Tagged Questions
Prime numbers, Diophantine equations, analytic number theory, algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
0
votes
0answers
81 views
Density of numbers whose prime factors all come from a fixed congruence class
Let $q$ be a positive integer greater than one, and let $a$ be an integer such that $\gcd(a,q) = 1$. Define
$$D(a,q) = \{n \in \mathbb{N} : p | n \Rightarrow p \equiv a \pmod{q} \}.$$
Do we know the ...
0
votes
0answers
35 views
Embedding rational simple algebras in the real quaternions [duplicate]
Is there any way to embed a rational division algebra of dimension higher than 4 over its center in the real quaternions ?
I think not, but I cannot prove it.
7
votes
1answer
295 views
Regularized sums of Mobius sequence
Do $\lim_{s \rightarrow \infty} \sum_{n \geq 1} \mu(n) e^{-n/s}$ and $\lim_{s \rightarrow \infty} \sum_{n \geq 1} \mu(n) e^{-n^2/s^2}$ both equal $-2$?
Experimentally this seems plausible (up through ...
4
votes
1answer
263 views
Group laws in class field theory
In the case of a quadratic imaginary number field one can construct its maximal abelian extension using torsion points of an elliptic curve with complex multiplication by this field.
In the case of a ...
6
votes
1answer
370 views
Adding sets not containing arithmetic progressions of length three by forcing
Consider the following forcing notion: conditions in $\mathbb{P}$ are pairs $(s, N),$ where:
1) $s\in 2^{<\omega}$,
2) $N\in \mathbb{N}$,
3) (by identifying $s$ with a subset of $lh(s)$) $s$ ...
0
votes
0answers
46 views
separating parameters in generalized quadratic Gauss sum
The normalized generalized quadratic Gauss sum is defined by
$$
G(a,b,c)=\frac{1}{c}\sum_{n=1}^ce\left(\frac{an^2+bn}{c}\right)
$$
where $e(x)=\exp(2\pi ix)$.
Under what conditions on $c$ can we ...
3
votes
0answers
104 views
Application of Stickelberger's Theorem to Quadratic field
I am trying to understand a proof of the Kronecker-Weber Theorem by Franz Lemmermeyer,[http://arxiv.org/pdf/1108.5671.pdf] in which he uses Stickelberger's Theorem applied to Kummer extensions. I can ...
7
votes
0answers
206 views
Can an abelian variety/Q have no points over Q_sol?
Let $A/\mathbb{Q}$ be an abelian variety. Must there be a finite solvable
extension $K/\mathbb{Q}$ such that $A(K)$ is nontrivial?
This follows from the conjecture that the maximal ...
-5
votes
0answers
73 views
Prove an equation is always false [on hold]
How can I prove an equation is always false?
For example:
b = b + 1
is false for all values of b. Very simple to see.
Now given a more complicated equation, such as:
b = sin(sin(b) - .56))
...
5
votes
1answer
115 views
Symmetry type of non-cohomological automorphic forms
By Katz-Sarnak philosophy a family of $L$-functions would have a symmetry type which would reflect the statistics of $L$-functions, such as low lying zeros and moments. Shin-Templier's paper on ...
5
votes
2answers
516 views
How close to an integer can a polynomial root be?
Suppose I have a polynomial $p(x) = a_n x^n + ... + a_0$ where $a_n, \dots, a_0$ are integers. I would like to show that any root of this polynomial is either an integer or is far from an integer. ...
2
votes
1answer
224 views
Is anything like $\phi(n)>\dfrac n{e^\gamma\log\log n},\ \sigma(n)<e^\gamma n\log\log n$ known/conjectured for the generalizations of these functions?
Is anything like $\dfrac n{\phi(n)}<\dfrac{\sigma(n)}n<e^\gamma\log\log n$ known/conjectured for the generalizations of these functions?
Let $n=p_1^{a_1}\cdots p_t^{a_t}$ be the canonical prime ...
0
votes
0answers
29 views
Find all integers x, y, and z such that 1/x + 1/y = 1/z [migrated]
Characterize all positive integers x, y, and z such that $\frac{1}{x} + \frac{1}{y} = \frac{1}{z}$. For example, $\frac{1}{x+1} + \frac{1}{x(x+1)} = \frac{1}{x}$
1
vote
0answers
95 views
Averages of $L(s,\chi)$
Let $(\frac{m}{n})$ denote the usual quadratic Jacobi symbol.
