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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Galois theory of modular functions

Let $\mathcal M_m$ be the set of $2$-by-$2$ primitive (relatively prime entries) matrices with determinant $m$. Let $\alpha \in \mathcal M_m$ and let $\Gamma\subset \operatorname{SL}_2(\mathbb Z)$. ...
Shimrod's user avatar
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Questions on Riemann's explicit formula

If we consider this version of the prime-counting function $$\pi_0(x) = \frac{1}{2} \lim_{h\to 0} (\pi(x+h) + \pi(x-h))$$ (with $\pi$ being the normal prime-counting function), then we can write $\...
tobias's user avatar
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1 vote
1 answer
192 views

Comparison of two integrals

Let $S(x)$ be continuous, differentiable, and such that $S(x)=O(x/\log x)$. Let $J(x)=\int_x^{\infty} \frac{S(y)(1+\log y)}{y^2\log^2 y}dy$ and let $K(x)=\int_x^{\infty}\frac{S(y)}{y^2}dy$. Let $K(2)...
EGME's user avatar
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2 votes
0 answers
159 views

A version of weak approximation with S-integers

Let $k$ be a finite field. Let $K$ a finite extension of $k(t)$. Let $S$ be a finite set of places of $K$. Let $$K_S = \prod_{v\in S} K_v$$ where $K_v$ is the completion of $K$ at $v$. For $v\in S$, ...
Will Dukeminier's user avatar
2 votes
1 answer
174 views

Centralizers of Cartan subgroups

Let $E$ be an elliptic curve with CM by an order $\mathcal O$ in an imaginary quadratic field $K$. Choose a basis for $E[N]$ to get an isomorphism $\operatorname{Aut}(E[N])\cong \operatorname{GL}_2(\...
Shimrod's user avatar
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2 votes
0 answers
162 views

Newman's conjecture of Partition function

(Sorry for my poor english....) Let $p(n)$ be a partition function and $M$ be an integer. Newman conjectured that for each $0\leq r\leq M-1$, there are infinitely many integers $n$ such that \begin{...
ililiil's user avatar
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11 votes
2 answers
937 views

Is there a proof of quadratic reciprocity using $p$-adic numbers?

I asked same question on MSE before, but I didn't get any answer yet. I know that the quadratic reciprocity can be regarded as a special case of Artin reciprocity (class field theory), and we can ...
Seewoo Lee's user avatar
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6 votes
1 answer
388 views

hook-length formula: "Fibonaccized": Part II

This is a natural follow-up to my previous MO question, which I share with Brian Hopkins. Consider the Young diagram of a partition $\lambda = (\lambda_1,\ldots,\lambda_k)$. For a square $(i,j) \in \...
T. Amdeberhan's user avatar
12 votes
2 answers
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how do we prove that a sum of two periods is still a period?

Kontsevich and Zagier define periods as the values of absolutely convergent integrals $\int_\sigma f$ where $f$ is a rational function with rational coefficients and $\sigma$ is a semi-algebraic ...
periods's user avatar
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0 answers
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A question about a recursive relation

let be $c(x)$ the function that counts the composite numbers less or equal to x. So for example $c(5)=1$. The number $128$ is a record because no composite number m less than 128 generates more ...
Enzo Creti's user avatar
9 votes
1 answer
968 views

Sums of two squares in arithmetic progressions

Let $r(n)$ denote the number of representations of $n$ as the sum of two squares. Are there any known results on $$\sum _{n\leq x\atop {n\equiv a(q)}}r(n)$$ and in particular is there an asymptotic ...
caws's user avatar
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3 votes
1 answer
382 views

Why are Poincare series defined as they are?

We know the Poincare series are defined as the following: The $m^{th}$ Poincare series of weight $k$ for $\Gamma$ is: $$ P_{m}^{k} (z) = \sum_{(c,d)=1} (cz+d)^{-k} e^{2 \pi in(\tau z)}. $$ The ...
Robert's user avatar
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9 votes
3 answers
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Is this a new Fibonacci Identity? [closed]

I have found the following Fibonacci Identity (and proved it). If $F_n$ denotes the nth Fibonacci Number, we have the following identity \begin{equation} F_{n-r+h}F_{n+k+g+1} - F_{n-r+g}F_{n+k+h+...
user918212's user avatar
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5 votes
0 answers
190 views

On the determinants $\det\left[(i\pm j)\left(\frac{i\pm j}p\right)\right]_{1\le i,j\le(p-1)/2}$

