All Questions
Tagged with reference-request nt.number-theory
1,408 questions
19
votes
5
answers
2k
views
Sum of the reciprocals of radicals
Recall that the radical of an integer $n$ is defined to be $\operatorname{rad}(n) = \prod_{p \mid n } p$.
For a paper, I need the result that
$$\sum_{n \leq x} \frac{1}{\operatorname{rad}(n)} \ll_\...
- 21.3k
1
vote
0
answers
89
views
Finiteness of elliptic curves with trivial conductor over function fields
Let $K=\mathbb{F}_q(C)$ be the function field of a smooth projective curve $C$ over a finite field $\mathbb{F}_q$ with $\text{cha}(K)>3$ and let $E$ be an elliptic curve over $E$. To $E$ we may ...
- 121
2
votes
0
answers
286
views
Is Sturm's theorem able to do these?
$\newcommand{\Ord}{\operatorname{Ord}}$Let $p$ be a positive integer and $F(q)=\sum A(m)q^m$ be a formal power series with integer coefficients. Then $\Ord_p(F(q))$ is defined by
$$\Ord_p(F(q)):=\min\{...
- 43.2k
34
votes
21
answers
11k
views
Applications of finite continued fractions
I know some applications of finite continued fractions. Probably you know more. Can you add anything? (For Applications of periodic continued fractions I have made a special topic.)
1) (Trivial) ...
2
votes
1
answer
148
views
A bijection between odd natural integers and compositions
Given an odd natural integer $2a-1$ with $a\geq 1$, associate to it recursively the composition $\psi(1)=\emptyset$ and $\psi(2^{-n}a)+(n+\delta_{>1}(m))$ if $a=2^n m$ with $m$ odd where $\delta_{&...
- 17.9k
10
votes
1
answer
550
views
Igusa's $\chi_{10}$ and Borcherds products
Igusa defined a genus 2 Siegel modular form $\chi_{10}$, which vanishes on the Humbert surface $G_{1}$ (the image of a "degenerate" Hilbert modular surface, the product of modular curves, ...
- 3,309
5
votes
2
answers
717
views
Sum of many squares modulo $n$
Let $n$ be a positive integer and $0 \leq i < n$. Define
$$
N(i) = \# \left\{ (x_1,\dots, x_s) \in [1, n]^s: x_1^2 +\dots + x_s^2 \equiv i \mod n \right\}.
$$
I am looking for a reference for ...
- 579
25
votes
1
answer
911
views
Reference request for a proof of the two-square Theorem
One can show (see below for a sketch of a proof) that every odd prime number $p$
can be written in exactly $(p+1)/2$ different ways as
$$p=a\cdot b+c\cdot d$$
with $a,b,c,d\in\mathbb N$ satisfying $\...
- 17.9k
2
votes
1
answer
243
views
$\pi$-adic Galois representations of attached to newforms at $p \nmid N$ are crystalline
Is [Scholl, Motives for modular forms, Theorem 1.2.4 (ii)] proven for any $p$ independent of the weight?
Concretely, let $f$ be a normalized eigenform of weight $w$. Let $p$ be a prime not dividing ...
user471019
3
votes
1
answer
156
views
$q$-series and Stirling of the 1st kind
Denote the (unsigned) Stirling numbers of the $1^{st}$-kind by ${n \brack k}$ and define
$$\mathbf{F}_a(q)=\sum_{m\geq1}\frac{q^{am}}{(1-q^m)^{2a}} \qquad \text{and} \qquad
\mathbf{G}_b(q)=\sum_{m\...
- 43.2k
4
votes
0
answers
149
views
Infinite family of monogenic cyclic quartic fields
I am currently reading a paper (masters' thesis, in fact) from 2008 where it is mentioned that it is not currently known whether there exist infinitely many monogenic cyclic quartic fields (i.e., ...
- 26.9k
15
votes
1
answer
484
views
Looking for a paper on transfinite diameter by David Cantor
I have been reading about transfinite diameter and its applications to number theory and have been hunting for the following paper for quite a while:
Cantor D.: On an extension of the definition of ...
