Questions tagged [nt.number-theory]
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
15,905
questions
3
votes
0
answers
282
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Reference request: basics about modular curves
Where can I find a reference (with carefully written proofs) for basic facts about modular curves? Namely:
Congruence subgroups
The open modular curve $Y_\Gamma$ admits the structure of a Riemann ...
2
votes
2
answers
455
views
Is the exponential version of Catalan-Dickson conjecture true?
The aliquot sum function $s:\mathbb{N}\rightarrow \mathbb{N}$ assigns to any natural number $n$ the sum of its proper divisors. Perfect numbers are fixed points of this function. The open conjecture ...
9
votes
0
answers
210
views
Surjectivity of reduction for Hilbert modular forms
Fix a totally real field $K$, a level $\mathfrak{n}$, a (parallel) weight $k\geq 2$ and a primitive ray class character $\chi$ modulo $\mathfrak{n}$.
Then one can form the space $S_k(\mathfrak{n},\...
3
votes
2
answers
613
views
Number of ways to write an integer as a sum of squares modulo $k$
Given a natural number $n$ and an element $k \in \mathbb{Z}_n$, how many solutions are there in $\mathbb{Z}_n$ to the equation $x^2+y^2 =k$? That is, I'm wondering whether there is a mod-$n$ version ...
9
votes
1
answer
316
views
A weak form of the Erdős-Turán conjecture
This question is motivated by the answer of Gowers to the question Erdos Conjecture on arithmetic progressions.
Question. (1)-Suppose $A \subset \mathbb{N}$ is such that
Lim$_n$ $log(n) \cdot |A \...
5
votes
1
answer
178
views
Is the function field of every congruence modular curve generated by $j,j\circ g$ for some $g\in\text{GL}_2(\mathbb{Q})^+$?
So I've heard in passing that for any congruence modular curve $X$ (over $\mathbb{C}$), there is a $g\in\text{GL}_2(\mathbb{Q})^+$ such that $X$ is birational to a plane curve in $\mathbb{C}^2$ given ...
9
votes
2
answers
374
views
How long iterations of $x \to (p \!\!\! \mod \!\! x)$ can be? [duplicate]
Suppose that $p$ is a prime number and $x_1$ is an integer in range $[1, p - 1]$. If $x_k \not = 1$,
define $x_{k + 1} := p \!\!\! \mod \!\! x_k$. Clearly, $x_{k + 1} < x_k$ and $x_{k + 1} > 0$
...
3
votes
0
answers
105
views
When does a number field have $p$-rank greater than $n$?
Consider $F/\mathbb{Q}$, a number field. Let $S$ be a finite set of primes of $F$ containing the Archimedean primes. Let $n$ be any natural number and $L_n$ be a finite extension of $F$ such that $\...
0
votes
0
answers
124
views
Summation formula for twisted L-function
Does any expert here know something about the summation formula of the Voronoi type for the sum $$\sum_{n\le X} a_{f}(n)\chi(n) e\left(\frac{an}{c}\right)?$$
Here $f$ is a newform of level $N$, $\chi$...
0
votes
1
answer
202
views
How can you find a divisor of the numerator of 1-1/2+1/3-1/4-...+1/47
How can you find a divisor of the numerator of the sum:
1-1/2+1/3-1/4-...+1/47
there are alternatives given such as 59, 61, 67, 71 and 79
using a calculator you get the answer is 71 but how can you ...
1
vote
0
answers
223
views
On the diophantine equations $x^n+n=y^m$ and $x^n-n=y^m$
Here I ask a question concerning the diophantine equations
$$x^n+n=y^m \quad \ \text{with}\ m,n,x,y\in\{2,3,\ldots\},\tag{1}$$
and
$$x^n-n=y^m \quad \ \text{with}\ m,n,x,y\in\{2,3,\ldots\}.\tag{2}$$
...
4
votes
1
answer
527
views
Reference for inequality for $\sum\limits_{d \mid n}\frac{\log d}{d}.$
Let $f(n)=\sum\limits_{d \mid n}\frac{\log d}{d}.$
It is not hard to see that $f(n)\ll(\log\log n)^2$. Is there any reference for this inequality?
