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

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1answer
133 views

Conjugate surfaces: informations about the orbits

Consider a complex algebraic variety $X$ (namely a $\mathbb C$-scheme, of finite type, geometrically integral and separated); if $\sigma\in\textrm{aut}(\mathbb C)$, then is well defined the complex ...
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0answers
57 views

Rank 2^n quadratic form with non trivial invariant e_n

I advise you to have a look this question of mine first Rank four quadratic Form with non trivial discriminant in I(k) From quadratic form theory its well known that for a field $k$ and the ...
7
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1answer
239 views

Cases where the number field case and the function field (with positive characteristic) are different

In number theory there is often an analogue between statements which holds over a number field (that is, a finite field extension $K/\mathbb{Q}$) and function fields (that is, finite extensions of the ...
5
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1answer
226 views

Motivic L-function vs motivic zeta function

Let $M$ be a pure motive over a field $k$. Roughly speaking, the L-function of $M$ is the product over all primes $p$ of $$L_p(M,s)=\det(I-Fr_p|_{M_\ell^I} N(p)^{-s})^{-1}$$ where $Fr_p$ is a ...
12
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2answers
396 views

Asymptotics for algebraic numbers of height less than one

The question. Is an asymptotic equivalent known or conjectured for the number $N(d)$ of $\alpha \in \bar{\mathbb{Q}}$ with $h(\alpha) < 1$ and $[\mathbb{Q}(\alpha):\mathbb{Q}] \leq d$? The rather ...
5
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0answers
204 views

Are all complex zeros of $\frac{\Gamma(s)}{z}Li_s(z) \, \pm \, \frac{\Gamma(1-s)}{z}Li_{1-s}(z)$ on the critical line for all $z \lt 1$?

Numerical evidence suggests that all complex zeros residing in the critical strip $0 < \Re(s) < 1$ of: $$\frac{\Gamma(s)}{z}Li_s(z) \, \pm \, \frac{\Gamma(1-s)}{z}Li_{1-s}(z)$$ are on the ...
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0answers
132 views

A question on linear forms

Let $K/\mathbb{Q}$ be a fixed number field of degree $n$, and let $\sigma_1, \cdots, \sigma_n$ denote the distinct embeddings of $K$ into $\mathbb{C}$. Suppose that $\alpha_1, \cdots, \alpha_{r}$ are ...
2
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2answers
202 views

Arbitrarily large number of representations

For a positive integer $k$ let $\gamma_k(n)$ be the number of representations of $n$ as a sum of strictly increasing perfect $k^{\text{th}}$ powers. For example $\gamma_k(2)=0$ for any $k$. Now is the ...
11
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1answer
246 views

A strengthening of base 2 Fermat pseudoprime

If $n$ is a prime then for all $k$ with $1 \le k \le [n/2]$, $k$ divides ${n-1 \choose 2k-1}$ because of the identity ${n-1 \choose 2k-1} \frac{n}{k}=2{n \choose 2k}$. My question is whether an ...
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0answers
116 views

a question on sum of Gaussian binomial coefficients

I was trying to calculate something and at some point I get the following sum: \begin{equation} \sum_{t=0,t \text{ even}}^{s}{s+3n \brack s-t}\sum_{i = 0}^{t/2}q^{2i^2}{t/2+2n-i \brack t/2-i}{n ...
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1answer
430 views

Subset of the integers with certain properties

How would one find the maximal $n$ such that there exists an $n$-subset $S$ of $\mathbb{Z}^+$ such that $\forall A\subseteq S, \sum_{a\in A}a$ is either a perfect square or a perfect cube, or can one ...
15
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418 views

The lonely molecule

Suppose $n$ air molecules (infinitesimal points) are bouncing around in a unit $d$-dimensional cube, with perfectly elastic wall collisions. Let $k=n^{\frac{1}{d}}$. For example, in 3D, $d=3$, with ...
2
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1answer
243 views

Does the set of automorphisms of a cyclic group exhibit some sense of randomness?

I prefer to proceed with a concrete example if I may. I appreciate that the answer might well be better explained with group theory, geometry and/or notions from probability theory, which I welcome. ...
10
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1answer
398 views

Newton polygons of modular polynomials

This is pretty much straightforward curiosity. Is there anything known about Newton polygons of classical modular polynomials (polynomial relations between $j(\tau)$ and $j(n\tau)$)? I understand that ...
3
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1answer
279 views

Lower bound for a prime gap occurring infinitely often

In his striking paper of may 2013, Zhang showed the existence of an even integer $g\lt 70,000,000$ such that $g$ is a prime gap occurring infinitely often. What is the best unconditional lower bound ...
11
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1answer
608 views

Tight prime bounds

This is a cross-post of this question on MSE. I would not usually do this, but have decided to in this case since it has had no responses having been posted as a bounty question. I did not delete the ...
2
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1answer
221 views

Integer points on $y^2=x^2-x^3+x^4$

Does the Diophantine equation $y^2=x^2-x^3+x^4$ have solutions other than $x=1,y=1$? Interestingly, the Diophantine equation $y^2=x^2-x^3+x^5$ has such solutions: $x=3,y=15$, $x=5,y=55$, ...
11
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1answer
263 views

