F}_p[T]$ with powers of linear polynomials

Let $n>1$ and $p$ be an odd prime with $p-1 \mid n-1$ such that $p^k - 1 \mid n-1$ does not hold for any $k>1$. Notice that, since $p-1 \mid n-1$, we have $T^p - T \mid T^n-T$ in $\mathbb{F}_p[T]...
1
vote
1answer
250 views

A type of principal ideal theorem of class field theory for ramified primes

Let $K$ be a number field and $\mathcal{O}_K$ be its ring of integers. Also let $p$ be a prime number, $\mathfrak{p}$ be a prime ideal of $\mathcal{O}_K$ and $\zeta_{m}$ be a primitive mth root of ...
-5
votes
0answers
72 views

How to homotopy groups will work on the prime numbers [closed]

Via this manner we can correspond every prime number by an subinterval of $[0.01,0.1)$: Suppose $r:\Bbb N\to (0,1)$ is a function given by $r(n)$ is obtained by putting a point at the beginning of $n$...
1
vote
2answers
77 views

Is the abscissa of convergence of $s\mapsto\sum_{n>0}(ng_{n}/2)^{-s}$ known?

The famous Polignac conjecture posits that the $n$-th prime gap $g_{n}:=p_{n+1}-p_{n}$ attains all even positive integral values infinitely many times, which implies $\displaystyle{\zeta_{Pol}:=s\...
5
votes
4answers
366 views

Request for an exact formula related to a partition in number theory

The Frobenius equation is the Diophantine equation $$ a_1 x_1+\dots+a_n x_n=b,$$ where the $a_j$ are positive integers, $b$ is an integer, and a solution $$(x_1, \dots, x_n)$$ must consist of non-...
14
votes
7answers
2k views

A special type of generating function for Fibonacci

Notation. Let $[x^n]G(x)$ be the coefficient of $x^n$ in the Taylor series of $G(x)$. Consider the sequence of central binomial coefficients $\binom{2n}n$. Then there two ways to recover them: $$\...
11
votes
3answers
1k views

What are the analytic properties of Dirichlet Euler products restricted to arithmetic progressions?

There are (at least) two ways of writing down the Dirichlet L-function associated to a given character χ: as a Dirichlet series $$\sum_{n=1}^\infty \frac{\chi(n)}{n^s}$$ or as an Euler product $$\...
8
votes
1answer
509 views

Classify all the fields with abelian absolute Galois group

I'm wondering if anyone has classified all the fields $K$ such that $Gal(\bar{K}/K)$ is abelian? The only examples I'm aware of are: finite fields, the real numbers $\mathbb{R}$ and $k((T))$ where $k$...
1
vote
0answers
245 views
+300

Sets of primes with a given Frobenius conjugacy class

Let $a$ and $b$ be two relatively prime positive integers. Denote by $S$ the set of all positive primes of the form $a+bn$ (where $n$ is an integer, possibly zero or negative). Let $S'$ be any set of ...
30
votes
4answers
3k views

Over which fields does the Mordell-Weil theorem hold?

According to a well-known theorem of Mordell, the group of rational points $E(\mathbf{Q})$ of an elliptic curve $E/\mathbf{Q}$ is finitely generated. Weil generalized this theorem to abelian varieties ...
0
votes
0answers
133 views

Determinant of a special matrix in characteristic $p$

Let $K$ be a field of characteristic $p > 0$. Choose $p^i$ numbers of elements $c_1,\ldots,c_{p^i} \in K$ and consider the determinant $D$ of the following matrix$\colon$ \begin{pmatrix}\label{...
5
votes
0answers
207 views

Does this equation have more than one integer solution?