- 739
17
votes
2
answers
940
views
Has the following problem, resembling the lonely runner conjecture, been studied?
Given $n$, what is the smallest value $\delta_n$ satisfying the following:
For any group of $n$ runners with constant but distinct speeds,
starting from the same point and running clockwise along the ...
- 1,209
1
vote
0
answers
127
views
an eigenvalue problem for Jacobi Forms
Assume $G(q,z)$ is a Jacobi form of a certain index k. It is known that $G$ can be expanded in a Taylor series with coefficients in the ring of quasi-modular forms (generators $E_2, E_4$ and $E_6$).
$\...
- 43.2k
0
votes
0
answers
201
views
Kato's explicit reciprocity law paper
Does anyone have a copy of Kato's article Generalized explicit reciprocity laws in Advanced Studies in Contemp. Math which is used heavily in his paper constructing his eponymous Euler system? I used ...
- 2,044
3
votes
0
answers
122
views
Description of $\operatorname{Gal}(K(E[n])/K)$ as a subgroup of $\operatorname{GL}_2(\mathbb{Z}/n\mathbb{Z})$ for a CM elliptic curve $E$
I am looking for a specific description of the Galois groups $\operatorname{Gal}(K(E[n])/K)$ as a subgroup of $\operatorname{GL}_2(\mathbb{Z}/n\mathbb{Z})$ for an elliptic curve $E$ with complex ...
- 309
4
votes
1
answer
601
views
Reference for a proof of Euclid's Theorem for the infinitude of primes
I would be curious to have a reference for the following proof
of Euclid's Theorem on the infinitude of primes:
Using Legendre's formula (also called de Polignac's formula) for
$p$-adic valuations of ...
- 17.9k
1
vote
1
answer
118
views
Product/quotient of factorials beget dyadic powers
I am writing up some notes and the following occurred to me and I would like to see if there are a variety of ways to prove it. Just for reference, the identity pops out of equality between constant ...
- 43.2k
1
vote
1
answer
393
views
Steuerwald's theorem
Background:
The perfect numbers are the positive integers $n$ such that $$\sigma(n)=2n,$$ where $\sigma(n)$ is the sum of divisors function.
The function $\sigma(n)$ is multiplicative and satisfies $$\...
user178594
1
vote
1
answer
204
views
Iterated exponential sums
Does anyone have any references for iterated exponential sums? That is, sums like
$$\sum_{1\leq n\leq X} e(f(n))\sum_{1\leq m\leq n} e(f(m)),$$
where $e(x)=e^{2\pi i x}$? I am looking for references ...
- 1,904
4
votes
0
answers
181
views
The order of the global Galois group
For any profinite group $G$, we can define the order of $G$ using the notion of supernatural number. Now let $K$ be a number field, $S$ a finite set of primes of $K$ and $ G_{K,S} $ the Galois group ...
- 863
3
votes
2
answers
713
views
Binomial coefficient congruence modulo $p^n$
I am interested in the following congruence
$$\binom{ap^n}{bp^n}\equiv \binom{a}{b}\pmod{p^n}$$
I am aware that by some reference in a book the above it should actually hold modulo $p^{3n}$; the ...
- 847
0
votes
1
answer
195
views
Fibonacci and product polynomials
The motivation for my current question arises from this MO post by R. Stanley. Caveat. There's a slight alteration.
With the convention $F_1=F_2=1$ for the Fibonacci numbers, define the polynomials $...
- 43.2k
10
votes
1
answer
669
views
The paper behind an olympiad problem
In IMO Shortlist 2013, there is a number theory problem:
Determine whether there exists an infinite sequence of nonzero digits $a_1,a_2,a_3,...$ and a positive integer $N$ such that for every integer $...
- 1,462
2
votes
0
answers
103
views
On equidistribution of primes in positive characteristic
In S. Lang's book "Algebraic Number Theory" (1986), page 317, Theorem 6 states essentially that given $P$ a set of primes, let $\tau:P\longrightarrow J$ be the typical idèle map taking ...