Borel-Serre compactification of $\mathbb{H}^3 / SL_2(\mathcal{O}_K)$

Let $\mathbb{H}^3$ be the three-dimensional hyperbolic space. Let $K$ be an imaginary quadratic number field and $\mathcal{O}_K$ its ring of integers. Then $SL_2(\mathcal{O}_K)$ acts on $\mathbb{H}^3$ ...
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0answers
96 views

Modular form, number of divisors [duplicate]

The Fourier expansion of Eisenstein series $E_k$ $(k \ge 4)$, which are modular forms, as well as the quasimodular $E_2$, involves powers-of-divisors $\sigma_{k-1}(n) = \sum_{d|n} d^{k-1}$. Is there ...
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0answers
208 views

Factors of the polynomial $X^n-a$

I am interested in the polynomial $X^n-a$ in $\mathbb{Q}[X]$, for some $a\in \mathbb{Q}^*$, and would like to know the irreducible factors of it. Is there something in the literature which gives a ...
14
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3answers
935 views

Not-lonely runners

The lonely runner conjecture has several formulations. They all involve a number $n$ runners running on a circular track, each with a different speeds, and the conjecture is that each runner is ...
11
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1answer
262 views

Hecke-module structure implicit in definition of automorphic forms in Borel-Jacquet's Corvallis article

Let $G$ be a connected reductive group over a number field $F$, $G_\infty=\prod_{v\mid\infty} G(F_v)$, $\mathbf{A}$ the adèles of $F$, $\mathbf{A}_f$ the finite adèles of $F$. Fix a maximal compact ...
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0answers
414 views

function field analogy and global/absolute geometry

The "function field analogy" seems to be a topic that is considerably bigger than any one existing writeup conveys. There are several old question on MO and and MathSE that ask for details. One of the ...
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0answers
51 views

How to test whether a distribution follows a power law? [closed]

I have the data of how many users post how many questions. For example, [UserCount, QuestionCount] [2, 100] [9, 10] [3, 80] ... ... it means each of the 2 users posts 100 questions, each of the 9 ...
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0answers
94 views

Interpretation of the Gross-Zagier formula for Green function

I am reading the paper of Gross and Zagier on heights of Heegner points and would like to check with the experts whether the following (meta?)mathematical statement makes sense. In the calculation of ...
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2answers
112 views

Asymptotics and error terms for an arithmetic function built upon $\omega$ and $\Omega$ functions

For any real number $x$, let's define $Om_{k}(x)$ as the number of positive integers $m$ below $x$ such that $\Omega(m)-\omega(m)=k$, where $\omega(n)$ is the number of distinct primes dividing $n$, ...
6
votes
3answers
489 views

Constructing quintic number fields with certain splitting behaviour

I am looking for number fields $K$ which satisfy the following properties: $[K:\mathbb{Q}]=5$. The Galois closure of $K$ has Galois group $S_5$. For each prime $p$ which ramifies in $K$, there ...
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3answers
1k views

Weil's Riemann Hypothesis for dummies?

Weil's Riemann Hypothesis is a deep result that I don't fully understand, but it has understandable corollaries which interest me. For example: (a) For any projective curve $X$ satisfying certain ...
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421 views

A letter from J. P. Serre

Which is the letter where J. P. Serre present "Analogues Kählériens de certaines conjectures de Weil" to Weil?
4
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2answers
280 views

Time-line until the publicaton of Weil of “Numbers of solutions of equations in finite fields”

In "On the history of the Weil Conjectures" Dieudonné says: "Appropriately enough, the story, as with so many problems in number theory, begins with Gauss...". C. F. Gauss, Disquisitiones ...
11
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0answers
338 views

Erdos multiplication problem revisited

The well-known problem is acquiring a cardinality of the set of distinct numbers in the multiplication table n x m. The very problem has been discussed in-depth and, as such, I require no further ...
3
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3answers
290 views

How to find an integer set, s.t. the sums of at most 3 elements are all distinct?

How to find a set $A \subset \mathbb{N}$ such that any sum of at most three Elements $a_i \in A$ is different if at least one element in the sum is different. Example with $|A|=3$: Out of the set $A ...
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2answers
209 views

overlap quadratic residues

Let $p$ be a prime number of form $4k+1$ and $M$ is its quadratic residue set. Let $M_i=\{i+x|\forall x\in M\}$ $\forall 0<i<p$. Does there exist a positive constant $\varepsilon$ such that ...
1
vote
1answer
116 views

General criterion to find a Z-basis in a fixed generating subset

Let $V=\mathbf{Z}^N$ be a free $\mathbf{Z}$-module of rank $N$. Let $S\subseteq V$ be a fixed finite subset. Consider the submodule $M:=\langle S\rangle\leq V$ generated by $S$. We know form the ...
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2answers
131 views

