Prime numbers, Diophantine equations, analytic number theory, algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
3
votes
1answer
123 views
Averages over integer points of the sphere
A paper of William Duke sketches a proof that integer points on the sphere are equidistributed.
$$ V_N = \{ (x,y,z) \in \mathbb{Z}^2 : x^2 + y^2 + z^2 = N \} $$
Up to reflections across the $x$, $y$ ...
2
votes
1answer
133 views
Dihedral extension of 2-adic number field
Sorry if the question is too long and maybe elementary.
I am reading a paper by Hirotada Naito on "Dihedral extensions of degree 8 over the rational p-adic fields". To generate dihedral extension ...
10
votes
0answers
106 views
Variety acquiring rational point over any quadratic extension
Does there exist a variety $X$ over $\mathbb{Q}$ (or a number field) such that it has no rational points over $\mathbb{Q}$ but acquires points over any quadratic extension $\mathbb{Q}(\sqrt{d})$?
If ...
5
votes
0answers
81 views
Degenerate linear recurrence sequences
Let $(u_n)_{n \geq 0}$ be a linear recurrence given by
$$u_n = a_1 u_{n-1} + \cdots + a_k u_{n-k} \quad \forall n \geq k ,$$
where $u_0, \ldots, u_{k-1}, a_1, \ldots, a_k \in \mathbb{Z}$. We recall ...
1
vote
0answers
72 views
dimension of a scheme and degree of an L-function [on hold]
I try to understand correctly the notion of scheme, as Serre in the second volume of his Oeuvres defines zeta and L-functions in this context. What seems interesting to me is that he states a theorem ...
1
vote
0answers
132 views
On Descartes / spoof odd perfect numbers
Descartes, Frenicle, and subsequently Sorli, conjectured that $k = 1$, if $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form (i.e., $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and ...
0
votes
0answers
25 views
How to find integer solution for bilinear transformation? [migrated]
Let y = (ax + b)/(cx + d), where a, b, c, d integer constants, is there any technique to find integer solution of x & y?
5
votes
0answers
80 views
Asymptotics of the multipartition function
Recall that the multipartition function $p_k(n)$ counts the number of $k$-tuples of partitions $\lambda^1,\ldots,\lambda^k$ of numbers $a_1,\ldots,a_k$ with $a_1+\cdots+a_k=n$. It has a generating ...
0
votes
0answers
99 views
Fermat's Theorem on p = a^2 + b^2 [migrated]
I have read that Fermat predicted that for an odd prime $p$, $p = a^2 + b^2$ iff $p = 1$ mod 4.
I heard that such a criterion could be possible for a given integer $n$ like
$p = a^2 + n b^2$
...
1
vote
0answers
105 views
Experimentation with partial Euler products
Richard Mathar $[1]\& [2]$ shows that
\begin{align}
&\zeta_{2}(s)\equiv\prod_{\Omega(n)=2}^{}\left(\dfrac{1}{1 - n^{s}}\right)^{-1}=\exp \left(\sum _{k=1}^n \frac{P(k s)^2+P(2 k s)}{2 ...
0
votes
0answers
87 views
Combinatorial support set in CRT
Is there a function $g(s)$ such that if there is a set of numbers $\{r_i\}_{i=1}^m$ such that $r_i\bmod p_j\in\{0,1\}$ at every prime in $\{p_j\}_{j=1}^n$ such that $2^t\bmod p_j\neq1$ at every ...
3
votes
0answers
148 views
Primes in integral domains
Let $\mathcal O_{\mathbb K}$ be the ring of integers in a number field $\mathbb K$. It is easy to prove that the number of (non-associated) primes in $\mathcal O_{\mathbb K}$ is infinite.
On the other ...
1
vote
0answers
121 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
37 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
343 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 ...
5
votes
1answer
307 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
405 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
54 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
113 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
227 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
81 views
Prove an equation is always false [closed]
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
121 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
542 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
234 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
98 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
211 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
40 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
207 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)$, ...
4
votes
1answer
193 views
Irrationality of Dedekind zeta values
For Riemann's zeta function, one knows that:
$\zeta(2n)$ is irrational (because a rational multiple of $\pi^{2n}$ is)
$\zeta(3)$ is irrational (proved by Apéry)
and a few other results like "there ...
2
votes
0answers
52 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
211 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,$$
for $x$ a positive real number, where $\mu(n)$ is the Möbius function. Let $\lambda(n)$ denote ...
7
votes
2answers
509 views
Most dense subset of numbers that avoids arbitrarily long arithmetic progressions
The famous Green-Tao theorem says that there exist arbitrarily long sequences of primes in arithmetic progression.
I am wondering: How dense can a subset $S \subset \mathbb{N}$ be and still avoid
...
0
votes
0answers
113 views
injective homogeneous polynomial functions $p(x,y) \in \mathbb{Z}[x,y]:{\mathbb{N}}^2 \to \mathbb{N}$
Related to this question Coefficients of $f(t)=(\sum_{m=0}^{+\infty}e^{2\pi im^4t})(\sum_{m=0}^{+\infty}e^{2 \pi inm^4t})$
I have the following question:
What is the set of homogeneous polynomials ...
1
vote
0answers
128 views
Coefficients of $f(t)=(\sum_{m=0}^{+\infty}e^{2\pi im^4t})(\sum_{m=0}^{+\infty}e^{2 \pi inm^4t})$
I want to prove that $\forall n \in \mathbb{N}$ at least one of the Fourier coefficients of $f(t)=(\sum_{m=0}^{+\infty}e^{2\pi im^4t})(\sum_{m=0}^{+\infty}e^{2 \pi inm^4t})$ is striclty greater than ...
10
votes
2answers
406 views
What is the Hausdorff dimension of this fractal?
Let $\sum_{i=h}^\infty d_i/b^i $ be the base $b$ representation of $x \geq 0,$ where $b>1$ and the $d_i$ are uniquely determined by the greedy algorithm. For fixed $c>1,$ let $f(x)= ...
1
vote
0answers
81 views
Best constant for Maier's theorem?
Maier proved that, for fixed $\lambda>1,$
$$
\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 ...
18
votes
1answer
603 views
Why would the roots of the generating functions of the number of k-almost primes less than x have negative real parts?
Specifically, I find it appealing to count only squarefree numbers having $k$ prime factors, so I define
$$\pi_k(x)=\#\{n\leq x: \omega(n)=k;\mu(n)\neq0 \}$$
and consider the generating functions
...
1
vote
2answers
309 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
122 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
248 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
257 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
811 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
196 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
692 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 ...