EDT 1: A possible answer is Analysis of the ...
7
votes
1
answer
549
views
Basel problem and inversive geometry
An interesting solution of the Basel problem is given in https://www.tandfonline.com/doi/abs/10.1080/00029890.2018.1452473 (Basel Problem: A Solution Motivated by the Power of a Point, by Kapil R. ...
5
votes
2
answers
666
views
Surjectivity of norm map on subspaces of finite fields
It is basic that the norm map $N:\mathbf{F}_{q^n}^* \to \mathbf{F}_q^*$ is surjective for finite fields. In fact $N(x) = x^{(q^n-1)/(q-1)}$. How well does this simple fact extend to subspaces?
A ...
0
votes
1
answer
133
views
A question in a function field of positive characteristic $p$
Let $K$ be a function field of positive characteristic $p$.
Let $b$, $u$, $u'$ be in $K\setminus K^p$.
I would like to show that there are only finitely many natural numbers $r$ prime to $p$ such ...
7
votes
1
answer
613
views
$(2x^2+1)(2y^2+1)=4z^2+1$ has no positive integer solutions?
Equation
$$(2x^2+1)(2y^2+1)=4z^2+1$$
has no solutions in the positive integers. Its true?
6
votes
0
answers
174
views
Approximating a ray with an integer lattice point
Take $X$ uniform on the unit sphere in $\mathbb{R}^n.$ For $r>0$, take $S_r=\{x\in \mathbb{Z}^n: \sum_i x_i^2 \leq r^2\}.$
With $\|\cdot \|$ the 2-norm, what is the distribution (or at least the ...
6
votes
7
answers
2k
views
Insights from disproofs after counterexamples have been given
Some conjectures are disproved by a single counter-example and garner little or no further interest or study, such as (to my knowledge) Euler's conjecture in number theory that at least $n$ $n^{th}$ ...
6
votes
1
answer
636
views
Faltings theorem and number of singularities
The Faltings theorem states that the number of rationals over an algebraic curve is finite if the genus is greater than 1. The genus decreases by increasing the number of singularities. My question is ...
10
votes
1
answer
439
views
Spacing of fractions with prime denominator
Let $X\ge 2$ be large. Let $$A = \left\{\frac{a}{q}: 1\le a < q\le X,\ (a, q) = 1\right\},$$ and let $$B = \left\{\frac{a}{p}: 1\le a < p\le X,\ (a, p) = 1,\text{ and $p$ is prime}\right\}.$$ ...
9
votes
2
answers
892
views
Kuznetsov trace formula, orthogonality of Bessel functions
Sorry if this is a vague question. I remember from my younger days that
before proving his trace formula, Kuznetsov had a pretty result on
orthogonality of Bessel functions. The formulas that I am ...
17
votes
1
answer
454
views
Centraliser of an absolute Galois group
Let $K$ be a finite extension of $\mathbb{Q}_p$.
Is the centraliser of $\operatorname{Gal}(\overline{K}/K)$ in $\operatorname{Gal}(\overline{\mathbb{Q}_p} / \mathbb{Q}_p)$ trivial ?
If yes, how can I ...
5
votes
1
answer
288
views
Additive basis of order 2
Can we find $\alpha>1$ such that $u=(\lfloor n^\alpha\rfloor)_{n\geqslant0}$ is an additive basis of order $2$ (i.e. $\forall x\in\mathbb{N}, \exists(n,m)\in\mathbb{N}^2, x=u_n+u_m$) ?
Remark : ...
13
votes
0
answers
447
views
The trace formula over function fields
There are many examples in number theory where an "arithmetic" problem (i.e. for number fields) has an easier analogue for function fields over finite fields. This is also true for questions ...
30
votes
2
answers
2k
views
When does doubling the size of a set multiply the number of subsets by an integer?
For natural numbers $m, r$, consider the ratio of the number of subsets of size $m$ taken from a set of size $2(m+r)$ to the number of subsets of the same size taken from a set of size $m+r$:
$$R(m,r)...
