All Questions
Tagged with reference-request nt.number-theory
1,409 questions
15
votes
1
answer
970
views
Counting lattice points inside a three-dimensional ellipsoid
I want to answer the following simple question:
Given a three-dimensional ellipsoid defined by $Q(x, y, z) \leq Z$ for a positive definite quadratic form $Q$, how many lattice points in $\mathbb{Z}^...
- 7,347
1
vote
1
answer
505
views
Fermat two square and Lagrange four square via Hardy-Littlewood circle method [closed]
Fermat two square: An odd prime p is expressible as
${\displaystyle p=x^{2}+y^{2},\,}$
with $x, y$ integers, if and only if
${\displaystyle p\equiv 1{\pmod {4}}.}$
Lagrange four square: Every ...
- 841
0
votes
0
answers
122
views
References for the extension of Euler's phi function to number rings
Can anyone post a self-contained reference concerning the extension of the Euler phi function to number rings and its basic properties (reminiscent of those that the classic Euler phi function has)? ...
- 123
4
votes
0
answers
313
views
Four-distance problem
A Problem due to Steinhaus is the following: do there exists a point in the plane at rational distance from the corners of the unit square?
Please give me a reference if this problem is solved.
- 367
3
votes
1
answer
719
views
Looking for concise books on automorphic L-functions, Eisenstein series on adelic homogeneous spaces
Indeed I am now trying to read a series of papers written by Einsiedler, Lindenstrauss, Michel and Venkatesh that study distribution of periodic torus orbits on homogeneous spaces. They make heavy ...
- 73
12
votes
2
answers
552
views
On the independence of lower and upper asymptotic and Banach densities
Given a set $X \subseteq \mathbf N^+$, denote by $\mathsf{d}_\ast(X)$ and $\mathsf{d}^\ast(X)$, respectively, the lower and upper asymptotic (or natural) density of $X$, viz. $$\mathsf{d}_\ast(X) := \...
- 10.5k
1
vote
0
answers
73
views
Reference request ( Conductor of Galois representation associated to Dirichlet character)
(Sorry for my poor english...)
Let $\chi$ be a Dirichlet character modulo $N$ and $\Psi_{\chi}$ be an one dimensional Galois representation such that
\begin{equation}
\Psi_{\chi}: \text{Gal}(\...
- 661
17
votes
0
answers
892
views
An elementary proof that, for every fixed $n \in \mathbf N^+$, there are infinitely many primes $\equiv -1 \bmod n$
This morning, I made a comment to a comment to a question of Ayman Moussa, only to point out that, among many others, there is an elementary proof of Dirichlet's theorem on the existence of infinitely ...
- 10.5k
32
votes
1
answer
4k
views
How should a number theorist learn a modest amount of algebraic geometry?
A little bit vague, but I hope useful for the entire community.
I am, by training, an analytic number theorist. I have managed to learn some algebraic geometry, by reading parts of Silverman's ...
3
votes
1
answer
561
views
The abscissa of convergence of the real part of a Dirichlet series
Let $L(s)=\sum_{n\ge1}\frac{a(n)}{n^s}$ be a Dirichlet series with a finite abscissa of convergence $\sigma_c.$ My question is the following :
On what condition the abscissa of convergence of $\sum_{...
- 93
7
votes
1
answer
508
views
What is the exact statement about uniform boundedness of rational points on curves of genus greater than one? Singular points can be unbounded
According to several sources, it is conjectured (or at least believed)
that the rational points of curves over the rationals of genus $g > 1$
are uniformly bounded by $g$. E.g. here p. 1.
Assuming ...
- 25.4k
3
votes
1
answer
189
views
The Golay-Rudin-Shapiro sequence as “Hankel transform”
Let ${\left( {{a_n}} \right)_{n \geqslant 0}}$ be a sequence of real numbers and ${H_n} = \det \left( {{a_{i + j}}} \right)_{i,j = 0}^{n - 1}$ the $n-$th Hankel determinant. The sequence ${\left( {{...
- 5,622
9
votes
2
answers
1k
views
runs of consecutive non squarefree integers
This question gained no attention at Math SE.
Call a sequence of $k$ consecutive naturals squary if each one of them is divided by a square > 1. The Chinese Remainder theorem trivially guarantees us ...
- 13.4k
2
votes
2
answers
257
views
Is this variant on set partition explored?
Let $[n]=\{1,2,\dots,n\}$ and fix $r\in[n]$. Define the set $\mathcal{B}_{n,r}$ as the set of all set partitions of $[n]$ into disjoint non-empty blocks such that each block $B$ satisfies:
$\,\,\,\,\,...
- 43.2k
0
votes
0
answers
116
views
Reference request for bounds of $n$-th composite
Motivation
I will briefly elaborate here my motivations for asking the question. If you are not interested in it then please go to the questions.
