**4**

votes

**1**answer

182 views

### Quotient of principal congruence subgroups

This is a direct follow-up to this question. What is the quotient $\Gamma(2)/\Gamma(2^n)?$ (the principal congruence subgroups are in $SL(2, \mathbb{Z}).$ It is a 2-group, but what else?

**5**

votes

**4**answers

672 views

### Consequences of the Inverse Galois Problem

Are there any papers written about the consequences of the Inverse Galois Problem in case it is proved to be true or false?
We know a lot of things that would be true if the Riemann Hypothesis holds. ...

**5**

votes

**0**answers

105 views

### How to assess the influence of a specific term in this telescoping series for $\zeta(s)$?

I like to expand on this (unanswered) MSE question.
Take the following, nicely symmetrical, telescoping series for $\zeta(s)$:
$$\displaystyle \zeta(s) = \frac{1}{2\,(s-1)} \left(1+\sum ...

**1**

vote

**0**answers

59 views

### Characterization of two sets

I posted this question on math.SE 10 days ago and had no answer, comment or vote. If an answer is not available, I could really use a reference point as well.
For the sake of completeness, I am ...

**0**

votes

**0**answers

52 views

### Is there any criterion when an interger solution of the equation $ax^2-by^2=c$ $(a,b,c\in\mathbb{N})$ exists? [migrated]

As title. In particular I'd like to know if it suffices that it's solvable in $\mathbb Z_p $ for $p$ being any prime number.

**0**

votes

**0**answers

132 views

### For which pairs of distinct positive primes $p$ and $q$, the integral closure of $\mathbb{Z}$ in $\mathbb{Q}[\sqrt{pq}]$ is a UFD?

For which pairs of distinct positive primes $p$ and $q$, the integral closure of $\mathbb{Z}$ in $\mathbb{Q}[\sqrt{pq}]$ is a UFD? I've proved that neither $p$ nor $q$ can be congruent to $1$ modulo ...

**2**

votes

**0**answers

99 views

### algorithm to find a new point of small height in a number field extension

By the height of an algebraic number $\alpha$, I mean the absolute, logarithmic (additive) Weil height $h(\alpha)$; e.g. $h(2^{1/n}) = (\log 2)/n.$
If $K$ is a number field, let $\delta(K)$ denote ...

**2**

votes

**1**answer

102 views

### Enumerating positive fractions (reference missing)

I remember that the recursion
$r(0)=0, \ \
r(n+1)=\frac{1}{2 [r(n)]+1-r(n)}$
produces a sequence of rational values $ 0 \mapsto 1 \mapsto 1/2 \mapsto 2 \mapsto 1/3 \mapsto ... $ which exausts the ...

**5**

votes

**1**answer

189 views

### Complete L-function and FE of Rankin-Selberg on GL(2)?

Let $f$ be a Maass cusp form of $\Gamma_0(N)$ on the upper half plane with character $\chi$ mod $N$ and eigenvalue $1/4+\mu^2$.
What is the complete $L$-function of the Rankin-Selberg product ...

**2**

votes

**1**answer

131 views

### A question on polynomial heights

For a given polynomial $f(x_1, \cdots, x_n) \in \mathbb{Z}[x_1, \cdots, x_n]$, define the height of $f$ as $H(f)$ as the maximum absolute value among its coefficients. We can also define the log ...

**0**

votes

**1**answer

195 views

### Find all possible rational values of the parameter of a parametric cubic such that it is reducible

Description: Given the following parametric cubic polynomials ${E}^{3}
- 15\, {\beta}_{\pm}\, {E}^{2} - 3 \left({71\, {\beta}_{\pm}^{2} + 352\, {\beta}_{\mp}}\right) E
+ 135\, {\beta}_{\pm} ...

**24**

votes

**2**answers

721 views

### Arranging numbers from $1$ to $n$ such that the sum of every two adjacent numbers is a perfect power

I've known that one can arrange all the numbers from $1$ to $\color{red}{15}$ in a row such that the sum of every two adjacent numbers is a perfect square.
$$8,1,15,10,6,3,13,12,4,5,11,14,2,7,9$$
...

**12**

votes

**1**answer

347 views

### Computer software for periods

Kontsevich and Zagier define a period as an integral of a rational function (over $\mathbb{Q}$) defined on a $\mathbb{Q}$-semialgebraic set. They conjecture that if two periods are equal, then the ...

**1**

vote

**0**answers

63 views

### Local duality for abelian varieties

Let $A$ be an abelian variety over a p-adic field $K$. Let $I$ be the inertia group of $K$. There is a Yoneda pairing $$H^n(\hat{\mathbb{Z}},A^I) \times Ext^{2-n}_{\hat{\mathbb{Z}}}(A^I,\mathbb{Z}) ...