- 675
7
votes
3
answers
933
views
In search of an alternative proof of a series expansion for $\log 2$
We all know the series expansion
$$\log 2=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}n. \tag1$$
I also am able to use the method of Wilf-Zeilberger to the effect that
$$\log 2=3\sum_{n=1}^{\infty}\frac{(-1)^{...
- 43.2k
20
votes
2
answers
4k
views
information-theoretic derivation of the prime number theorem
Motivation:
While going through a couple interesting papers on the Physics of the Riemann Hypothesis [1] and the Minimum Description Length Principle [2], a derivation(not a proof) of the Prime Number ...
- 3,871
9
votes
2
answers
1k
views
Extracting constant terms: is there a direct way?
$\DeclareMathOperator\CT{CT}$
Let $\CT_t(f(t))$ denote the constant term of the Laurent polynomial of $f(t)$.
Define the two functions $F(x_1,\dots,x_n)$ and $G(y)$ by
$$F:=\prod_{i=1}^nx_i^{-1}(1-x_i)...
- 43.2k
0
votes
1
answer
112
views
Statistics of action of Galois group of number field on primes over unramified rational primes
Let $p \in \mathbb{Z}$ be prime and $K / \mathbb{Q}$ be a finite Galois extension. The Galois group $G$ of $K$ acts on the primes of $\mathcal{O}_K$ over $p$. Do we know any statistical information ...
- 658
7
votes
2
answers
789
views
Reference request for the explicit formula for $\sum_{n\leq x} \Lambda(n)n^{-s}$
Denote by $\Lambda(n)$ the von Mangoldt function, which is equal to $\log p$ if $p\geq 2$ is a prime, and $0$ otherwise. Let $\rho$ denote a complex zero of the Riemann $\zeta$-function. If I recall ...
- 1,019
8
votes
1
answer
410
views
An unpublished note by Bloch-Kato on p-divisible groups and Dieudonné crystals
I wonder if anyone could find the following unpublished paper of Bloch-Kato:
Spencer Bloch and Kazuya Kato, $p$-divisible groups and Dieudonné crystals, unpublished.
A similar question is here ...
- 105
26
votes
1
answer
2k
views
The "stubborn" solutions to sums of three cubes
It is conjectured (see [1]) that for any integer $k\not\equiv \pm 4\pmod 9$ there are infinitely many integer solutions to
$$
a^3+b^3+c^3=k.
$$
Numerical investigations of this conjecture show that ...
- 7,193
13
votes
2
answers
803
views
Two interpretations of a sequence: an opportunity for combinatorics
The sequence that is addressed here is resourced from the most useful site OEIS, listed as A014153, with a generating function
$$\frac1{(1-x)^2}\prod_{k=1}^{\infty}\frac1{1-x^k}.$$
In particular, look ...
- 43.2k
14
votes
7
answers
3k
views
A special type of generating function for Fibonacci
Notation. Let $[x^n]G(x)$ be the coefficient of $x^n$ in the Taylor series of $G(x)$.
Consider the sequence of central binomial coefficients $\binom{2n}n$. Then there two ways to recover them:
$$\...
- 43.2k
1
vote
1
answer
364
views
Good references to study Baker's theory
I am studying diophantine equations and I need the theory of Bakers, Can you advise me about good books, or lectures on Baker's theory?
- 17
3
votes
1
answer
419
views
Counting cubic residues mod p
Given a prime $p=3m+1$, $(p-1)/3$ of the residues mod $p$ are cubic residues. So heuristically, for any given integer $k>1$ not a perfect cube, we would expect that about 1/3 of the primes $\equiv1\...
- 9,114
1
vote
0
answers
132
views
Are the binary digits of the sequence of the prime numbers correlated?
Let $p_n\geq 3$ be the $n$th prime number with the binary expansion $p_n = \sum_{k=0}^{\infty} b_{nk}2^k$ ($b_{nk}\in\{0,1\}$). Let's write $q_{nk} = 1-2b_{nk}$.
Question: Is it true that for $k,l\...
- 2,605
26
votes
0
answers
567
views
Elliptic analogue of primes of the form $x^2 + 1$
I have a project in mind for an undergraduate to investigate next quarter -- a curiosity really, but I'm surprised I can't find it in the literature. I do not want a detailed analysis here... but ...
