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

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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 ...
modular's user avatar
  • 31
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 ...
Morteza Azad's user avatar
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},\...
fretty's user avatar
  • 562
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 ...
Dave Gaebler's user avatar
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 \...
Mohammad Golshani's user avatar
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 ...
Will Chen's user avatar
  • 10k
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 $x_{k + 1} > 0$ ...
Kaban-5's user avatar
  • 543
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 $\...
debanjana's user avatar
  • 1,161
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 $N$, $\chi$...
Fei's user avatar
  • 303
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 ...
Ricardo Espino Lizama's user avatar
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 $$x^n-n=y^m \quad \ \text{with}\ m,n,x,y\in\{2,3,\ldots\}.\tag{2}$$ ...
Zhi-Wei Sun's user avatar
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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 ...
Alexey Ustinov's user avatar
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. ...
Zurab Silagadze's user avatar
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 ...
Sean Eberhard's user avatar
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 ...
Tony's user avatar
  • 289
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?
witek's user avatar
  • 73
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 ...
Christian Chapman's user avatar
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 ...
Alm's user avatar
  • 1,159
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 $$B = \left\{\frac{a}{p}: 1\le a < p\le X,\ (a, p) = 1,\text{ and $p$ is prime}\right\}.$$ ...
Mayank Pandey's user avatar
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 ...
Henri Cohen's user avatar
  • 11.5k
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 ...
Pierre21's user avatar
  • 385
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 : ...
uvdose's user avatar
  • 593
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 ...
user125639's user avatar
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)...
Greg Egan's user avatar
  • 2,852
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 ...
Kaban-5's user avatar
  • 543
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 ...
P Gibson's user avatar
  • 115
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}_{\...
rime's user avatar
  • 445
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\...
Gabe Conant's user avatar
  • 3,204
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 ...
Sylvain JULIEN's user avatar
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^...
john mangual's user avatar
  • 22.6k
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 ...
anna abasheva's user avatar
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.
Pablo's user avatar
  • 11.2k
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 $C$ ...
Pietro Paparella's user avatar
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 ...
Ramin's user avatar
  • 1,362
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 ...
Taras Banakh's user avatar
  • 40.8k
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 $2^n$ ...
kodlu's user avatar
  • 10.1k
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 ...
Jz Pan's user avatar
  • 153
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 ...
Anton's user avatar
  • 1,573
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 ...
Kurisuto Asutora's user avatar
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,\...
kodlu's user avatar
  • 10.1k
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} ...
Petro Kolosov's user avatar
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 ...
gualterio's user avatar
  • 1,043
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 $\frac{1}{2}<\Re(s)\leq 1$ ?
Q_p's user avatar
  • 824
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 $0\leq a<b \leq 1$?
Milin's user avatar
  • 395
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 ...
Malkoun's user avatar
  • 4,981
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 ...
ljk's user avatar
  • 105
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 ...
Y. Zhao's user avatar
  • 3,317
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 ...
Jojo's user avatar
  • 293
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)...
Đào Thanh Oai's user avatar

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