# Tagged Questions

**1**

vote

**0**answers

13 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 ...

**2**

votes

**1**answer

120 views

### Varieties with few monomials and the n-conjecture

The n-conjecture
is a generalization of abc and basically says that the if
$a_1 + \ldots + a_n=0$, no proper subsum vanishes and $a_i$
are coprime, then the radical of $a_1\cdots a_n$ can't be
too ...

**7**

votes

**1**answer

184 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,
...

**10**

votes

**2**answers

374 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)= ...

**4**

votes

**2**answers

364 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. ...

**16**

votes

**1**answer

425 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: ...

**5**

votes

**1**answer

96 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 ...

**-4**

votes

**0**answers

62 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))
...

**1**

vote

**1**answer

172 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 ...

**1**

vote

**0**answers

97 views

### Explicit description/calculation of norm group of ideles of characteristic $p$ global field

I posted the same question earlier in stack exchange,
(http://math.stackexchange.com/questions/1130391/algebraic-proof-of-2nd-inequality-of-global-class-field)
thinking it is most definitely not a ...

**1**

vote

**1**answer

230 views

### Constant group scheme and torsors

Let $X$ be a scheme and $G$ a (commutative) constant group scheme. Consider a $G$-torsor $Y$ for $X$, by which I mean that there is a canonical isomorphism:
$$g_Y \colon Y \times_X Y \cong Y ...

**0**

votes

**0**answers

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, $\frac{1}{x+1} + \frac{1}{x(x+1)} = \frac{1}{x}$

**6**

votes

**4**answers

604 views

### Multiplicity one conjecture

I recently became interested in Maass cusp forms and heared people mentioning a "multiplicity one conjecture". As far as I understood it, it says that the dimension of the space of Maass cusp form for ...

**11**

votes

**3**answers

769 views

### Probability of coprime polynomials

Given positive integer $N$, we choose $m_1, m_2, n_1, n_2$ independently and with equal probabilities from $\{0,1,\ldots,N\}$, and let
$f_1 = x^{m_1} + (1+x)^{n_1}$ and $f_2 = x^{m_2} + (1+x)^{n_2}$ ...

**15**

votes

**1**answer

904 views

### Singmaster's conjecture

Has any work been done on Singmaster's conjecture since Singmaster's work?
The conjecture says there is a finite upper bound on how many times a number other than 1 can occur as a binomial ...

**1**

vote

**0**answers

91 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

**1**answer

90 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 ...

**8**

votes

**2**answers

1k views

### Eigenvalues of nonnegative integer matrices

Edit
I realized that the key piece of information that I need is question 1, and so I'd like to rephrase this post:
What are the possible eigenvalues of nonnegative integer matrices?
Any answer ...

**3**

votes

**0**answers

198 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 $\Gamma_1(N) \cap \Gamma_0(\ell)$, ...

**0**

votes

**0**answers

70 views

### Integer points on Elliptic Curves [on hold]

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 ...

**4**

votes

**2**answers

184 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 ...

**-5**

votes

**0**answers

39 views

### Quadratic residue [on hold]

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.

**2**

votes

**0**answers

117 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

**1**answer

208 views

### Is there a non-tempered representation of U(2)?

I am wondering why the first well known example of irreducible admissible representation of $p$-adic group $U(n)$ should be $U(3)$. Because, Gelbart and Rogawski suggested the non-tempered ...

**25**

votes

**2**answers

959 views

### What are the possible motivic Galois groups over $\mathbb Q$?

Let $E$ be a motive over $\mathbb Q$. (I should precise, that by a motive I mean
here a pure motive over $\mathbb Q$, with coefficients in $\mathbb Q$, that I see here as a conjectural object which ...

**7**

votes

**2**answers

483 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
...

**1**

vote

**0**answers

48 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 ...

**0**

votes

**0**answers

111 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

**0**answers

124 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 ...

**1**

vote

**0**answers

78 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 ...