Recently during trying to understand and prove the ...
user57432
2
votes
0
answers
154
views
Categorical representations of absolute Galois groups
I am looking for interesting examples of categorical representations of absolute Galois groups of arithmetic fields. Pointers to the literature would be appreciated.
user127738
2
votes
1
answer
191
views
Synchronised $\beta$-shifts
I have been reading some papers recently, in particular, Blanchard's paper $\beta$-expansions and symbolic dynamics which state that a $\beta$-shift $S_{\beta}$ is a synchronised shift if and only if ...
4
votes
1
answer
372
views
How close do partitions get to perfect squares?
This comes purely out of curiosity and experiments. I'm not sure if the literature has any coverage.
Let $p(n)$ be the number of integer partitions of $n$. Then, we have the well-known generating ...
- 43.2k
10
votes
1
answer
1k
views
Rank of Elliptic Curves
Recently, I have heard of some heuristics that would suggest that the rank of elliptic curves are bounded (specifically in the congruent number family). I always though that the best way to prove ...
- 103
5
votes
1
answer
444
views
Mazur's Galois Deformations paper for non-residually irreducible case
In Barry Mazur's paper introducing Galois deformations, he hints at having a general theory for representations which are not residually Schur, but with more complicated statements. Does anyone know ...
- 2,429
2
votes
0
answers
519
views
Good place to learn about arithmetic schemes?
Where is a good place to learn about arithmetic schemes? There is discussion in Eisenbud-Harris's book The Geometry of Schemes (and also Mumford's red book) and I hear that there is discussion in Liu'...
- 21
5
votes
0
answers
317
views
Elliptic curve sequences needed for universal forgery
Elliptic Curve Digital Signature Algorithm (ECDSA) admits universal forgery (UF) if the Attacker can solve the equation
$$z=\frac{f_{k-1}(x,y)f_{k+1}(x,y)}{f_{k}(x,y)^2},$$
where $k$ is unknown, $f_{k}...
- 12.3k
13
votes
2
answers
931
views
Did Erdős publish his proof of the multiplicative version of the Erdős-Turán conjecture?
I read in an article of Erdős ("Extremal problems in number theory") that he had a proof of the multiplicative version of the Erdős-Turán conjecture. The statement of this theorem is
Let $a_1 < ...
- 85.6k
12
votes
1
answer
3k
views
Number theory underlying Euler's theory of music
I've recently been studying Euler's theories on music, and I came across Euler's concept of gradus suavitatis or 'degree of pleasure' of a rational number representing the ratio of two tones. (I found ...
- 3,359
41
votes
0
answers
2k
views
What does the theta divisor of a number field know about its arithmetic?
This question is about a remark made by van der Geer and Schoof in their beautiful article "Effectivity of Arakelov divisors and the theta divisor of a number field" (from '98) (link).
Let ...
- 2,029
6
votes
1
answer
302
views
Reference request to proof that H$^2(\Gamma, \mathbb{Q}/\mathbb{Z}) = 0$
Does anyone maybe have a reference to the proof of the following result by Tate?
Let $\Gamma$ be the absolute Galois group of the rationals. Then the second cohomology group (for trivial $\Gamma$-...
- 63
2
votes
0
answers
100
views
Function equation over general number fields
Let $\chi$ be a Hecke character on a number field $k$, where could I find a precise reference for the function equation of the $GL(1)$ L-functions
$$L(s, \chi)?$$
I only find references for the case ...
- 937
5
votes
1
answer
243
views
How to find a set of integers that satisfy certain linear conditions
Suppose I have a sequence of non-negative integers $J=\{j_1,j_2,\ldots,j_n\}$
and want to find (if possible) a set of integers $I=\{0=i_1<i_2< \cdots < i_m\}$
such that $j_t$ counts the ...
- 51
14
votes
3
answers
1k
views
How many sequences of rational squares are there, all of whose differences are also rational squares?
After commenting on a
question
of Joseph O'Rourke's, I thought it interesting that a number theory result (artihmetic progressions of rational squares cannot be arbitrarily long) had applications to ...
- 13k
3
votes
0
answers
201
views
Growth Rate of the Square-Free Part
In the course of considering this Diophantine equation, I convinced myself that the following question is interesting:
If $n$ is large, must it be the case that the square-free part of $2^n-1$ is ...
- 153
3
votes
2
answers
306
views
Asymptotics for the number of digits of the ratio of binomial coefficients
Let $a$ and $b$ be distinct positive real numbers. Let $(a_n)$ and $(b_n)$ be sequences of natural numbers such that $a_n\sim an$ and $b_n\sim bn$. All the limit relations here are for $n\to\infty$. ...
- 128k
5
votes
0
answers
775
views
A conjecture about the degrees of special polynomials
Define the congruence "modulo m" on exponential Taylor series as
$$
\sum_{n=0}^\infty \frac{a_n}{n!}x^n \equiv \sum_{n=0}^\infty \frac{b_n}{n!} x^n \mod m \iff \forall n: \frac{a_n-b_n}{m}\in \mathbb{...
