All Questions
Tagged with reference-request nt.number-theory
1,408 questions
9
votes
1
answer
1k
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 ...
- 143
7
votes
1
answer
474
views
Fibonacci embedded in Catalan?
Given a partition $\lambda$ and its Young diagram $\pmb{Y}_{\lambda}$, we say $\lambda$ is a $(t,s)$-core partition provided that neither $t$ nor $s$ is a hook length in $\pmb{Y}_{\lambda}$. We now ...
- 43.2k
32
votes
2
answers
3k
views
The Erdős–Turán conjecture or the Erdős conjecture?
This has been bothering me for a while, and I can't seem to find any definitive answer. The following conjecture is well known in additive combinatorics:
Conjecture: If $A\subset \mathbb{N}$ and $$\...
- 11.4k
4
votes
1
answer
463
views
Generalizations of the Brun-Titchmarsh theorem
Let $\pi(x;q,a)$ count the number of primes $\leq x$ congruent to $a$ mod $q$. The Brun-Titchmarsh Theorem states that for all $q< x$, $(a,q)=1$, we have
$$
\tag{1}
\pi(x;q,a) \leq \frac{2x}{\...
- 1,990
8
votes
1
answer
465
views
Smooth numbers in short intervals
Let $\psi(x,y)$ be the number of positive integers up to $x$ which are $y$-smooth, that is, integers whose prime factors are at most of size $y$. There has been, for a few decades now, a lot of ...
- 14.6k
9
votes
0
answers
462
views
Who realized the finite fields $\mathbb F_{p^n}$ first? Gauss or Galois?
Let $p$ be a prime, and let $n$ be a positive integer. The finite field $\mathbb F_{p^n}$ is often called a Galois field and denoted by $\mathrm{GF}(p^n)$ by researchers on coding theory.
On the other ...
- 15.6k
3
votes
1
answer
200
views
Maximum number of edges in a "coprime graph"
Let's define a coprime graph as a simple graph (undirected graph without any self-loops or multiple-edges) in which for all edges $(𝑢, 𝑣)$, the property $\gcd(\mathrm{degree}_u, \mathrm{degree}_v) = ...
13
votes
2
answers
1k
views
Question about a lesser-known "class number formula" of Gauss
My question refers to article 301 of section 5 of Gauss's D. A. - there Gauss gives an asymptotic formula for the mean number of classes of forms with positive discriminant ($D>0$):
$$h(D) = \frac{...
- 2,099
10
votes
1
answer
591
views
Are there numbers whose binary and ternary representations simultaneously have few digit transitions? How frequent are those numbers?
For a natural number $n$, let $c_b(n)$ denote the number of digit transitions in the representation of $n$ in base $b$. By a digit transition, we mean a pair of successive, unequal digits: for ...
- 183
3
votes
1
answer
280
views
Congruence residues of integer partitions
Consider the number of integer partitions $p(n)$ of $n$ whose (product) generating function reads
$$\sum_{n\geq0}p(n)\,x^n=\prod_{k\geq1}\frac1{1-x^k}.$$
There are many congruences for $p(n)$ ...
- 43.2k
6
votes
1
answer
636
views
The history and original paper of the Rosser–Iwaniec sieve
I'm trying to find Rosser's original paper where he introduces his eponymous sieve. I've already found https://arxiv.org/pdf/math/0505521 (where the reference isn't given, but where it is indicated ...
- 559
1
vote
0
answers
87
views
Doubly log-concave or doubly log-convex
Suppose $(a_k)_{k\geq0}$ is a sequence of real numbers. Consider the operator $\mathcal{L}a_k=a_k^2-a_{k-1}a_{k+1}$.
We say $(a_k)_k$ is log-concave (resp. log-convex) provided $\mathcal{L}a_k\geq0$ (...
- 43.2k
3
votes
0
answers
221
views
Reference request Re Vinogradov's ternary Goldbach proof
I believe that I.M. Vinogradov's proof of the ternary Goldbach conjecture used the observation that the number of ways $n$ can be written as a sum of three primes equals
$$
\int_0^1 \sum_{p , q , r \...