**4**

votes

**1**answer

193 views

### Constructing a family of convergents from continued fractions formed by a set of prime partial quotients

For a given real number $x$, the continued fraction representation $x = [a_0; a_1, a_2, \cdots]$ where $(a_n)_{n \geq 0}$is defined by setting $x = \alpha_0$, then $a_i = \lfloor \alpha_i \rfloor$, ...

**9**

votes

**1**answer

1k views

### Convergence of the series $\sum_p p^{-s}$ ($p$ prime and $s>1$)

I know that $\sum_p p^{-s}$, $s>1$, converges. Now, I define $J(s) = \sum_p p^{-s}$. Are there any "well known" values for $J(2)$, $J(3)$, $J(4)$, etc? We all know that $\zeta(2)= \frac{\pi^2}{6}$, ...

**1**

vote

**2**answers

306 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

**1**answer

118 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

**1**answer

243 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 ...

**9**

votes

**2**answers

790 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
$$\sum_{i=0}^{n}\binom{n}{i}^3$$ ?

**4**

votes

**1**answer

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) ...

**6**

votes

**1**answer

143 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: ...

**3**

votes

**1**answer

254 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?

**16**

votes

**4**answers

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 ...

**27**

votes

**2**answers

2k views

### What is known about the sum x^{n^2}/n?

It follows from a general theorem of Honda that the formal group with the logarithm
$$
x+x^{2^s}/2+x^{3^s}/3+x^{4^s}/4+\cdots
$$
has integer coefficients. I became interested in it because its ...

**0**

votes

**0**answers

65 views

### Regular Magic Squares [closed]

I am currently studying magic squares and ran into a bit of trouble. Basically it deals with something I came up with, a regular square. Below are the conditions of a regular square.
We can say that ...

**3**

votes

**1**answer

690 views

### Efficient algorithms to determine the roots of: $p(x) = r^x $ in the finite field $GF(q)$, where $r$ a primitive root of the field

I need to make sure that no efficient (i.e., polynomial time) algorithm exists for the following problem:
Exponentiating Polynomial Root Problem (EPRP)
Let $p(x)$ be a polynomial with $\deg(p) \geq ...

**8**

votes

**2**answers

685 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 ...

**6**

votes

**1**answer

397 views

### Which L-functions are not “Langlands-Shahidi L-functions”?

The Langlands-Shahidi method, among other things, obtains certain L-functions from the constant term of Eisenstein series attached to so-called $(G,M)$ pairs, where $G$ is a reductive group, $M$ a ...

**2**

votes

**2**answers

404 views

### Number of solutions of a binary quadratic form.

Given a binary quadratic form with negative discriminant, such as $3x^{2} + y^{2} = k$, is there an efficient algorithm to compute a value $k$ (or all the values) for which the form has exactly $n$ ...

**16**

votes

**1**answer

713 views

### Primes that are sums of two squares with constraints on the squares

It is well known that there are infinitely many primes of the form $a^2+b^2$ (namely all primes congruent to $1$ modulo $4$). On the other hand, Euler raised the problem as to whether there are ...

**33**

votes

**4**answers

3k views

### Fermat's last theorem over larger fields

Fermat's last theorem implies that the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}$ is finite.
Is the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}^{\text{ab}}$ finite?
Here ...

**2**

votes

**2**answers

160 views

### Relations of eisenstein series with eta quotient

Theorem 1.67 On page 19 of Ken Ono's book The Web of Modularity says:
Every modular form on $SL_2(\mathbb{Z})$ may be expressed as a rational function in $\eta(z)$, $\eta(2z)$ and $\eta(4z)$.
The ...

**4**

votes

**0**answers

350 views

### Is it possible to find explicit formula for the product $\prod_{\substack{d\mid n \\ d>1}} (1-\mu(d)/\varphi(d))^{\varphi(d)}$? [closed]

I am trying to calculate the following product
$$
\prod_{\substack{d\mid n \\ d>1}} \left( 1-\frac{\mu(d)}{\varphi(d)} \right)^{\varphi(d)}
$$
where the functions $\varphi$ and $\mu$ are Euler's ...