**4**

votes

**3**answers

394 views

### $2^n$-1 consisting only of small factors

I've checked the factorization of $2^N - 1$ up through N = 120 for the largest prime factor, and it looks like the largest value of N where $2^N-1$ has a largest prime factor under 2500 is N = 60 ...

**1**

vote

**0**answers

77 views

### Sparse sets of numbers with the Goldbach property

Let's say that a subset $A$ of $\mathbb{N}$ has the Goldbach property if every even number $\geq4$ is the sum of two numbers of $A$.
Are there any results and examples of low density sets with these ...

**2**

votes

**1**answer

135 views

### integral basis for the Lie algebra of the Neron model of an abelian variety

Let $A$ be an abelian variety over a number field $K$. Let $\mathcal{A}$ be the Neron model of $A$ over $O_K$. Let $\Omega_{\mathcal{A}/O_K}$ be the sheaf of invariant differential forms on ...

**10**

votes

**0**answers

187 views

### Are there integral representations of the Mertens constant?

It is well-known that the Euler constant $\gamma=\lim\limits_{n\to \infty}\biggl( \sum\limits_{k\le n}\dfrac{1}{k}-\ln{n}\biggr ) $ has a bunch of integral representations, e.g. ...

**4**

votes

**1**answer

208 views

### Permutation polynomials mod $p$ of the form $(x+1)^n-x^n$

Among some permutation polynomials I've been studying, $f(x)=(x+1)^n-x^n$ is one of the polynomials that I cannot grasp.
Question : For a given odd prime $p$, how can we find every positive ...

**0**

votes

**0**answers

74 views

### Polynomial congruences with respect to a large prime (power)

Suppose that $f(x) \in \mathbb{Z}[x]$ is a polynomial of degree $d$ such that for all primes $p$, there exists $x_0 \in \mathbb{N}$ such that $p^2 \nmid f(x_0)$. Further, suppose that $f$ is ...

**5**

votes

**1**answer

270 views

### A set of integers whose factorial can be written as a product of two factorials

I am trying to collect informations concerning the set
$$\mathcal{A}=\left\{n\in\mathbb{N} \mid (\exists k,l\in\{2,3,\dots,n-2\})(n!=k!l!)\right\}.$$
It seems not much is known about the set ...

**1**

vote

**2**answers

520 views

### Numbers greater than Skewes's whose existence can be found in number theoretic proofs

Skewes has proved (without assuming RH) that $\pi(x)<Li(x)$ is violated below $e^{e^{e^{e^{7.705}}}}$ which is clearly a very large number.I was wondering if somewhere else some greater number than ...

**11**

votes

**0**answers

219 views

### Most discriminants are almost squarefree

Write, for $f(x) = x^d + a_2 x^{d-2} + \cdots + a_d\in \mathbb{Z}[x]$, $H(f) := \max(|a_i|^{\frac{1}{i}})$.
Does anyone know of a reference that would allow me to show that the proportion of $f$ with ...

**-3**

votes

**1**answer

263 views

### Asymptotic formula for $\prod_{p\leq x} (1-p^{-1})$ [closed]

Does there exists a good asymptotic formula for
$$A(x) := \prod_{p\leq x}(1-\frac 1p).$$
By using a heuristic argument one can guess:
$$A(x) \sim \frac{1}{2\,\mathrm{ln}(x)}.$$
Here is the ...

**8**

votes

**2**answers

561 views

### Sum of consecutive cubes

I'm investigating when the sum of $n$ consecutive cubes equals a cube, i.e., for which $n$ does
$$\sum_{i=0}^{n-1} (k+i)^3 = k^3 + (k+1)^3 + \cdots + (k+n-1)^3 = Y^3 $$
have nontrivial solutions ...

**7**

votes

**2**answers

349 views

### Averages over integer points of the sphere

A paper of William Duke proves 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 $x$, $y$ and $z$ ...

**3**

votes

**1**answer

196 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 ...

**16**

votes

**2**answers

405 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

**0**answers

118 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

**0**answers

97 views

### dimension of a scheme and degree of an L-function [closed]

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

**0**answers

154 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

**0**answers

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?

**6**

votes

**0**answers

124 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

**0**answers

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

**0**answers

123 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 ...

**3**

votes

**0**answers

163 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

**0**answers

134 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

**0**answers

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.

**9**

votes

**1**answer

402 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

**1**answer

423 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

**1**answer

428 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

**0**answers

58 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

**0**answers

121 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 ...

**9**

votes

**0**answers

301 views

### Can an abelian variety/Q have no non-trivial 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

**1**answer

172 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

**2**answers

551 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

**1**answer

286 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

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

**1**

vote

**0**answers

110 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

95 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 ...