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

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1
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0answers
42 views

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

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 ...
-4
votes
0answers
36 views

Which of the following conversion is wrong in the number system? [on hold]

Here is a diagram. which of the 2 boxes' conversion is wrong and please provide me a reason for it. I am doing the same conversion in 2 different procedure. where am i making the mistake.Please help ...
1
vote
0answers
62 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 ...
1
vote
2answers
75 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 $n$, ...
4
votes
2answers
287 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 ...
0
votes
0answers
35 views

Suitable algorithm for selecting /matching a set of memory [on hold]

I am looking for a standard algorithm that addresses the following problem. Does any such exist? if not, is there any suitable approach for this problem. I have a set of N memory locations available. ...
6
votes
2answers
945 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 ...
0
votes
0answers
390 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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1answer
171 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 ...
10
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0answers
284 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 ...
-1
votes
0answers
74 views

Help with extension of f (mentioned below) to f: Zp -> Zp ,continuous function [on hold]

This is a (different version to) question from Serre 'A Course in Arithmetic'.Let p be an odd prime number. $\forall n\geq 1$ (n positive integer), $f$ is defined by: $$f(n)=(-1)^n\prod_{1\le k\le n ...
2
votes
2answers
236 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 ...
1
vote
2answers
200 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
108 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 ...
-4
votes
0answers
104 views

What is the use of arithmetic groups? [on hold]

I want to ask a question that what is the relation between arithmetic group and number theory? We take a lot efforts to prove some kinds of lattics are arithmetic, do we get some bonus from the ...
0
votes
2answers
123 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
1answer
151 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
votes
0answers
102 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 ...
18
votes
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
176 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
votes
2answers
323 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, ...
14
votes
0answers
174 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 ...
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votes
0answers
46 views

ray class field [closed]

Many texts give definition of ray class fields without actually computing them for few base fields by way of examples. I want to explicitly see how the ray class field of K=Q(i) with modulus 3 is ...
6
votes
1answer
479 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) $ ...
-2
votes
0answers
101 views

A Very Simple Question of Number Theory [closed]

$n$ is a positive integer. Express $x$ in terms of $n$ where $x$ is the number of decimal digits of $2^{16 n}$. n = 1, 2^16 = 65536, x = 5 n = 2, 2^32 = 4294967296, x = 10 n = 3, 2^48 = ...
3
votes
1answer
143 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
votes
2answers
134 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
votes
0answers
72 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 ...
1
vote
2answers
130 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 ...
-2
votes
0answers
74 views

2-torsion points on Frey-Hellegouarch curve. [closed]

Let us consider the so-called Frey-Hellegouarch curve $E: Y^2 = X(X-a^l)(X+b^l)$. Q: What are two independent ${\Bbb Q}$-rational points of order $2$ on $E$ ?
0
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0answers
276 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
votes
0answers
18 views

Integer Solutions To Linear Equation [migrated]

$$a*q_1+b*q_2=c$$ $$a*q_3+b*q_4=f$$ $q_1, q_2, q_3, q_4$ rational numbers, $c,f$ integer Given $q_1, q_2$ can you construct all solutions $(a,b)$ where $c,f$ is intenger I made an edit since the ...
0
votes
2answers
218 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
votes
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
votes
0answers
164 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 ...
-1
votes
0answers
93 views

evaluate $\prod_{k=0}^{\lfloor \log_{2}(n+1) \rfloor - 1}\frac{n-(2^{k}-1)}{2^k}$? [closed]

$$\prod_{k=0}^{\lfloor \log_{2}(n+1) \rfloor - 1}\frac{n-(2^{k}-1)}{2^k}$$ I think it might be okay also to ignore the floor function business and merely assume that $n$ is such that $\log_{2}(n+1)$ ...
5
votes
1answer
265 views

Connection of Galois representation and arithmetic geometry

This is might be a dumb question. There are lots of Galois representations which arise naturally from geometric objects, for example, Galois representations attached to elliptic curves. I know that ...
3
votes
0answers
222 views

Independent units in pure number fields $\mathbb{Q}(\sqrt[p]{t})$

Theorem: In this paper of Frei and Levesque, they correct the proof of a result of Halter-Koch and Stender: Define the real pure algebraic number field $\mathbb{K}=\mathbb{Q}(\omega_n)$ for ...
4
votes
0answers
82 views