Consider the following diophantine equation $$n = (3^x - 2^x)/(2^y - 3^x),$$ where $x$ and $y$ are positive integers and $2^y > 3^x$. Does $n$ have any other integer solutions besides the case ...
0
votes
1answer
261 views

A sufficient condition for a set of primes to be the set of reducibility of an integer polynomial

Let $P$ be the set of all positive primes. Let $S$ an arbitrary infinite subset of $P$ satisfying the following assumption: there exists a finite Galois extension $K$ of $\mathbb{Q}$ and a conjugacy ...
5
votes
1answer
199 views

The abc-conjecture over the positive rationals and Levy-Schoenberg kernels?

I am continuing the "abc-adventure" and have a specific question, which needs some explanation: In this paper by Gangolli, the term "Levy-Schoenberg" kernel is defined (Definition 2.3). Consider the ...
1
vote
1answer
287 views

System of congruences

I have a system of $n$ congruences. the generic $m$_th congruence of the system ($m = 1,\dots,n$) is in the form: $(p_n\#)^2(\displaystyle\sum_{\substack{i=1\\i\neq m}}^n{\frac{x_iy_m}{p_ip_m}}+\...
5
votes
1answer
380 views

Primes mod 4 and integer polynomials

I have asked these questions as comments here (these are related to the question there). The questions are: Let $S$ be one of the following sets of primes: All primes of the form $4k+1$ ; All primes ...
2
votes
2answers
139 views

For what automorphic representations is Ramanujan-Petersson known?

I had in mind that Ramanujan-Petersson conjecture was essentially unknown in the case of number fields. I however recently heard that If an automorphic representation on $GL(2)$ is ramified at a ...
29
votes
16answers
31k views

Good algebraic number theory books

I have just finished a master's degree in mathematics and want to learn everything possible about algebraic number fields and especially applications to the generalized Pell equation (my thesis topic),...
1
vote
1answer
161 views

Schur polynomials with zeros in an infinite geometric progression

Let $a_1,...,a_n\in \overline{\mathbb{Q}}^\times$ be algebraic numbers, such that $\frac{a_i}{a_j}$ is not a root of unity for $i\neq j$. Furthermore, let $m\in\mathbb{N}$, $m>1$. Question: Is ...
9
votes
0answers
593 views

The Möbius function as eigenvalues

Let the $N$ by $N$ matrix $A$ be defined by the tetration: $$\Large \text{If } \gcd(n, k)=1 \text{ then } A(n,k)= \underbrace{e^{e^{\cdot^{\cdot^{e^{\Re\left(\frac{1}{n^s}\right)}}}}}}_m \text{ else }...
1
vote
0answers
56 views

On Kellner's result and the Erdos-Moser equation

Let $m, k$ be positive integers. Consider the Erdos-Moser expression $S_{k}(m) = 1^k + 2^k + ... + (m-1)^k$. By a result of Kellner, we know that if $m | S_{k}(m)$, then $m|B_k$, where $B_r$ denotes ...
6
votes
1answer
175 views

Distribution of signs of automorphic forms

Let's say we have an automorphic form $f$ on $GL(2)$ that is self-dual. In particular, the associated L-function $L(s,f)$ satisfies a functional equation with sign $\varepsilon_F = \pm 1$. Is it ...
-1
votes
0answers
56 views

Bijection map between equation and integer point [closed]

Suppose that we have a equation E define by $n!=m^2(m^2-2)+1$ and we want to solve it in $\mathbb{N}$ , and transforme it by the bijection map $\phi $ : $ (n,m ) $ $ \rightarrow $ $ (n,2m) $ to ...
2
votes
1answer
87 views

$(a-1)m>D(a,S(a-1,m))$?

Let $D$ be the function define as $D(b,n)$ be the sum of the base-$b$ digits of $n$. Example: $D(2,7)=3$ means $7=(111)_2\implies D(2,7)=1+1+1=3$ Define $S(a,m)=1^m+2^m+3^m+...+a^m$ where $a,m\in\...
2
votes
0answers
42 views

Relations between spectral parameters of automorphic representations

Let $\pi$ be an automorphic representation (say, trivial central character) of $GL(2)$. Let $\alpha(p)$ and $\beta(p)$ denote its spectral parameters at the place $p$, that is to say the associated ...
9
votes
0answers
336 views
+200

Classification of finite flat group schemes over integers?