- 171
21
votes
2
answers
2k
views
Applications of number theory in dynamical systems
I am looking for references (or ways to find references) on significant and/or recent applications of techniques in number theory to problems in the areas of dynamical systems and nonlinear dynamics.
...
- 760
7
votes
1
answer
293
views
On a certain double integral appearing in the Fourier series coefficients of $\mathrm{SL}_2(\mathbb{C})$-Eisenstein series
The following integral appears naturally within the computation of the Fourier series coefficients of a real analytic $\mathrm{SL}_2(\mathbb{C})$-Eisenstein series:
\begin{align*}
\int_{-\infty}^{\...
- 7,609
-2
votes
1
answer
139
views
Congruence modulo 4 for a generating function leads to perfect squares? [duplicate]
Consider the number of integer partitions $p(n)$ of $n$ whose generating function is
$$\sum_{n\geq0}p(n)\,x^n=\prod_{k\geq1}\frac1{1-x^k}.$$
Also, the number of partitions into distinct parts $Q(n)$ ...
- 43.2k
-1
votes
1
answer
162
views
Convergence to a constant or not? Reference request [closed]
Consider the function
$$f(n) = \log n /(n\ \log\theta(p_n)),$$
where $\theta$ is the first Chebyshev function and $p_n$ is the $n$-th prime. Does $f$ converge to a constant as $n$ grows to infinity, ...
- 1,018
5
votes
2
answers
525
views
A modern reference to the Zsigmondy Theorem
I need to cite the classical Zsigmondy Theorem, which was proved in 1892.
Is there any modern reference to this theorem?
I mean some standard textbook in Number Theory containing this theorem together ...
- 42k
4
votes
0
answers
272
views
The Gamma-transform and $p$-adic $L$-functions
I'm currently reading the paper "On the $\mu$-invariant of the $\Gamma$-transform of a rational function" by W Sinnott. In this paper, he gives an alternate proof that $\mu=0$ for abelian ...
- 1,949
7
votes
5
answers
941
views
Perfect powers in the solutions of a certain Pell equation
The fundamental solution of the Pell equation $$x^{2}-3y^{2}=1$$ is $2+\sqrt{3}$.
It seems that if $x_{n}+y_{n}\sqrt{3}$, $x_{n}, y_{n} \in \mathbb{N}$, is a solution of the said Pell equation and $x_{...
- 87
7
votes
2
answers
948
views
Reference of J.L. Waldspurger's paper on Shimura correspondence
I want to find reference of Waldspurger's paper referred at "Sur les coefficients de Fourier des formes modulaires de poids demi-entier" J. Math. Pures Appl. (9) 60 (1981), no. 4, 375–484 (...
- 73
8
votes
2
answers
1k
views
Lower bound on exponential sums
Let $k\geq 2$. Consider the following norm of exponenetial sum:
$$
I(N,p,k)=\int_0^1\int_0^1 \left|\sum_{n=0}^N e^{2\pi i (n x+n^k y)}\right|^p dxdy.
$$
Bourgain mentioned on Page 118 of
https://...
- 751
4
votes
2
answers
579
views
Hardy-Littlewood circle method for non-diagonal quadratic forms
In short, the question is for any references describing how to use the Hardy-Littlewood circle method to find an asymptotic for the number of solutions to $F(x_1, ..., x_s) = k$ for $(x_1, ..., x_s) \...
- 813
0
votes
1
answer
180
views
Name of conjectures similar to Goldbach conjecture
Consider the following """easier""" conjectures:
C1. every sum of two semiprimes $n = pq + rs$, $p,q,r,s$ primes, can be expressed as $n = (a + b)/2$; with $a,b$ primes.
...
- 1,602
3
votes
0
answers
158
views
What can be said about the primality of Zsigmondy numbers?
I am cross-posting this from math.stackexchange, as it has received upvotes but no comments/answers after a couple months.