Density of p-ordinary modular forms

Fix an odd prime $p$. For concreteness, let $N$ be coprime to $p$, and let $2 \leq k \leq p$. Let $S^+(N,k)$ be the newforms in $S_k(\Gamma_1(N))$. Let $f = \sum a_n q^n \in S^+(N,k)$. We say that ...
3
votes
0answers
122 views

Algorithm to compute a common denominator of a finite set of rational numbers

Let $x_1,\dots,x_n \in \mathbb{Q} \cap (0,1)$ be fixed, but unknown, and assume that we know a number $K \in \mathbb{N}$ such that there is a number $N \in \{1,\dots,K\}$ such that $x_1 N,\dots,x_n N ...
0
votes
0answers
32 views

$B_k[1]$ sets with smallest possible $m = max B_k[1]$ for given $k$ and $n = |B_k[1]|$ elements

Sidon sets are sets $A \subset \mathbb{N}$ such that for all $a_j,b_j \in A$ holds $$a_1+a_2=b_1+b_2 \iff \{a_1,a_2\}=\{b_1,b_2\}$$ Thus if you know the sum of two elements, you know which elements ...
15
votes
0answers
307 views

Spencer's “six standard deviations” theorem - better constants?

This question is about Joel Spencer's famous "six standard deviations" theorem. The theorem says that when $$ L_i(x_1,\dots,x_n) = a_{i1} x_1 + \dots + a_{in} x_n, \quad 1 \leq i \leq n, $$ are $n$ ...
8
votes
2answers
356 views

Incomplete Kloosterman sum

I am interested in an upper bound on the following incomplete Kloosterman sum $$ \sum_{\substack{x=1 \\ x+_{_{\bf Z}}x^{-1}>p}}^{p-1}e\left(\frac{x+x^{-1}}{p}\right).$$ Using the Weil's bound it ...
2
votes
1answer
117 views

Regarding the set up of a geometry of numbers lemma

I have a question related to geometry of numbers, which although seems quite basic, I was rather confused by it so I decided to ask here. Let $\Lambda$ be a lattice in $\mathbb{R}^n$. Let $R_1, ..., ...
6
votes
0answers
126 views

Power series defined by Witt vectors / Teichmüller representatives of p-adics

Let $K$ be $\mathbb{Q}_p$ for some prime $p$ (or more generally an unramified extension $W(\mathbb{F}_q)$ of $\mathbb{Q}_p$). If $\xi \in K$, we can write it in a unique way in the form $\sum a_i ...
1
vote
0answers
151 views

Number of Orbits of symmetric group acting on $(\mathbb{Z}/n)^{l}$ [migrated]

I have encountered a problem that I suspect has been thoroughly studied but I have not been able to find references. Can anyone point me to a published reference dealing with this or a closely related ...
2
votes
0answers
89 views

Bloch Kato Exponential as formal lie group exponential

Let $K$ be a $p$-adic field and $V$ a $p$-adic representation. In their paper on tamagawa numbers of motives, Bloch and Kato define an exponential map as the connecting homomorphism $$DR(V) ...
1
vote
0answers
124 views

Arguments for the second Hardy–Littlewood conjecture being false?

Assume that $x,y > 2$, and that $x<y$. Then the second Hardy–Littlewood conjecture states that $$\pi(x + y) - \pi(y) \leq \pi(x).$$ We can easily justify this heuristically, since $$ ...
2
votes
0answers
57 views

Asymptotic formula for restricted partition function

Let $p(n)$ be the partition function. Hardy and Ramanujan - and Uspensky, independently proved the asymptotic formula $$(1) \quad p(n) \sim \frac1{4\sqrt{3}} \frac{e^{c_0\sqrt{n}}}{n} \text{ as } n ...
17
votes
1answer
593 views

The conjecture of Montgomery and Soundararajan on primes in short intervals: Empirical inconsistencies?

Assume that $y/ \log x \rightarrow \infty$ and that $y/x \rightarrow 0$. Then, from a conjecture by Montgomery and Soundararajan, we expect the number of primes in the interval $[x,x+y]$ to be ...