Let $p$ be an odd prime and define $$D_p^+:=\det\left[(i+j)\left(\frac{i+j}p\right)\right]_{1\le i,j\le(p-1)/2}$$ and $$D_p^{-}:=\det\left[(i-j)\left(\frac{i-j}p\right)\right]_{1\le i,j\le(p-1)/2},$$ ...
Zhi-Wei Sun's user avatar
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3 votes
1 answer
179 views

A question on Bernoulli polynomials

Denote by $B_r$ the $r$-th Bernoulli polynomial. Are there any positive integers $r, x$ such that. $B_r(x)$ divides $B_r(x+1)$ or vice versa ?
Q_p's user avatar
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1 vote
1 answer
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Why it is so hard to find a prime of the form $a^b+b\cdot c+c^d$?

Why it is so hard to find a prime $p$ of the form: $a^b+b\cdot c+c^d$? where $a$, $b$, $c$ and $d$ are four consecutive primes $a<b<c<d$? Can such a prime exist?
Enzo Creti's user avatar
15 votes
2 answers
1k views

hook-length formula: "Fibonaccized" Part I

Consider the Young diagram of a partition $\lambda = (\lambda_1,\ldots,\lambda_k)$. For a square $(i,j) \in \lambda$, define the hook numbers $h_{(i,j)} = \lambda_i + \lambda_j' -i - j +1$ where $\...
T. Amdeberhan's user avatar
1 vote
0 answers
131 views

About the sum of prime reciprocals

Let $B$ be the Meissel-Mertens constant., $\theta$ the Chebyshev theta. Let $S(x)=\sum_{p\le x}1/p$, $p$ prime. Let $M(x)=S(x)-\log\log x-B$. Robin, and later Diamond and Pintz, showed that $M(x)$ ...
EGME's user avatar
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1 vote
1 answer
119 views

Does the intercept converge if we fit a best fit line to points with prime coordinates?

A few months ago I asked this question on Mathematics Stack Exchange but it has received little attention. Perhaps the question is more applicable here. Let $p_k$ denote the $k$th prime such that $...
TheSimpliFire's user avatar
0 votes
0 answers
211 views

Positivity of an integral

Let $f(x)=x-\theta(x)$, and for $x\ge 2$ let $J(x)=\int_x^{\infty}\frac{f(y)(1+\log y)}{y^2\log^2 y}dy$. Furthermore, $J(2)>0$. Suppose $J(x)>0$ when $f(x)=0$. Does it follow that $J(x)>0$ ...
EGME's user avatar
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7 votes
1 answer
434 views

What numbers can simulate 1/2?

Given two numbers $p,q\in(0,1)$, we say that $p$ can simulate $q$ if, given a biased coin with probability $p$, we can toss it a bounded number of times and use the results to simuate a biased coin ...
Erel Segal-Halevi's user avatar
2 votes
0 answers
76 views

Accelerating convergence of a product by multiplying by zeta values: history?

Let $R(s_1,\dotsc,s_n) = \prod_p r(p^{-s_1},\dotsc,p^{-s_n})$, where $r$ is a rational function on $n$ variables. Say we want to compute the value of $R(s_1,\dotsc,s_n)$ for some choice of $s_1,\dotsc,...
Nell's user avatar
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7 votes
0 answers
445 views

Integral models of perfectoid modular curves

Scholze constructed perfectoid modular curve and its canonical and anticanonical part in his paper On torsion in the cohomology of locally symmetric varieties (Annals of Mathematics 182 (2015) pp 945–...
Zariski93's user avatar
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4 votes
0 answers
141 views

Local behaviour of fractions with bounded denominator / Was it already studied?

My question is about a point process that I feel it would be natural to study, but that I have never heard of… This point process would represent, morally, the local behaviour of the set of fractions ...
Rémi Peyre's user avatar
1 vote
1 answer
353 views

Relating the shortest vector of a lattice to the orthogonal complement of the lattice

By a lattice we mean sub-lattice of $\mathbb{Z}^n \cap V$, where $V$ is a subspace of $\mathbb{R}^n$ defined over $\mathbb{Q}$. We say that a lattice $\Lambda$ is primitive if a basis of $\Lambda$ can ...
Stanley Yao Xiao's user avatar
20 votes
1 answer
2k views

Circle $x^2 + y^2 = n!$ doesn't hit any lattice points for any $n$ except for $0$, $1$, $2$ and $6$ or does it?

