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

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4
votes
1answer
286 views

Does the Euler product for $L(s,\chi_4)$ also converge in the right half of the critical strip?

This question expands on this one from MSE. In the literature about Dirichlet $L$-series, I found that their Euler products: $$L(s, \chi) =\prod_p \bigg(\frac {1}{1-\frac{\chi(p)}{p^s}} \bigg)$$ ...
6
votes
1answer
117 views

Monte-Carlo computation of the Smith normal form

Quite some time ago I saw an article where a Monte-Carlo algorithm for computing the Smith normal form of an integer matrix was described. In this article the following problem was posed: Suppose $P, ...
8
votes
2answers
630 views

Is $f(x,y)=\sum_{n\in\mathbb{Z}\backslash\{0\}}\frac{1}{n}e^{2\pi i(xn+yn^2)}$ essentially bounded?

Let $$f(x,y)=\sum_{n\in\mathbb{Z}\backslash\{0\}}\frac{1}{n}e^{2\pi i(xn+yn^2)} $$ Is it true that $\|f\|_{L^{\infty}(\mathbb{R}^2)}<\infty$? i.e. is $f$ essentially bounded?
4
votes
0answers
363 views

Best known bounds on certain exponential sums

What are the best bounds currently known for the following exponential sum: $$\sum_{x < p \le 2x} e(\alpha p^k)$$ for values of $\alpha$ far from a rational with small denominator. ($p$ refers ...
3
votes
2answers
309 views

Geometry, Number Theory and Graph Theory of n-gon, permutation and graph labeling?

Given $n$ and $t$ lengths $ l_i, 1\leq l_1<l_2<\cdots<l_t\leq n-1$, of directed diagonals within an $n$-gon such that $l_1+\cdots+l_t\neq 0 \pmod n)$. Does it exist a directed path within ...
3
votes
0answers
103 views

Swan conductor, representation Weil group

Let $F$ be a non-archimedean local field and $\mathcal W_F$ its Weil group. We consider a linear representation $\sigma$ of $\mathcal W_F$. Could someone explain to me the definition of the Swan ...
38
votes
1answer
881 views

Transitivity on $\mathbb{N}_0$ — a 42 problem

Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$. Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition $\tau_{r_1(m_1),r_2(m_2)}$ be the ...
3
votes
1answer
448 views

Is this polynomial irreducible over the rationals?

Prove (or disprove): Define $T_n(x)$ as the Chebyshev polynomial of the first kind with degree $n$ . If $p$ is an odd prime, then $\sqrt{\frac{T_p(x)-1}{x-1}}$ is an irreducible polynomial over the ...
32
votes
2answers
869 views

Can we find two positive integers $n$ and $m$ ($n,m>1$) such that $n^\pi = m$? [duplicate]

I came across this apparent random question in some math questions website. At first, i thought it was easy to show that there are not integer solutions to this equation, but then i realized that the ...
9
votes
1answer
489 views

Do relaxed Liouville functions violate Chowla's conjecture?

Let $\lambda$ be the Liouville function. One version of Chowla's conjecture says that for each set of distinct natural numbers $h_1 , \dots , h_k$, $$\sum_{n\leq x} \lambda(n+h_1) \dots ...
2
votes
1answer
71 views

On finding the region $R$ for which the multi-variable sequence converges [closed]

Find the region $(x,y) \in R$ for which the following sequence converges $$\lim_{n \to \infty} \; \;\left| e^n\frac{(\sqrt{y}-\sqrt{x})^{2n}}{x^n} \right| = 0$$ I am currently doing number theory ...
3
votes
1answer
280 views

Is this differential equation for zeta on the critical line? One can compute it from its derivative and simpler functions

Looks like on the critical line one can compute $\zeta(1/2+it)$ from $\zeta^{'}(1/2+it)$ and simpler functions. Let $$ \begin{aligned} f(t)= & 2\, \left( {\frac { \left( \left| \zeta^{'} ...
5
votes
1answer
237 views

Do small prime (bounded) gaps imply larger (but still bounded) gaps?