What is the
abscissa of convergence
of the double Dirichlet series ?
$$
\sum_{\substack{m,n \in \mathbb{N} \\ \gcd(m,n)=1 \\m,n\equiv 1 ...
1
vote
1answer
91 views
Q(\sqrt{-l_0}) satisfies Heegner hypothesis for an Elliptic curve of conductor C implies C is a square
Trying to understand the proof of Corollary 2.3 in the following paper,
http://arxiv.org/pdf/1312.3884.pdf
I would like to be able to justify that the root number of the quadratic twist ...
0
votes
0answers
70 views
Integer points on Elliptic Curves [closed]
It's easy to prove that equation
y^2=x(x-a)(x-b)
with a and b integer has integer solution, other than (0,0), (a,0) and (b,0), if a and b jointly admits the representation
a=(r-s)r (1)
b=(r-t)t ...
7
votes
1answer
202 views
Geometric interpretation of Cusps for general groups?
Let $\mathrm{G}$ be a reductive group over a number field $F$, but for simplicity we can think about $\mathrm{G}=\mathrm{GL_n}$ for $n>2$ and $F =\mathbb{Q}$.
Then for an automorphic form,
...
-5
votes
0answers
39 views
Quadratic residue [closed]
P is prime. Prove -1 is a quadratic residue mod p if and only if either p = 2 or else p =1(mod4)
I tried using quadratic reciprocity proofs to solve, but am unable to discover the answer.
3
votes
0answers
202 views
The $\ell = p$ case of Ihara's lemma
Let $N \ge 1$ and let $\ell$ and $p$ be primes not dividing $N$.
The classical Ihara lemma says that if $Y_1(N, \ell)$ is the modular curve attached to the subgroup $\Gamma_1(N) \cap \Gamma_0(\ell)$, ...
2
votes
0answers
121 views
irrationality of Dedekind zeta values
For Riemann's zeta function, one knows that
$\zeta(2n)$ is irrational (because $\pi$ is)
$\zeta(3)$ is irrational (proved by Apéry)
and a few other results like "there are infinitely many ...
1
vote
0answers
49 views
Euler series with milder divergence
Theorema 19 in Euler's memoir "Variae observationes circa series infinitas" says
The sum of the reciprocals of the prime numbers is infinitely great but is infinitely times less than the sum of the ...
4
votes
2answers
198 views
Bound on a scaled sum of the Liouville function
Terence Tao has shown see his blog post that
$$\left| \sum_{n\leq x} \frac{\mu(n)}{n} \right|\leq 1,$$
for $x$ a positive real number, where $\mu(n)$ is the Möbius function. Let $\lambda(n)$ denote ...
7
votes
2answers
500 views
Most dense subset of numbers that avoids arbitrarily long arithmetic progressions
The famous Green-Tao theorem says that there exist arbitrarily long sequences of primes in arithmetic progression.
I am wondering: How dense can a subset $S \subset \mathbb{N}$ be and still avoid
...
0
votes
0answers
112 views
injective homogeneous polynomial functions $p(x,y) \in \mathbb{Z}[x,y]:{\mathbb{N}}^2 \to \mathbb{N}$
Related to this question Coefficients of $f(t)=(\sum_{m=0}^{+\infty}e^{2\pi im^4t})(\sum_{m=0}^{+\infty}e^{2 \pi inm^4t})$
I have the following question:
What is the set of homogeneous polynomials ...
1
vote
0answers
126 views
Coefficients of $f(t)=(\sum_{m=0}^{+\infty}e^{2\pi im^4t})(\sum_{m=0}^{+\infty}e^{2 \pi inm^4t})$
I want to prove that $\forall n \in \mathbb{N}$ at least one of the Fourier coefficients of $f(t)=(\sum_{m=0}^{+\infty}e^{2\pi im^4t})(\sum_{m=0}^{+\infty}e^{2 \pi inm^4t})$ is striclty greater than ...
10
votes
2answers
398 views
What is the Hausdorff dimension of this fractal?