Let $\mathcal{Z}(n,a,b)=\frac{\Phi_n(a,b)}{\gcd (\Phi_n(a,b),n)}$ be the $n$-...
- 101
6
votes
1
answer
302
views
A 3rd formula for the central Delannoy numbers?
There are several in the literature proving the two alternative formulas for the (diagonal) Delannoy numbers; namely that
$$d_n=\sum_{k=0}^n\binom{n}k\binom{n+k}k=\sum_{k=0}^n\binom{n}k^22^k.$$
Each ...
- 43.2k
2
votes
0
answers
245
views
Ambiguity about the exact definition of coefficients of modular forms
You can see the parts after my questions in the boxes. I received the answer to my first question in the comments.
I am confused about the definition of $a_n$ and $b_n$ in Part II below. I know the ...
9
votes
2
answers
2k
views
Any simple concrete proof of Faltings theorem?
Are there simple proofs of some concrete special cases of Faltings's theorem? Any help would be appreciated.
- 3,094
5
votes
1
answer
2k
views
Is there a relation between the number of lattice points lie within these circles
Suppose we have a circle of radius $r$ centered at the origin $(0,0)$. The number of integer lattice points within the circle, $N$, can be bounded using Gauss circle problem.
Suppose that another ...
- 225
3
votes
1
answer
206
views
asymptotic growth of a sum involving partitions
Let $\lambda\vdash n$ denote the integer partition of $n$. Define the product $\mathcal{N}(\lambda)=\lambda_1\lambda_2\cdots\lambda_r$ when $\lambda=(\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_r>0)...
- 43.2k
5
votes
1
answer
365
views
Power of $2$ dividing a specialized Mittag-Leffler polynomial
While studying the so-called Mittag-Leffler Polynomials, denoted $M_n(x)$, I was looking into the sequence $\frac1{n!}M_n(n)$ which takes the following form
$$a_n:=\sum_{k=1}^n\binom{n-1}{k-1}\binom{n}...
- 43.2k
41
votes
2
answers
9k
views
What should I read before reading about Arakelov theory?
I tried reading about Arakelov theory before, but I could never get very far. It seems that this theory draws its motivation from geometric ideas that I'm not very familiar with.
What should I read ...
16
votes
1
answer
2k
views
Reference to a Don Zagier result and the congruent number problem
I was looking for a reference/explanation as to how Don Zagier managed to find the side lengths of a rational right triangle with area 157. There have been many literature references to the fact that ...
- 163
1
vote
1
answer
186
views
Connection between central factorial numbers and the Stern–Brocot tree
Consider the central factorial numbers of even indices formed by
$$U(n,k)=\frac1{(2k)!}\sum_{i=0}^{2k}(-1)^i\binom{2k}i(k-i)^{2n}.$$
Let $u(n,k):=U(n,k)\mod 2$. Define the triangle of numbers
$$A(r,j)=...
- 43.2k
5
votes
3
answers
843
views
Is the number of solutions of $\phi(x)=n!$ bounded? If yes, what is its bound?
Pillai showed in 1929 that the function $A(n)$ giving the number solutions of the equation $\phi(x)=n$ is unbounded in (S. Pillai, On some functions connected with $\varphi(n)$, Bull. Amer. Math. Soc. ...
5
votes
2
answers
1k
views
Error term in Mertens' third theorem
Mertens' third theorem states that:
$$\prod_{\substack{
p \leq x \\
\text{p prime}
}} \left( 1 - \dfrac{1}{p} \right) \sim \dfrac{e^{-\gamma}}{\log(x)}$$
Question: what is the best functions (...
- 521
60
votes
2
answers
11k
views
What is a good roadmap for learning Shimura curves?
I am interested in learning about Shimura curves. Unlike most of the people who post reference requests however (see this question for example), my problem is not sorting through an abundance of books ...
user1073
12
votes
1
answer
526
views
Equidistribution of $\{\alpha p\}$ for $p$ in an arithmetic progression
Let $\alpha$ be irrational. A famous theorem of Vinogradov says that $\{ \alpha p\}$ is equidistributed in $[0,1]$ as $p$ runs over all primes.
