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
15,907
questions
6
votes
1
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727
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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 $\Gamma\subset \operatorname{SL}_2(\mathbb Z)$. ...
1
vote
0
answers
184
views
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 $\...
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)...
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 $v\in S$, ...
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(\...
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{...
11
votes
2
answers
937
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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 ...
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 \...
12
votes
2
answers
2k
views
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 ...
0
votes
0
answers
88
views
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 ...
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 ...
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 ...
9
votes
3
answers
2k
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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+...
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 $$D_p^{-}:=\det\left[(i-j)\left(\frac{i-j}p\right)\right]_{1\le i,j\le(p-1)/2},$$
...
3
votes
1
answer
179
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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 ?
1
vote
1
answer
292
views
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?
15
votes
2
answers
1k
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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 $\...
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 $M(x)$ ...
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 $...
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 $J(x)>0$ ...
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 ...
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,...
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–...
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 ...
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 ...
20
votes
1
answer
2k
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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 ...
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 ...
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 $\theta$. ...
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}} \...
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 $a_{12}$ =...
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 ...
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 [...
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, ...
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!=...
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 ...
6
votes
2
answers
1k
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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 ...
-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) ...
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 ...
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 ...
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:
$...
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:
$$X_N:=\left\{(c_{ij})_{i,j=1}^4\in M_4(\mathbb{Z}_{\geq 0})\bigg| \sum_j c_{1j} = \sum_i ...
3
votes
0
answers
155
views
How to express the cuspidal form in terms of Poincare series?
Sorry to disturb. I recently read Blomer's paper. There is a blur which needs the expert's help here. Blomer's paper says "For any cusp form $f$ of integral weight $k$, level $N$, $f$ can be written ...
1
vote
0
answers
126
views
How many solutions to $p_i|a_{i,1} p_1 + \dotsc + a_{i,n} p_n$?
Consider a system of $n$ divisibility conditions on $n$ prime variables:
$$p_i|a_{i,1} p_1 + \dotsc + a_{i,n} p_n,\;\;\;\;\;1\leq i\leq n,$$
where $a_{i,j}$ are bounded integers. How many solutions ...
0
votes
1
answer
146
views
Power series rings and the formal generic fibre
Let $S = K[[S_1,\ldots,S_n]]$ and consider $d$ elements
\begin{equation*}
f_1,\ldots,f_d \in S[[X_1,\ldots,X_d]]
\end{equation*}
and the prime ideal ${\frak P} \colon\!= (f_1,\ldots,f_d)$ generated ...
2
votes
0
answers
205
views
Satake correspondence for groups over finite field
I asked the same question in MSE, but I didn't get any answer. So I decided to post it here, too.
In Langlands' program, Satake correspondence gives a correspondence between unramified ...
3
votes
0
answers
97
views
Supremum of certain modified zeta functions at 1
Let $D$ be an integer number and let $\chi$ be the Dirichlet character defined by
$$\chi(m) = 0 \text{ if $m$ even, } \chi(m) = (D/m) \text{ if $m$ odd,}$$
where $(D/m)$ denotes the Jacobi symbol. ...
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 ...
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 ...
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 (...