The best result concerning bounded gaps between primes, whose existence was first proved by the seminal work of Yitang Zhang two years ago, are to my knowledge all of the form $\liminf_{n \rightarrow ...
2
votes
1answer
372 views

Number of twin primes

Consider number of twin primes less than $x$. We know that this number less than $\frac{Cx}{\log^2 x}$ for some constant $C$. Denote by $p_n$ the $n$-th prime number. Do we have the same result ...
5
votes
1answer
240 views

Fourier expansion of Eisenstein Series

I have been reading a bit about the Fourier expansion of Eisenstein series (weight 1/2). I came across the fact that the coefficients contain Modified Bessel functions. Further reading I found ...
36
votes
0answers
2k views

How to prove this polynomial always has integer values at all integers?

Let $m$ be any positive integer. $$ P_m(x)=\sum_{i=0}^{m}\sum_{j=0}^{m}{x+j\choose j}{x-1\choose j}{j\choose i}{m\choose i}{i\choose m-j}\frac{3}{(2i-1)(2j+1)(2m-2i-1)}. $$ Question: $P_m(x)$ always ...
10
votes
2answers
939 views

A new generalisation of Fermat's little theorem?

I would like to share a personal result, which I interpret as a "generalisation of Fermat's little theorem". I know that there exist already several generalisations (e.g. Euler's) but I would like to ...
3
votes
0answers
130 views

Algebraic integers with conjugates of fixed norm

Let $n$ be a positive integer and let $Q_n^d$ be the set of algebraic integers $\zeta$ which live in a degree $d$ extension of the rationals and such that any Galois-conjugate $\zeta^\sigma$ of ...
2
votes
2answers
571 views

If $\binom{2p}{p}$ is $(-1)^{p-1} \bmod 2p+1$ is then $2p+1$ prime?

Let $p$ be a positive integer; if $2p+1$ is prime then it is easily checked that $$(2p+1)\mid\left(\binom{2p}{p}+(-1)^{p-1}\right);$$ conversely I conjecture that if the above divisibility assumption ...
4
votes
1answer
211 views

Ref. request: Additive probability measure on $\mathcal P({\bf N})$ supplies subset of $\mathbf R$ without Baire property

ZFC proves, among the other things, the existence of a (finitely) additive probability measure $\theta: \mathcal P(\mathbf N) \to \mathbf R$ on the power set of $\mathbf N$ such that $\theta(X) = 0$ ...
4
votes
0answers
140 views

Are there infinitely many zeros of $\chi(s)+ \dfrac{2^{s}- 2^{2s-1}}{2^{s}-1}$ on the critical line?

Take $\chi(s)= 2^s\,\pi^{s-1}\,\sin\left(\frac{\pi\,s}{2}\right)\,\Gamma(1-s)$, so that $\zeta(s)=\chi(s)\,\zeta(1-s)$. The zeros of $\chi(s)=-1$ and the non-trivial zeros $\rho$ of $\zeta(s)$, seem ...
4
votes
0answers
210 views

Every integer, and every positive integer?

Construct $A_k = (a_1, …, a_k)$ and $D_k = (d_1, …, d_k)$ inductively from $a_1 = d_1 = 0$, starting with $k = 1$, as follows: let $h$ be the least integer $>-a_k$ such that $h \not\in D_k$ and ...
2
votes
0answers
158 views

Unirationality and the Hasse principle

Is there an example of a quasiprojective variety $X$ defined over ℚ such that $X$ is unirational over all finite fields, and $X$ is unirational over $\mathbb{R}$, and $X$ is not unirational over ...
2
votes
1answer
104 views

Representation numbers of numerical semigroups

I've been playing around with numerical semigroups lately. I'm pretty new to this stuff, so I apologize in advance if my notation is non-standard. Fix positive integers $x_1,\dots,x_r$ with ...
2
votes
2answers
237 views

Is it possible that $(a,b,c)$, $(x,y,a)$, $(p,q,b)$ are Pythagorean triples simultaneously? [closed]