Let $a,q$ be natural numbers with $\gcd(a,q) = 1$. Then ...
- 21.3k
3
votes
1
answer
270
views
Distance formula for continued fractions
In the book Neverending fractions from Borwein, van der Poorten, Shallit and Zudilin, there is the so called distance formula (Theorem 2.45, p. 43) stated:
$$\alpha_1\alpha_2\cdot...\cdot\alpha_n=\...
- 406
7
votes
2
answers
191
views
Character sums concerning $a^x-1$
Let $p$ be a prime number and $\mathbb{F}_p$ be a finite field with $p$ elements. Let $\chi$ be a multiplicative character from $\mathbb{F}_p^{\times}$ to $\mathbb{C}$, where $\mathbb{F}_p^{\times}=\{...
user125345
19
votes
2
answers
2k
views
Who first proved the generalization of Bertrand's postulate to (2n,3n) and (3n,4n)?
In Wikipedia's page for Bertrand's postulate, it is said that its (2n,3n) version was proved by El Bachraoui in 2006. Seems likely that it was first proved way before than that! Can anyone point to ...
- 2,992
9
votes
1
answer
388
views
$π(x+y) - π(x) ≤ c·y/\ln(y)$ for some constant $c$?
(I posted this question on Math SE but it has had no answer for a year now so I would like to ask if anyone here can provide one.)
Thinking about the prime number theorem, I wondered whether it is ...
- 2,912
6
votes
2
answers
715
views
Origin and variations of problem on $4xy-x-y$ being square
One of the forms in which the Diophantine equation in question can be found in the literature is this:
Solve the equation \begin{eqnarray}z^{2} = 4xy-x-y \qquad \qquad (\ast)\end{eqnarray} in ...
- 8,842
1
vote
2
answers
635
views
Is there a database about the particular values of $j$-invariant?
Is there a database that has all the known particular values of the $j$-invariant?
- 2,971
2
votes
0
answers
156
views
A question on terminology for sequences satisfying $\gcd(a_m,a_n)=a_{\gcd(m,n)}$
How do you refer to those sequences $\{a_{n}\}_{n \in \mathbb{Z}^{+}}$ of integers that satisfy the condition $\text{gcd}(a_{m}, a_{n}) = a_{\text{gcd}(m,n)}$ for every $(m,n) \in \mathbb{Z}^{+} \...
- 87
2
votes
0
answers
150
views
Closeness of a rational approximation
What is
$$p_*:=\inf\big\{p\in\mathbb R\colon\,\inf_{n\in\mathbb N}n^p\,\inf_{k\in\mathbb N}
|2\sqrt{3n}-9\pi/4-k\pi|>0\big\},$$
where $\mathbb N:=\{1,2,\dots\}$?
In other words, I would like to ...
- 128k
5
votes
0
answers
104
views
Exponential sums with monomials with divisor-function coefficients
In their paper "Exponential Sums with Monomials," Fouvry and Iwaniec study exponential sums roughly of the form
$$
\sum_{m_1 \sim M_1} \cdots \sum_{m_r \sim M_r} c_1(m_1) \cdots c_r(m_r) e\...
- 1,990
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 ...
- 43.2k
0
votes
0
answers
196
views
Sum of squares squared in an arithmetic progression
Let $r(n)$ be the number of ways to write $n$ as a sum of two squares and $(a,q)=1$.
What is known about
$$
\sum_{n \le x,n \equiv a (\text{mod} \, q)} r(n)^2 \quad?
$$
I am looking for uniform ...
- 130
25
votes
2
answers
4k
views
Primes of the form $x^2+ny^2$ and congruences.
The answer of following classical problem is surely known, but I can't find a reference
For which positive integer $n$ is the set $S_n$ of primes of the form $x^2+n y^2$ ($x$, $y$ integers) ...
- 26k