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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69 views

What are the consequences if the Birch–Swinnerton-Dyer conjecture is assumed to be false

The Birch–Swinnerton-Dyer (BSD) conjecture (https://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture) describes the set of rational solutions to equations defining an elliptic curve. This ...
-1
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1answer
214 views

$\exists k$ s.t. $k^m\le 1^m+2^m+…+(k-1)^m <2\cdot k^m$?

Can it be shown that,for all $m\in\mathbb{Z}_+$ there exists at least one $k$ with respect to $m$ such that $$1\le \frac{\sum_{i=1}^{k-1}i^m}{k^m}<2$$ Example: let $m=1$ then $k=\{3,4\}$ This ...
4
votes
1answer
161 views

Estimating certain short Kloosterman sums

Recall that for the classical Kloosterman sum $$ K(a,b,p^t):= \sum_{x \in (\mathbb{Z}/ p^t \mathbb{Z})^* } \psi \left(\frac{ax+bx^{-1}}{p^t} \right),$$ where $\psi(x)=e^{2\pi ix}$, $a,b,t$ are natural ...
18
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0answers
215 views

Geometry of algebraic curve determined by point counts over all number fields?

Let $C$ be a smooth irreducible projective curve of genus $g>1$ over a number field $K$. The Mordell conjecture (first proved by Faltings) says that for any finite field extension $L/K$ (say, ...
2
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0answers
86 views

Analogues over finite fields of certain integers defined multiplicatively in $\mathbb Z$

For any irreducible polynomial $f$ of degree $d$ in finite field $\mathbb F$ we have that $$x^{|\mathbb F|^d}-x\equiv0\bmod f$$ and thus the polynomial $x^{|\mathbb F|^d}-x$ is the analog of factorial ...
2
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0answers
176 views

Generating all graphs of order 4 with the help of Collatz

Given a set of positive integers, its common divisor graph ( CD-graph) is the graph whose vertices are the integers, two of which are joined by an edge if (and only if) they have a common divisor ...
2
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1answer
105 views

Numerical estimates for a function relating to twin primes :

Consider the following function : $$F(s)= \sum_{\text{$p,\ p+2$ are primes}} \left({\frac{1}{p^s}}+{\frac{1}{(p+2)^s}}\right).$$ Brun's theorem tells us that $F(1)$ is finite. We are looking for ...
1
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2answers
88 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\...
11
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1answer
315 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]...
5
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0answers
217 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 ...
5
votes
1answer
209 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
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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 ...
5
votes
1answer
393 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 ...
1
vote
1answer
169 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 ...
6
votes
1answer
181 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 ...
2
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0answers
45 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 ...
-1
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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
2answers
148 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 ...
0
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1answer
173 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 ...
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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 ...
0
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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 $m\neq 1$) $$H_{N,m}=\sum_{k=1}^N k^{-m}\qquad ?$$ ...
5
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0answers
235 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-...
8
votes
2answers
329 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 ...
1
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0answers
53 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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0answers
47 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: $$...
2
votes
1answer
84 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 ...
6
votes
1answer
137 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)\...
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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_{...
0
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0answers
52 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]....
0
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0answers
109 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
1answer
191 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 ...
1
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0answers
251 views

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 ...
2
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0answers
117 views

Action on the upper half plane of the double coset of an irrational matrix in $\mathrm{SL}_2(\mathbb{R})$

Consider the action of $\mathrm{SL}_2(\mathbb{R})$ on the upper half plane $\mathbb{H}$ by Möbius transformations. Denote by $\Gamma$ the group $\mathrm{SL}_2(\mathbb{Z})$. It is known that if $z \in ...
0
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0answers
128 views

On the difference $\operatorname{Li}(\theta(x))-\pi(x)$

In G. Robin's paper, more precisely in Lemme12, how does he use formula (39) to prove formula (36)? [1] Robin, Guy, "Estimation de la fonction de Tchebychef θ sur le k -ième nombre premier et ...
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-...
-3
votes
2answers
471 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
80 views

On some integral involving the Liouville sum

Define $L(x) = \sum_{n\leq x} \lambda(n)$, where $\lambda$ denotes the Liouville function. If $c$ is the supremum of the real parts of the zeros of the Riemann zeta function and $\lfloor x\rfloor$ ...
6
votes
0answers
200 views

What is known about “almost orthogonal vectors”?

Motivation: Suppose we have a kernel $k(a,b)$ defined over the natural numbers. Then by the Moore–Aronszajn theorem, we can embedd the natural number $a$ in some Hilbert space $\mathbb{H}$, which we ...
4
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0answers
206 views

Current status on Deligne's conjecture of special values of L-functions

In the paper "Valeurs de Fonctions $L$ et périodes d’intégrales" in Proceedings of Symposia in Pure Mathematics 33, (1979), Part 2, 313-346, Deligne formulated his famous conjecture relating the ...
1
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0answers
134 views

Questions about a certain sequence of naturals generated by primorials

I'm working on the following sequence of naturals (which is NOT listed in OEIS) $$3,5,11,17,23,29,59,89,119,149,179,209,419,629,839,1049,1259,1469,1679,...$$ whose elements are generated this way $$3=(...
2
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0answers
91 views

On the set $\{n>0:\ n\ \text{is a quadratic nonresidue modulo the}\ n\text{th prime}\}$

Let $S$ denote the set of positive integers $n$ with $n$ a quadratic nonresidue modulo the $n$th prime $p_n$. The first 20 elements of $S$ are $$2,\, 3,\, 6,\, 7,\, 8,\, 10,\, 11,\, 13,\, 15,\, 18,\, ...
0
votes
1answer
263 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 ...
6
votes
2answers
220 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}\...
0
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0answers
99 views

Exact and simple expressions for the (inverse) of the $n$-th Bernoulli number

Of course, there are many known equations and approximations relating to the $n$-th Bernoulli number like $$B^{-{}}_m = \sum_{k=0}^m \sum_{v=0}^k (-1)^v \binom{k}{v} \frac{v^m}{k + 1}$$ or $$|B_{2 n}| ...
2
votes
0answers
38 views

To choose a set of $n$ rectangles which together form largest number of rectangular layouts

Question: Given a number $n$, find that set of $n$ rectangular tiles of any area and perimeter (the tiles in the set could be any type of rectangles; one could choose some of them as identical, some ...
1
vote
0answers
30 views

Asymptotic behaviour of $\sum_{k=1}^{n}\frac{R_{k+1}+R_k}{R_{k+1}-R_k}$, where $R_k$ are the Ramanujan primes

I was inspired in the following post from this Math Overflow that was asked yerterday Asymptotic behavior of a certain sum of ratios of consecutives primes to ask about a similar question for ratios ...
6
votes
1answer
126 views

Representation of primes of the form $4m+3$ with double radicals

Let $\,q\,$ be a prime of the form $\,4\, m_q+3$. I ask if it is always possible to find two primes $\,p_1$ and $\,p_2$ of the form $\,4\, m_p+1$ such that $$q=\sqrt{p_1+\sqrt{p_2+q}}$$ E.g. $$3=\...
0
votes
0answers
41 views

Exponential sums involving completely multiplicative functions

It is well-known that exponential sums is used as a tool from the analytic number theory to optimize or to compute asymptotic formulas. My question is the following: Given a completely multiplicative ...
0
votes
0answers
135 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{...

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