9
votes
2
answers
324
views
Algebraic power series over $\mathbb{F}_2$ as roots of polynomials of special form
Let $F = \mathbb{F}_2$ be the field with two elements. I will denote the rings of polynomials and formal
power series over $F$ as $F[t]$ and $F[[t]]$ respectively. Suppose that $x \in F[[t]]$ is ...
1
vote
1
answer
150
views
Simultaneous rational approximation to transcendental and algebraically independent numbers
I’m interested in the problem of simultaneous rational approximation to $k\geq 2$ numbers $\alpha_1,…,\alpha_k$ in the generic case where the $\alpha_j$ are transcendental and algebraically ...
2
votes
1
answer
509
views
Points of infinite level modular curve
Let us consider the anticanonical adic tower of modular curves which is described in Peter Scholze's paper "On torsion in the cohomology of locally symmetric varieties", and let us call $\mathcal{X}_{\...
0
votes
1
answer
138
views
Solutions to linear equations from recurrence sequences with no repeated roots
Let $U=(u_n)_{n=0}^{\infty}\subseteq\mathbb{C}$ be a sequence enumerated by a linear homogeneous recurrence relation with constant coefficients, i.e., there is some $d\geq 1$ and $a_1,\ldots,a_d\in\...
4
votes
1
answer
411
views
Is there some numerical evidence that $ \pi(x+x^{1/e})-\pi(x)\geq 1 $ for any large enough $ x $?
Disclaimer : the following reasoning is a physicist's one and as such may not be suitable for this website. Still it may give rise to potentially interesting insights so I ask it anyway.
I stumbled ...
8
votes
2
answers
462
views
Which primes $\mathfrak{p} \in \mathbb{Z}[i]$ can be represented as $\mathfrak{p} = x^2 + 2y^2$?
Consider the quadratic form $x^2 + 2y^2$ over the ring $\mathbb{Z}[i]$. It is irreducible, so I wanted to know which primes could be represented by this quadratic form.
$$ \mathfrak{p} = x^2 + 2y^...
19
votes
3
answers
998
views
Points of elliptic curves over cyclotomic extensions
Let $E$ be an elliptic curve over $\mathbb Q$. Let's look at the group of points of this elliptic curve over $\mathbb Q(1^{1/\infty})$ which we get after adding all roots of unity to $\mathbb Q$. It ...
3
votes
0
answers
199
views
Cancellation in this exponential sum?
I would like to know whether it is possible to obtain cancellation in the sum
$$\sum_{p \leq X} e^{{2\pi iX}/{p}}$$
where $X$ is a real number that goes to $\infty$, and $p$ denotes a prime number.
4
votes
0
answers
98
views
Rank of binary matrix related to the number of positive squarefree integers less than $n$
I posted this question at the Mathematics SE, but received no response there so I am posting it here.
The following fact is stated in the comments-section of sequence A013928 in the OEIS.
Let $C$ ...
5
votes
0
answers
213
views
Isomorphism classes of lattices
Suppose we have a $4 \times 6$ matrix $A$ of rank $4$ whose entries are rational numbers. Define
$$
V = \{x \in \mathbb R^6 \mid A \cdot x = 0\}
$$
and
$$
\Lambda = \{x \in \mathbb Z^6 \mid A \cdot ...
10
votes
1
answer
258
views
The partial preorder on $\mathbb N$ generated by the finite axioms of choice
Let $\mathsf C_n$ denotes the statement:
for any family $\mathcal F$ of $n$-element sets there exists a choice function (i.e., a function $f:\mathcal F\to\bigcup\mathcal F$ such that $f(F)\in F$ for ...
2
votes
0
answers
124
views
Number of partitions of $\{1,2,\ldots,n\}$ whose blocks are arithmetic progressions of length $t$ or more
This question was inspired by the earlier question here,where no lower bound on arithmetic progression size was given. In particular, $t\geq 3,$ is assumed here.
The set $\{1,\ldots,n\}$ has $2^n$ ...
9
votes
1
answer
534
views
Are all totally ramified $\mathbb{Z}_p$-extensions of local fields come from (relative) Lubin-Tate formal groups?
