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

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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
1answer
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
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1answer
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
2answers
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
2answers
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
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1answer
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
1answer
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
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0answers
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
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1answer
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
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0answers
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
1answer
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
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, $\frac{1}{x+1} + \frac{1}{x(x+1)} = \frac{1}{x}$
6
votes
4answers
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
3answers
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
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1answer
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
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0answers
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
1answer
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
2answers
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
0answers
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
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0answers
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
2answers
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 ...
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0answers
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
0answers
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
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1answer
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
2answers
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
2answers
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
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0answers
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
0answers
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 ...
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0answers
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
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0answers
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
1answer
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
1answer
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
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2answers
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
1answer
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
1answer
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
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2answers
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
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) ...
6
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1answer
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
1answer
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
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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 ...
27
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2answers
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
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0answers
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
1answer
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
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2answers
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
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1answer
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
2answers
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
1answer
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
4answers
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
2answers
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
0answers
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 ...