I stumbled across the following problem in high school:$$ x^2 + y^2 = n! $$ I tested it within my laptop capabilities, watched a 3b1b video Pi in prime regularities, where he explains how to find the ...
Betydlig's user avatar
  • 343
9 votes
2 answers
1k views

Dynamics of Riemann zeta function

Has the dynamics of the Riemann zeta function been studied? By dynamics I mean the limiting behavior of the sequence of iterates $s, \zeta (s), \zeta (\zeta (s)), \zeta (\zeta (\zeta (s)))\dots $ for ...
user137686's user avatar
3 votes
3 answers
353 views

Oscillations of $\theta(x)-x$, for the Chebyshev $\theta$ function

Is anything known about the relative "periodicity" of the oscillations of $\theta(x)-x$, that is, how frequent, in general terms, are the sign changes? Here, $\theta(x)$ is the Chebyshev $\theta$. ...
EGME's user avatar
  • 1,008
3 votes
2 answers
519 views

1/2 Wilson's theorem

During my research I came across this question : Question: What's the value of $x_p=(\dfrac{p-1}{2})! \mod p$ when $p>3$ is prime ? Remark: It's easy to see $x_p^2 \mod p=(-1)^{\dfrac{p+1}{2}} \...
Dattier's user avatar
  • 3,737
6 votes
3 answers
379 views

Strings of consecutive integers divisible by 1, 2, 3, ..., N

For each n, let $a_n$ be the least integer, greater than n, such that the numbers $a_n$, $a_n$+ 1, $a_n$+ 2, ..., $a_n$+ (n – 1) are divisible, in some order, by 1, 2, 3, ..., n. For example $a_{12}$ =...
Bernardo Recamán Santos's user avatar
1 vote
0 answers
127 views

Is the Upper Banach density always zero with respect to some sequence of Finite subset

The following question came to me while reading the paper 'Density in Arbitrary Semigroups' by Hindman and Strauss. Question: Given an infinite subset $A$ of $\mathbb{N}$ such that $A^c$ is also ...
Surajit's user avatar
  • 73
6 votes
2 answers
366 views

Hyperelliptic Jacobians with (or without) CM

Let $C$ be a hyperelliptic curve $y^2 = f(x) $ defined over $\mathbb{Q}$, where $f(x) \in \mathbb{Q} [x]$ is a polynomial of degree $n=5$ or $6$, and $J = Jac(C)$ its Jacobian. I know Zarhin's result [...
Kazuki  Sato's user avatar
2 votes
0 answers
90 views

A character sum $\sum_{0<n\leq Y}\chi_4(n)\chi_0^{(k)}(n)$ estimate

I'm reading the paper 'Jutila, Matti. "On the Mean Value of $L (1/2, χ)$ FW Real Characters." Analysis 1.2 (1981): 149-161.' Let $\chi_4(n)$ be the real primitive nonprincipal character of modulo 4, ...
LWW's user avatar
  • 653
5 votes
1 answer
300 views

"Oddity" of Fibonacci-Catalan numbers

As a follow up to my previous two MO questions, here and here, let's consider the below inquiry. Define the Fibonacci-Catalan numbers by $FC_n=\frac1{F_{n+1}}\binom{2n}n_F$ where $F_0=0, F_1=1, F_0!=...
T. Amdeberhan's user avatar
3 votes
0 answers
103 views

Numbers with large algebraic independency

Define the measure of algebraic independence of the numbers $a_1, \ldots, a_n \in \mathbb{C}$ as $\Phi(a_1, \ldots, a_n; m, H) = \min |P(a_1, \ldots, a_n)|$, where the minimum is taken over all ...
Alexey Milovanov's user avatar
6 votes
2 answers
1k views

Products and sum of cubes in Fibonacci

Consider the familiar sequence of Fibonacci numbers: $F_0=0, F_1=1, F_n=F_{n-1}+F_{n-2}$. Although it is rather easy to furnish an algebraic verification of the below identity, I wish to see a ...
T. Amdeberhan's user avatar
-4 votes
1 answer
93 views

How to solve a Diophantine equation in six variables? [closed]

Find all the integer solutions $a, b, c, d, e, f$ satisfying the equation $a^2b^3 + c^2d^3 = e^2f^3$. Note that if we prove that there are no such solutions with the condition $\text{gcd}(ab, cd, ef) ...
DSop's user avatar
  • 9
14 votes
8 answers
2k views

Applications of the idea of deformation in algebraic geometry and other areas?