Proof of equidistribution theorem for exponential coefficients

Can anyone provide a proof of the equidistribution theorem using Weyl's criterion for the case of $c*a \,\,\, \text{(mod 1)}$ where $c=2^n: \,\,\, \forall n \in N_0$ for irrational algebraic $a$? The ...
5
votes
3answers
368 views

Bounds on horizontal minima of the Riemann zeta function

It is known that $\zeta(s)$ has an infinity of zeros in the strip $0<\sigma<1$ and that those zeros become closer together as $t\rightarrow\infty$. More precisely, Littlewood showed that there ...
4
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0answers
115 views

On Skinner and Urban's $p$-adic L functions

We know that if $\pi$ is an automorphic representation of $GL_2$ over a number field, the most important point for its $L$-function $L(s,\pi)$ is $s=\frac{1}{2}$. More specifically, in Skiner and ...
21
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2answers
1k views

For $x,y\ge 2$ does $x^4+x^2y^2+y^4$ ever divide $x^4y^4+x^2y^2+1$?

For a problem in combinatorics, it comes down to knowing whether there exist integers $x,y\ge 2$ such that $$ x^4+x^2y^2+y^4\mid x^4y^4+x^2y^2+1. $$ Note that ...
4
votes
1answer
198 views

How frequently is 3 a cubic residue mod primes in an arithmetic progression?

Suppose $(a,3q)=1$ and $a\equiv 1\pmod 3$. Are there infinitely many primes $p\equiv a\pmod {3q}$ such that $3$ is a cubic nonresidue modulo $p$? Or, an equivalent formulation using quadratic forms: ...
10
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2answers
342 views

Lebesgue measure of a set of irrational numbers

Let $I_{\lambda},$ $\lambda>0$ be a subset of all irrational numbers $\rho=[a_{1},a_{2},...,a_{n},...]\in(0,1)$ such that $a_{n}\leq \text{const}\cdot n^{\lambda}.$ Here, ...
15
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0answers
238 views

Are there any integers which can't be written as a sum of two fourth powers minus a cube?

To be precise, I am asking: Does there exist an integer $k$ such that there do not exist (possibly negative) integers $x,y,z$ satisfying $x^4+y^4=z^3+k$? Heuristically the answer must be yes, in ...
6
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1answer
497 views

Is this weak asymptotic Goldbach's conjecture open?

Let $\tau(x)$ be the number of even numbers $2<2n<x$ which can't be written as a sum of two primes. Goldbach's conjecture: $\tau(x) = 0$ Asymptotic Goldbach's conjecture: $\tau(x) = O(1) $ ...
3
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1answer
149 views

If there are two primes at an even distance $k$ is there another disjoint pair of primes also at distance $k$?

Suppose $p,q$ are two primes at even distance $k$. Must there necessarily exist a different pair $p',q'$ composed of entirely different numbers such that $p'$ and $q'$ are also at distance $k$? Edit: ...
5
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2answers
146 views

Covolume of the row span of a matrix and of the kernel of a matrix

Let $L$ be a $k$-dimensional lattice in $\mathbb{R}^n$. The covolume $\hbox{CoVol}(L)$ of $L$ is the $k$-dimensional volume of a fundamental domain for $L$, i.e., the volume of the parallelopiped ...
4
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0answers
82 views

minimal conductors among elliptic curves with a fixed CM type

Let $K$ be a quadratic imaginary field. To simplify my life, let us assume that $K$ has class number one. Consider the following infinite set: $S_1:=$ $\{$ $E\subseteq\mathbf{P}^2(\mathbf{C})$ is an ...
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2answers
142 views

On the conductor of the Groessencharacter of a CM elliptic curve

Let $K$ be a quadratic imaginary field. Let $L$ be a number field which contains $K$ and let $E/L$ be an elliptic curve defined over $L$ with complex multiplication by $K$, i.e. such that ...
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0answers
301 views

Conjectures on fractions where each digit appears once in numerator and denominator

This is a highly redacted version of a question that was asked before. Please see Criteria of considering relevance of the question to the domain of research topics for details. Some numerical ...
0
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2answers
229 views

Goldbach-type problem: the valuation of irreducible elements of a subgroup of $\mathbb{Q}_{> 0}$

Let $\phi : \mathbb{Q}_{>0} \to \mathbb{Z}$ be the group morphism defined by $\phi(p) = p$ for $p$ a prime number. It follows that $\phi(\prod_i p_i^{n_i}) = \sum_i n_i p_i$, with $p_i$ a prime ...
2
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0answers
230 views

Fractional Part Problem

Suppose we have coprime integers $a$ and $b$ with $p \mid a$ but $p^2 \nmid a$ for some prime $p\geq 5$. We can write $a=px$ and $b=pr+\hat{b}$. Suppose also that $a$ and $\hat{b}$ are coprime; that ...
11
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0answers
180 views

What is the lowest-weight non-cyclotomic Galois representation in $\overline{\mathcal M}_{g,n}$?

I want to know about low-weight Galois representations in $H^i(\overline{\mathcal M}_{g,n}, \overline{\mathbb Q}_\ell)$ that aren't cyclotomic. This should be equivalent to finding $p,q$ such that ...