Let $\sum_{i=h}^\infty d_i/b^i $ be the base $b$ representation of $x \geq 0,$ where $b>1$ and the $d_i$ are uniquely determined by the greedy algorithm. For fixed $c>1,$ let $f(x)= ...
1
vote
0answers
80 views
Best constant for Maier's theorem?
Maier proved that, for fixed $\lambda>1,$
$$
\limsup_{x\to\infty}\frac{\pi(x+\log^\lambda x)-\pi(x)}{\log^{\lambda-1}x}>1
$$
and in particular
$$
\limsup_{x\to\infty}\frac{\pi(x+\log^\lambda ...
16
votes
1answer
464 views
+50
Why would the roots of the generating functions of the number of k-almost primes less than x have negative real parts?
There is a 50 point bounty on this question.
Specifically, I find it appealing to count only squarefree numbers having $k$ prime factors, so I define
$$\pi_k(x)=\#\{n\leq x: ...
1
vote
2answers
307 views
Examples that the Fermat-Catalan conjecture does not cover
The Fermat-Catalan conjecture states that there are only finitely many sex-tuples $(a, b, c, d, e, f)$ of positive integers such that
(1) $a^d + b^e = c^f$,
(2) $\gcd(a, b, c) =1$,
(3) ...
2
votes
1answer
119 views
Irreducibility of Faulhaber-like Polynomials over $\mathbb Q[x]$
Motivation: Inspired by the famous Faulhaber polynomials $F_k(N)=\displaystyle\sum_{n=0}^Nn^k,$ I decided to study their alternating versions, $\Phi_k(M)=\displaystyle\sum_{n=0}^M(-1)^nn^k$.
For ...
4
votes
1answer
246 views
Square-free grows as $6n/\pi^2$: $k$-th free?
The asymptotic number of
square-free numbers
$\le n$ is $Q(n) = 6n/\pi^2 + O(\sqrt{n})$.
Because
$\zeta(2)=\pi^2/6$,
$Q(n) \approx n/\zeta(2)$.
OEIS A004709
says that cube-free numbers have ...
3
votes
1answer
255 views
Invertibility of a matrix whose entries are certain binomial coefficients
Let $l$ be a positive integer. Does the matrix
$$
M_l \ := \ \left( \binom{l-(2p+1)}{j} \right)_{0\leq p,j \leq[(l-1)/2]}
$$
have nonzero determinant?
1
vote
1answer
242 views
Is there a non-tempered representation of U(2)?
I am wondering why the first well known example of non-tempered irreducible admissible representation of $p$-adic group $U(n)$ should be $U(3)$. Because, Gelbart and Rogawski suggested the ...
9
votes
2answers
800 views
What is $\sum_{i=0}^{n}\binom{n}{i}^3$?
We know that $$\sum_{i=0}^{n}\binom{n}{i}=2^n$$
and that
$$\sum_{i=0}^{n}\binom{n}{i}^2= \binom{2n}{n}$$
what about
$$\sum_{i=0}^{n}\binom{n}{i}^3$$ ?
6
votes
1answer
187 views
When are toral orbits in buildings the difference of fixed-sets?
Let $L$ be a $p$-adic field, let $G$ be a reductive group over $L$ (I'm even okay assuming semisimplicity for now). Let $T$ be a maximal torus of $G$. Let $B$ be the building for $G(L)$. (Edit 1: ...
4
votes
1answer
196 views
Counting couples of square-free polynomials over finite fields
I have a curve defined by the following equations over the finite field $\mathbb{F}_q$ with $q=p^r$ with $p \geq 3$:
$$C_{h_1,h_2}:\begin{cases} y_1^2=h_1(t) & \\
y_2^2=h_2(t) ...
16
votes
4answers
2k views
A question on Collatz's conjecture
Let $C$ : ${\mathbb N}\longrightarrow {\mathbb N}$ be Collatz's map defined by $C(n) = 3n+1$ if $n$ is odd, and $C(n)=n/2$ if $n$ is even. Then according to Collatz's conjecture, we should have $C^k ...
8
votes
2answers
689 views
What is known about primes of the form x^2-2y^2?
David Cox's book Primes of The Form: $x^2+ny^2$ does a great job proving and motivating a lot of results for $n>0$. I was unable to find anything for negative numbers, let alone the case I am ...