The idea of proving something by deforming the general case to some special cases is very powerful. For example, one can prove certain equalities by regarding both sides as functions/sheaves, and show ...
2 votes
1 answer
430 views

Are there infinitely many prime p, such that p=1296k^2+36k+7? [closed]

I encountered a number theory problem when doing my research: 1.I want to know whether or not there are infinitely many primes $p$ satistying $gcd(\frac{p-1}{6},6)=1$, such that $6$ is a cubic ...
Zuo Ye's user avatar
  • 29
2 votes
0 answers
148 views

theta function with a low bound in the sum

I encounter a sum similar to the Jacobi Theta function except there is a lower bound $-J$ with $J\geq 0$: \begin{equation} f(q,x)=\sum_{n=-J}^\infty q^{n^2}x^n. \end{equation} My question is whether ...
color's user avatar
  • 99
5 votes
1 answer
1k views

Partial sums of primes

$2+3+5+7+11+13...$ is clearly the sum of the primes. Now I consider partial sums such: $2+3+5+7+11=28$ which is divisible by $7$ My question is: are there infinitely many partial sums such that: $...
Enzo Creti's user avatar
2 votes
2 answers
286 views

Calculating the number of solutions of integer linear equations

Let $N$ be a natural number. Consider the following set of matrices whose entries are non-negative integers: $$X_N:=\left\{(c_{ij})_{i,j=1}^4\in M_4(\mathbb{Z}_{\geq 0})\bigg| \sum_j c_{1j} = \sum_i ...
Ehud Meir's user avatar
  • 4,969
3 votes
0 answers
155 views

How to express the cuspidal form in terms of Poincare series?

Sorry to disturb. I recently read Blomer's paper. There is a blur which needs the expert's help here. Blomer's paper says "For any cusp form $f$ of integral weight $k$, level $N$, $f$ can be written ...
Fei's user avatar
  • 303
1 vote
0 answers
126 views

How many solutions to $p_i|a_{i,1} p_1 + \dotsc + a_{i,n} p_n$?

Consider a system of $n$ divisibility conditions on $n$ prime variables: $$p_i|a_{i,1} p_1 + \dotsc + a_{i,n} p_n,\;\;\;\;\;1\leq i\leq n,$$ where $a_{i,j}$ are bounded integers. How many solutions ...
Nell's user avatar
  • 535
0 votes
1 answer
146 views

Power series rings and the formal generic fibre

Let $S = K[[S_1,\ldots,S_n]]$ and consider $d$ elements \begin{equation*} f_1,\ldots,f_d \in S[[X_1,\ldots,X_d]] \end{equation*} and the prime ideal ${\frak P} \colon\!= (f_1,\ldots,f_d)$ generated ...
Pierre's user avatar
  • 563
2 votes
0 answers
205 views

Satake correspondence for groups over finite field

I asked the same question in MSE, but I didn't get any answer. So I decided to post it here, too. In Langlands' program, Satake correspondence gives a correspondence between unramified ...
Seewoo Lee's user avatar
  • 1,911
3 votes
0 answers
97 views

Supremum of certain modified zeta functions at 1

Let $D$ be an integer number and let $\chi$ be the Dirichlet character defined by $$\chi(m) = 0 \text{ if $m$ even, } \chi(m) = (D/m) \text{ if $m$ odd,}$$ where $(D/m)$ denotes the Jacobi symbol. ...
Davide Cesare Veniani's user avatar
5 votes
1 answer
603 views

Modified Pascal's triangle

I posted this question to Mathematics Stack Exchange but got no answers. I hope that this question is advanced enough for this forum: In Pascal's triangle, each number is the sum of the two numbers ...
We Pretty's user avatar
3 votes
1 answer
262 views

Upper bounds for $|\theta(x)-x|$ assuming Riemann Hypothesis

What are the best currently known upper bounds for $|\theta(x)-x|$ assuming the Riemann Hypothesis, where $\theta(x)$ is the Chebyshev theta, and can someone provide the reference for this (not ...
EGME's user avatar
  • 1,008
8 votes
0 answers
285 views

A006517: Integers with $n\mid 2^n+2$

The following question was asked at Math StackExchange but, having attracted some attention, didn't get solved. Problem 323 from the Mathematical Excalibur Vol. 14, No. 2, May-Sep. 09, linked here (...
W-t-P's user avatar
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