Do there exist postive integers $a,b,c,x,y,p,q$ such $(a,b,c)$, $(x,y,a)$, $(p,q,b)$ are all Pythagorean triples? That is, does the system $$\begin{cases} a^2+b^2=c^2\\ x^2+y^2=a^2\\ p^2+q^2=b^2 ...
9
votes
2answers
554 views

Iwaniec-Kowalski Exponential Sum for Quadratic Function

I am reading about 'Exponential Sums' in the book 'Analytic Number Theory' by Iwaniec and Kowalski. On page 199 they mention the bound: $$|S_f(N)|^2 \le N +2N^2q^{-1}+4(N+q)\log q \tag{1}$$ where, ...
2
votes
2answers
320 views

Lifting projective Galois representation with condition

Let $\bar{\rho}: G_K\to PGL_n(\mathbb{C})$ be projective representation of the absolute Galois group of a number field $K$ and $\varphi\in Aut(G_K)$. A theorem of Tate tells us that we can always ...
2
votes
1answer
88 views

Is there a systematic relation between the generating functions for the rows vs that for columns of infinite sized Carleman-matrices?

(I've asked this in MSE but nobody had an idea since dec 14...) (Roughly related, but generalizing, of this earlier MSE question) Background: ...
3
votes
1answer
186 views

Product of $\tau(k)$

How do we show that $$\prod\limits_{k=1}^{n} \tau(k) = 2^{n (\log \log n + C) + \phi(n)},$$ where $\tau(k)$ is the number of divisors of $k$, the constant $C$ is given by $$C = \gamma + \sum_{\nu ...
1
vote
1answer
168 views

Finding the “minimum norm” of points in projective space above a prime field

I am working on doing explicit computations finding class groups of quaternions over $\mathbb{Q}$, and the following question (with $n=3$) was a clear choking point: Given a point $P = [x_0, \dots, ...
0
votes
0answers
170 views

p-adic asymptotic analysis

There is a huge field of asymptotic expansions and such over the real and complex fields (see Bender and Orszag, or Copson, or Whittaker and Watson). How different is the theory over p-adic fields? ...
5
votes
0answers
110 views

Furtwangler's Principal ideal theorem in number fields

Does anyone know a simple proof, using cohomological method of the fact that the verlagerung from a finite group G. to its commutator subgroup G', i.e. $$G/G'->(G')^{ab}$$ vanishes? The simplest ...
13
votes
1answer
580 views

Is the set of multiplicatively even numbers thick?

A positive integer is multiplicatively even (odd) if, when decomposed into primes, the sum of the exponents is even (odd). A subset of the integers is thick if it contains arbitrarily long intervals ...
4
votes
2answers
840 views

Find all rational solutions of this diophantine-equation?

Now, today, my friend tell me this problem was posted by American Mathematical Monthly (Vol. 111, No. 2 Feb., 2004), p. 165 by Wu wei Chao ,and It is said that this problem is unsolved, until now. ...
17
votes
2answers
519 views

divisors of $p^4+1$ of the form $kp+1$

In group theory the number of Sylow $p$-subgroups of a finite group $G$, is of the form $kp+1$. So it is interesting to discuss about the divisors of this form. As I checked it seems that for an odd ...
8
votes
2answers
434 views

Existence of polynomials of degree $\geq 2$ which represent infinitely many prime numbers

To my knowledge it is open so far whether the polynomial $x^2+1 \in \mathbb{Z}[x]$ takes infinitely many prime numbers as values. Is it known so far whether there is at all any polynomial $P \in ...
1
vote
0answers
97 views

A question on exponential sums

Let $F(x_1, \cdots, x_n) \in \mathbb{Z}[x_1, \cdots, x_n]$ be a homogeneous polynomial, irreducible over $\mathbb{Q}$. Let $A$ be a positive constant, and let $B$ be a positive real number understood ...
10
votes
3answers
569 views

About the prime divisors of values of polynomials

Let $P(x)$ be a polynomial having integer coefficients (and degree $\geq 3$), and let $p_1<p_2<\dots$ be the prime divisors occurring in the set of values $\{P(n):\ n\in\mathbb{Z}\}$. Is it ...
2
votes
1answer
204 views

A computation about Whittaker functions and Eisenstein series

I have some questions about the computation of Eisenstein series and Whittaker functions in the book. The question is on page 29, Theorem 4.3. My questions are in the following. (1) I think that ...
5
votes
2answers
603 views

When is $(-1+\sqrt[3]{2})^n$ of the form $a+b\sqrt[3]{2}$?