8
votes
0answers
214 views
Geometric vs combinatorial motives over Spec Z
Consider the category of reduced schemes of finite type over $\mathbb{Z}$. Take the Grothendieck group of this category, i.e. the free abelian group on isomorphism classes, modulo the usual "syzygy" ...
2
votes
1answer
260 views
$n^3 | \sum_{i=1}^{n-1}\binom{n}{i}^2$ => $n | \sum_{i=1}^{n-1}\binom{n}{i}$?
For $n\in \mathbf{N}$ is $$n^3 \text{ divides } \sum_{i=1}^{n-1}\binom{n}{i}^2=\binom{n}{1}^2+\cdots +\binom{n}{n-1}^2$$ impling $$n \text{ divides } ...
1
vote
0answers
67 views
Spectrum of primitive nonnegative integer matrices
Let $P(X) = a_nX^n + \cdots + a_1X + a_0$ with $a_i \in \mathbb Z$.
Question 1. Is there an efficient criterion on the $a_i$ to decide if there exists a primitive nonnegative integer matrix with ...
-1
votes
0answers
265 views
Is there a polynomial that generates only primes or semi-primes? [migrated]
I know that no non-constant polynomial function $P(n)$ with integer coefficients exists that evaluates to a prime for every integer value of $n$.
My question is - does there exist a non-constant ...
21
votes
2answers
3k views
A possibly surprising appearance of Fibonacci numbers
Let $H(n) = 1/1 + 1/2 + \dotsb + 1/n,$ and for $i \leq j,$ let $a_1$ be the least $k$ such that
$$H(k) > 2H(j) - H(i),$$
let $a_2$ be the least k such that
$$H(k) > 2H(a_1) - H(j),$$
and ...
2
votes
0answers
94 views
Are all zeros of $\xi(a\,s) \pm \xi\left(a\,(1-s)\right)$ on the critical line for $\forall a \in \mathbb{R}/0$?
This question expands on this one and seems to have a stronger result.
Take the Riemann $\xi$-function $\xi(s) =\frac12 s\,(s-1) \,\pi^{-\frac{s}{2}}\, \Gamma\left(\frac{s}{2}\right)\, \zeta(s)$. We ...
4
votes
0answers
281 views
Conjectures on perfect squares
There exist infinitely many sets of three positive integers $(a,b,c)$ where $a\not=b,b\not=c$ and $c\not=a$ such that each of $$ab-1,\ bc-1,\ ca-1,\ ab-a-b+c,\ bc-b-c+a,\ ca-c-a+b$$
is a perfect ...
4
votes
1answer
558 views
How good can we approximate with algebraic curve the egg shaped vanishing of $\Re \zeta(s)$ near the origin?
Related to this question
where degree $2$ algebraic curve is good approximation to vanishing of
the real part of expression involving zeta.
Near the origin, $\Re \zeta(s)$ vanishes in egg shaped ...
6
votes
0answers
100 views
How comes vanishing of the real part of function involving zeta is very well approximated by algebraic curve?
In this question
Agno asked about the zeros of $\zeta\left(\frac{s}{a}\right) \pm \zeta\left(\frac{1-s}{a}\right)$.
I fixed $a=2$ and the minus sign and defined:
$$
f(s)=\Re \left( ...
0
votes
0answers
134 views
Does the Euler product converge at $s=1$ for the Dirichlet $L$ function?
For the Riemann Zeta function, the Euler product converges on $\{Re(s)=1\}$ except at $s=1$.The zeta series diverges everywhere on $\{Re(s)=1\}$. But the $L$ series converges on $\{Re(s)>0\}$. What ...
14
votes
0answers
230 views
Is the absolute Galois group of the rationals Hopfian?
Is every continuous epimorphism from the absolute Galois group of $\mathbb{Q}$ to itself injective?
7
votes
1answer
230 views
Set of integers having finite intersection with the image of any polynomial of degree $\geq 2$
Is there a set $A$ of positive integers such that
$\sum_{n \in A} \frac{1}{n} = \infty$, and
there is no polynomial $f \in \mathbb{Z}[x]$ of degree at least $2$
which takes infinitely many values in ...