One can classify (commutative) finite flat group schemes (with order of $p$-powers) over $\mathbb Z_p$ using semi-linear algebraic datas such as Breuil-Kisin modules. And we can fix the special fiber ...
6
votes
2answers
218 views

Around the diophantine equation $\frac{a}{2b+3c}+\frac{b}{2c+3a}+\frac{c}{2a+3b}=\text{odd integer}$, over positive integers

I am interested to know if a similar theorem that shows this answer of the post Estimating the size of solutions of a diophantine equation (this MathOverflow, January 5th 2016) is feasible for a ...
20
votes
1answer
1k views

The abc-conjecture as an inequality for inner-products?

The abc-conjecture is: For every $\epsilon > 0$ there exists $K_{\epsilon}$ such that for all natural numbers $a \neq b$ we have: $$ \frac{a+b}{\gcd(a,b)}\,\ <\,\ K_{\epsilon}\cdot \text{rad}\...
2
votes
1answer
138 views

Characters and weights of Hilbert modular forms

I'm trying to learn about weights of Hilbert modular forms, viewing the weights as characters $Res_{\mathcal{O}_F/\mathbb{Q}}\mathbb{G}_m \to \mathbb{G}_m$. For simplicity, assume that our totally ...
0
votes
1answer
167 views

Which Hilbert's 10th polynomials are known to have solutions?

The Diophantine equation $$x^3 + y^3 + z^3 = 42$$ was recently solved by Booker and Sutherland: Sum of three cubes for 42 finally solved. Is there a clean partition of the form of those polynomial ...
5
votes
3answers
754 views

Are there Heronian triangles that can be decomposed into three smaller ones?

Is there anything known about the existence of Heronian triangles ABC (i.e. with rational side lengths and rational area) that can be decomposed into three Heronian triangles ABD, BCD, CAD? ...
5
votes
0answers
234 views

Analytic continuation of $f(x)=\sqrt{\frac{1-x}{1-x^p}}$ over the p-adics

Consider the power series $f(x)=\sqrt{\frac{1-x}{1-x^p}}$ over the algebraic closure of $\mathbb{Q}_p$, defined by $f(0)=1$. What can be said about an analytic continuation "in the form of Mittag-...
-1
votes
0answers
59 views

Asking for reference request to study the proof of a result which is used in atleast 4 papers to prove existance of irrational odd zeta values

I am studying a research paper of T. Rivoal and Wadim Zudilin , "a note on odd zeta values " and I am unable to think how a result implies the theorem to be proved. So, I began to read other paper of ...
2
votes
1answer
82 views

Distribution of partial quotients of $\sqrt{d}$ with $X < d \leq 2X$

Recall that for a real number $\alpha$, it has a continued fraction expansion usually written as $$\displaystyle \alpha = [a_0; a_1, a_2, \cdots].$$ Moreover, $\alpha$ is rational if and only if its ...
9
votes
1answer
258 views

Second moment estimates for $\zeta(s)$: different methods?

What are some different ways to achieve the bound $$\int_0^T \left|\zeta\left(\frac{1}{2} + i t\right)\right|^2 dt = T \log \frac{T}{2 \pi} + (2 \gamma - 1) T + E(T)$$ with an error term $E(T) = O(T^{\...
6
votes
1answer
136 views

“Sub-logarithmic” zero-free regions from Deuring-Heilbronn/Linnik's repulsion theorem

For each $n\in\mathbb{N}$, let: $\chi_n\pmod{q_n}$ a real non-principal Dirichlet character ($q_1 < q_2 < \cdots$), $\beta_n$ the largest real zero of $L(s,\chi_n)$, $\delta_n := (1-\beta_n)\...
112
votes
13answers
23k views

Why are modular forms interesting?