When is $(-1+\sqrt[3]{2})^n$ of the form $a+b\sqrt[3]{2}$ ($n$ being an integer) , i .e., when does $(-1+\sqrt[3]{2})^n$ not have a non-zero term in $\sqrt[3]{4}$. As you might have noticed, I'm ...
1
vote
1answer
128 views

How to solve a quadratic diophantine equation [closed]

I'm trying to solve $y^2=3x^2+3x+1$ for integers, which transforms into $(2y)^2-3(2x+1)^2=1$. I know how to solve pell's equation, but how can we extract only (odd,even) pair from the solutions of the ...
7
votes
0answers
168 views

Almost rational point

Let $X$ be a variety over a number field $K$. Let $S$ be a finite set of places of $K$. Is there a notion of a point $p \in X(\overline{K})$ to be "almost rational" in the following sense?: $p$ and ...
4
votes
0answers
122 views

Is $K^{ur} K^{\pi} = L$?

Let $L/K$ be a finite extension of $p$-adic fields, $\pi$ a uniformizer of $K$, $\theta = (-, L/K)$ the local Artin map $K^{\ast} \rightarrow Gal(L/K)$, $E$ be maximal unramified extension of $K$ ...
14
votes
1answer
257 views

Geometric Mean of $L(1,\chi)$ for quadratic Dirichlet characters

Let $S = \{D_1, D_2, D_3, \ldots \}$ be the set of all prime discriminants (or positive prime discriminants) of quadratic number fields. For such a discriminant let $\chi_j(n) = (\frac{D_j}n)$ be ...
0
votes
2answers
138 views

Special type Diophantine equations with integer solutions

The following problem on Diophantine equation is still solved or not I don't know. However, I found few solutions by trail and error method. Problem: $X^2 - X = Y^5 - Y$ has integer solutions or not? ...
10
votes
3answers
740 views

Find all solution $a,b,c$ with $(1-a^2)(1-b^2)(1-c^2)=8abc$

Two years ago, I made a conjecture on stackexchange: Today, I tried to find all solutions in integers $a,b,c$ to $$(1-a^2)(1-b^2)(1-c^2)=8abc,\quad a,b,c\in \mathbb{Q}^{+}.$$ I have found some ...
1
vote
1answer
125 views

From an eigenfom with $\mathbf{Q}$-coefficients to $j$-invariants

Given a cuspidal, classical eigenform $f\in S_2(\Gamma_0(N))$ of weight $2$ and with $\mathbf{Q}$-coefficients is there a way of describing the set $J_f$ of $j$-invariants of the elliptic curves lying ...
3
votes
0answers
97 views

Do the roots of this equation involving two Euler products all reside on the critical line?

This question loosely builds on the second part of this one. Take the Riemann $\xi$-function: $\xi(s) =\frac12 s\,(s-1) \,\pi^{-\frac{s}{2}}\, \Gamma\left(\frac{s}{2}\right)\, \zeta(s)$. Numerical ...
4
votes
0answers
88 views

Uniqueness of cohomological holomorphic discrete series representation

In Claus Sorenson's PhD thesis, he proves a theorem about level lifting of paramodular forms whose associated automorphic representation has component $\pi_{\infty}$ that is the "cohomological ...
2
votes
0answers
104 views

Nonnegative integers represented by $\prod_{i=1}^m \sum_{j=1}^n a_{i,j} x_j $, where the $a_{i,j}$ are positive integers

Fix $m, n \in \mathbf N^+$ with $m+n \ge 3$, and let $A = (a_{i,j})_{1 \le i \le m, 1 \le j \le n}$ be an $m$-by-$n$ matrix of positive integers. What is known about the asymptotic behavior of the ...