The setup is as follows:
$k/\mathbb{Q}_p$ is a finite extension, $\mathfrak{p}$ is the maximal ideal of $\mathcal{O}_k$, $q=\#(\mathcal{O}_k/\mathfrak{p})$
$k'/k$ is a finite unramified extension of ...
5
votes
4
answers
760
views
Solutions to the Diophantine equation $x^2+3y^2+3z^2=n$
For a fixed positive integer $n$, the Diophantine equation
$$x^2 + y^2 + z^2 = n$$
was studied by Gauss in Disquisitiones Arithmeticae. As is known, this equation is intimately connected to the ...
1
vote
0
answers
181
views
Sieving beyond threshold
This is a follow-up to the question here: Sum of divisors below threshold. User "Lucia" gave an excellent answer there, and probably the question below is very closely related. Still, since I am not ...
0
votes
1
answer
265
views
The minimum of the maximum of a sequence of sinc functions
I apologise if this is trivial or well known to be impossible:
Can one find a finite set of integers $2\leq a_1<a_2<\ldots<a_m<\infty$
such that for the function defined as
$$
f_{a_1,\...
1
vote
1
answer
496
views
Coefficients $U_m(n,k)$ in the identity $n^{2m+1}=\sum\limits_{0\leq k \leq m}(-1)^{m-k}U_m(n,k)\cdot n^k$
Review the main result of mathoverflow.net/questions/297900, that is the identity
\begin{equation}\label{f1}
n^{2m+1}=\sum\limits_{1\leq k \leq n}\sum\limits_{j\geq0}A_{m,j}k^j(n-k)^j,
\end{equation}
...
2
votes
1
answer
352
views
A question on a proof in the Ralph Greenberg's paper "On a Certain l-Adic Representation"
I'm currently reading the paper "On a Certain l-Adic Reprersentation" written by Ralph Greenberg.(Inventiones 1973) And I'm stuck with a proof of the Proposition 2.
Here $k$ is a totally imaginary ...
0
votes
4
answers
686
views
On the real part of the Riemann zeta function inside the critical strip
Denote by $\zeta$ the Riemann zeta function. Does $\Re\zeta(s)$ ever vanish for $\frac{1}{2}<\Re(s)\leq 1$ ?
4
votes
2
answers
438
views
Is Riemann zeta function injective in some strips $a<\Re(s)<b$, where $0\leq a<b \leq 1$?
Or more generally, are L-functions injective in some strips $a<\Re(s)<b$, where $0\leq a<b \leq 1$?
6
votes
1
answer
307
views
Is this generalization of the Hopf map for quadratic field extensions surjective?
Let $k$ be a field, and let $L$ be a quadratic extension of $k$. Denote by $\sigma$ the non-trivial element of $\operatorname{Gal}(L/k)$. Let $M_2(L)$ be the vector space over $L$ of two-by-two ...
1
vote
0
answers
142
views
Fejer-Jackson-like inequality with divisor sum
A question was recently asked about a generalization of the Fejer-Jackson inequality $$\sum_{k=1}^n \frac{\sin kx}{k}\gt 0 \quad \forall\: n\in\mathbb{Z}^+\: \text{and}\: 0\lt x\lt\pi$$
to ...
6
votes
1
answer
465
views
Strengthened supercongruences for Ramanujan-type formulas for $1/\pi^k$
The question below is again a follow-up of an old question.
Motivation: Zhi-Wei Sun listed a number of supercongruences attached to Ramanujan-type $1/\pi$ formulas in the arXiv paper which can be ...
26
votes
1
answer
4k
views
Why did Euler consider the zeta function?
Many zeta functions and L-functions which are generalizations of the Riemann zeta function play very important roles in modern mathematics (Kummer criterion, class number formula, Weil conjecture, BSD ...
1
vote
1
answer
151
views
Is $P_{2n} \ge 2P_n$ and $\pi(2x) \le 2\pi(x)$ inconsistent to the Prime k-tuple conjecture?
I posed a conjecture as follows that is a special case of Second Hardy–Littlewood conjecture:
For $n, x \ge 2$ be two integers then:
$$P_{2n} \ge 2P_n$$
and
$$\pi(2x) \le 2\pi(x)...