- 13.3k
30
votes
9
answers
10k
views
Diophantine equation with no integer solutions, but with solutions modulo every integer
It's probably common knowledge that there are Diophantine equations which do not admit any solutions in the integers, but which admit solutions modulo $n$ for every $n$. This fact is stated, for ...
- 10.3k
2
votes
0
answers
228
views
Ramanujan's theta functions and hook lengths?
Given an integer partition $\lambda\vdash n$ of $n$, one may associate a Young diagram $Y(\lambda)$ to it followed by a computation of hook length $h_{\square}$ for each cell $\square=(i,j)$ in $Y(\...
- 43.2k
4
votes
0
answers
97
views
"Convolving" a general Catalan with classical Catalan
Consider what is sometimes known as generalized Catalan sequence
$$\mathcal{{\color{red}C}}_{a,b}:=\frac{2b+1}{a+b+1}\binom{2a}{a+b}.$$
Observe that $\mathcal{{\color{red}C}}_{n,0}$ reduces to the ...
- 43.2k
2
votes
1
answer
305
views
Level spacing statistics for primes
In the preprint "Level Spacing Statistics for Primes", we have found some patterns of prime spacings, which may provide new insights on the distribution of primes:
We would like to know ...
- 149
2
votes
1
answer
289
views
Known upper bounds for $\sum_\limits{p\equiv a \pmod q, p\leq x}\frac{1}{p}$
I am trying to deduce an upper bound for $\sum_\limits{p\equiv a \pmod q, p\leq x}\frac{1}{p}$ where $p$ is a prime and $(a,q)=1$. Using Theorem 1 of Carl Pomerance's article on amicable numbers (...
- 309
14
votes
3
answers
2k
views
Norms in quadratic fields
This should be well-known, but I can't find a reference (or a proof, or a counter-example...). Let $d$ be a positive square-free integer. Suppose that there is no element in the ring of integers of $\...
- 38k
6
votes
2
answers
1k
views
Reference for universal elliptic curves
I've seen the following sentence come up in a few papers:
Consider the modular curve $Y_1(N)$ and let $E$ be the universal elliptic curve over $Y_1(N)$.
This comes up in Deligne's construction of ...
- 1,949
2
votes
2
answers
308
views
Reference for zero sum estimates of Dirichlet L functions
Let $\chi$ be a primitive character mod $p$ (prime) and $\rho = \beta + i \gamma$ be a non-trivial zero of $L(s, \chi)$.
I am reading a paper by Ihara and Murty where they use following estimate :
$\...
- 41
7
votes
1
answer
723
views
Alternate algorithms for Chinese remainder theorem
I was teaching Discrete this semester and set the students loose on a system of linear congruences. One of them came up with this solution. Say $$ x \equiv 1 \textrm{ mod } 3 $$ $$ x \equiv 3 \textrm{ ...
- 525
5
votes
1
answer
386
views
Is there a simple expression for $\sum_{k =1}^{(p-3)/2} \frac{1\cdot 3\cdots (2k-1)}{2\cdot 4 \cdots 2k\cdot(2k+1)} \bmod p$?
Let $p \equiv 1 \pmod 4$ be a prime and $E_n$ denote the $n$-th Euler number. While investigating $E_{p-1} \pmod{p^2}$ I have encountered this summation (modulo $p$)
\begin{align*}
\sum_{k =1}^{\frac{...
- 707
7
votes
1
answer
707
views
How hard is it to find the first layer of this basic $\mathbb{Z}_p$-extension?
$\DeclareMathOperator\Gal{Gal}$Let $p$ be a prime number and $\zeta_{p^n}$ be a primitive $p^n$-th root of unity. We know that there is a unique subfield $\mathbb{Q}_1$ of $\mathbb{Q}(\zeta_{p^2})$ ...
- 707
1
vote
0
answers
98
views
Existence of countable dense normal subgroups of global Galois group
Let $K$ be a number field, $S$ a finite set of primes of $K$ and $ G_{K,S} $ the Galois group of the maximal extension of $ K $ (inside a fixed algebraic closure of $K$) unramified outside $ S $. In ...
- 863