# 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

11,542
questions

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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

**1**answer

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

**1**answer

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**

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**0**answers

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

**2**answers

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

**4**answers

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

**7**answers

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

**3**answers

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

**1**answer

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 $k$...

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vote

**0**answers

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

**4**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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

**16**answers

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

**1**answer

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

**0**answers

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 }...

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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

**1**answer

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 ...

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**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**3**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**13**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**0**answers

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**

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**0**answers

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

**1**answer

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

**0**answers

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

**2**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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 ...