Well, I'm aware that this question may seem very naive to the several experts on this topic that populate this site: feel free to add the "soft question" tag if you want... So, knowing nothing about ...
8
votes
2answers
323 views

Why is modular forms applicable to packing density bounds from linear programming at $n\in\{8,24\}$?

Sphere packing problem in $\mathbb R^n$ asks for the densest arrangement of non-overlapping spheres within $\mathbb R^n$. It is now know that the problem is solved at $n=8$ and $n=24$ using modular ...
10
votes
1answer
351 views

An upper bound for the length of the continued fraction expansion of $\sqrt d$

Let $d\ge 2$ and let $$ \sqrt d =[a_0; \overline{a_1,\dots, a_\ell, 2a_0}] $$ be its continued fraction expansion. Clearly, if $d=n^2+1$, then $\ell=0$, which gives the lower bound for $\ell$. ...
0
votes
1answer
84 views

Limit of a ratio of harmonic numbers?

Is there any way to find the following limit $$R(n,m)=\lim_{N\to\infty}\frac{H_{nN,m}}{H_{N,m}}$$ which involves harmonic numbers (generalized if $m\neq 1$) $$H_{N,m}=\sum_{k=1}^N k^{-m}\qquad ?$$ ...
1
vote
0answers
48 views

Nature of Fourier coefficient of a modular form after applying a certain map (trace operator)

Asking this here because of no response at (MathStackExchange). Let $N|M$, and consider the trace operator $Tr^M_N$ defined on $M_k(\Gamma_0(M))$ - vector space of modular forms of weight $k$ for ...
0
votes
0answers
46 views

Proportionality constant for the growth rate of $\zeta\left(s\right)$ along the imaginary axis

Going through what it says on page 95 of Titchmarsh's book on the Zeta function, and using his remark about $\mu$ at the bottom of that page, I conclude that there is some constant $C>0$ so that: $$...
6
votes
1answer
296 views

Hecke operator which changes character

In This MO question, Werner said that Hecke operator "changes" characters. I'm looking for any explicit theory of this kind, about Hecke operator with characters. Actually, there are somewhat ...
-1
votes
0answers
98 views

Is this integral an $O_{\varepsilon}(x^{\varepsilon})$?

This question is a follow-up to Is the imaginary part of $t\mapsto\zeta(1/2+it)$ close to the derivative of its real part? As I asked in a comment to the answer, does the following hold? $$\int_{x_{...
-3
votes
2answers
467 views

When is $x^3+x+1$ reducible mod $p$? [closed]

The original question with polynomial $x^2+ x +1$ turned out to be trivial. Let us change the polynomial. For which primes $p$ the polynomial $x^3+x+1$ is reducible mod $p$? It is irreducible mod 2, ...
1
vote
0answers
50 views

Brocard's conjecture for Ramanujan primes

Brocard's conjecture is a conjecture about the expected number of prime numbers $p_k$ between the squares of two consecutive prime numbers, I add as reference the Wikipedia Brocard's conjecture or [1]....
4
votes
1answer
189 views

Is the imaginary part of $t\mapsto\zeta(1/2+it)$ close to the derivative of its real part?

Plotting $t\mapsto\zeta(1/2+it)$ on Wolfram alpha, it seems that the maxima of its real part are close to the zeros of its imaginary part, while the maxima of the latter seem close to the inflection ...
0
votes
0answers
108 views

Neukirch's theorem on absolute Galois groups in English [duplicate]

Is there a paper or book available in English that proves the result of Neukirch on absolute Galois groups of number fields? I'm having a hell of a time with the German originals.
4
votes
0answers
123 views

Growth of the number of fixed points of a $p$-adic group under natural filtrations

Let $G$ be a $p$-adic reductive group, so by definition as a locally profinite group it's the group of $\mathbb Q_p$ points of a connective reductive group over $\mathbb Q_p$, $K$ be a parahoric ...

1 2 3 4 5 231