- 237
5
votes
1
answer
278
views
Are there primitive quartic CM fields whose norms of units give all totally positive units of the real quadratic subfield?
Let $K$ be a primitive (i.e. not biquadratic) quartic CM-field. That is we have $[K:\mathbb{Q}]=4$ and let $K_0=\mathbb{Q}(\sqrt{d})$ be the totally real quadratic subfield, here $d> 1$, that is we ...
- 1,025
4
votes
1
answer
647
views
Did Erdős prove there are two primes $4a+1, 4b+3$ between between $n$ and $2n$?
http://mathworld.wolfram.com/ChoquetTheory.html
Is the claim in the link true? Here's the reference given there:
https://www.renyi.hu/~p_erdos/1934-01.pdf
Erdős proved that there exist at least one ...
- 193
17
votes
1
answer
882
views
Reference Request: Conductors of Twists of Hyperelliptic Curves
It is my understanding that if I twist a hyperelliptic curve of genus 2 whose Jacobian has conductor $N$ by a prime $p$ with $p\nmid N$, that the conductor of the Jacobian of the twist is expected to ...
- 715
8
votes
1
answer
571
views
Subsets of [1..N] with no three-term arithmetic progressions and no large gaps
Let S be a subset of [1..N] containing no three-term arithmetic progression, and let h(S) be the size of the largest gap between two consecutive elements of S. By Roth's theorem, h(S) has to grow ...
- 19.2k
11
votes
1
answer
2k
views
Where in the literature does the anticyclotomic $\mathbf{Z}_p$-extension of an imaginary quadratic field first appear?
If $K$ is an imaginary quadratic field, then the $\mathbf{Z}_p$-rank of $K$ is $2$, meaning that the Galois group of the compositum of all the $\mathbf{Z}_p$-extensions of $K$ in an algebraic closure $...
- 3,380
7
votes
2
answers
1k
views
What is a random number? (poll experiment) [closed]
Imagine the following experiment: you wait say at a subway exit, and ask everyone passing "please tell me a number" (positive integer, of course). You do this day after day, until you reach say 1M ...
- 1,363
4
votes
0
answers
190
views
Restricted Iwasawa theory
Let $p$ be a prime number, let $K$ be a number field, and let $L$ be a Galois extension of $K$ such that the Galois group $\Gamma$ of $L$ over $K$ is (continuously) isomorphic to the (additive) group $...
- 11.3k
6
votes
1
answer
674
views
Good references for K-theory of modular curves?
The title says it. I am looking for a good exposition on the K-theory of the curves $X_{i}(N)$, $Y_{i}(N)$, where $i\in\{0,1\}$.
I have some background in $K$-theory and also some background in ...
- 1,805
8
votes
2
answers
532
views
Integer numbers of the form $m = x^n + y^n$
First of all, I am no number theorist, so this question may be a little dummy.
The two squares theorem imply that $m = x^2 + y^2$ for some (possible zero) integer numbers $x,y$ iff $m$ factors as $m = ...
- 800
6
votes
1
answer
360
views
Friable Numbers In Short Intervals: Density Estimates?
I am hoping for explicit numerical estimates like the following sample (with made up numbers, though it might be true): for every $n \gt 10^6$ and every $b$ with $b^2 \lt n \lt b^3$, the number of ...
- 13k
7
votes
1
answer
468
views
Vinogradov's method for sums of more than three primes
In Hardy-Littlewood's 1923 paper "Some problems of 'Partitio Numerorum' III" it is proven, assuming a weak version of GRH (namely that there is $\varepsilon>0$ s.t. all zeroes of $L(s,\...
- 825
6
votes
3
answers
939
views
Uniformly distributed sequence in $\mathbb{R}$
We say that a sequence $(x_n)_{n=1}^\infty \subseteq \mathbb{R}$ is "uniformly distributed in $[a,b]$", with $a < b$, if $(x_n)_{n=1}^\infty \cap [a,b] \neq \varnothing$ and
$$\lim_{N \to \infty} \...
- 61
3
votes
2
answers
706
views
Expository articles on Algebraic Number Theory
I am about to start learning Algebraic Number Theory and thus was looking for some expository articles on this subject. So far I have found two such articles:
Dickson, L. E.. (1917). Fermat's Last ...
- 422
9
votes
3
answers
3k
views
Elliptic Curves over Global Function Fields
I am currently thinking about a problem, and I feel that by knowing more about elliptic curves over extensions of $\mathbb{F}_q(T)$, for $q$ a power of $p$ say, might lead to insight. I am also ...
- 831
13
votes
1
answer
334
views
Elementary prime-generating sequences
A student of mine keeps coming again and again and telling "I've found a formula $n\mapsto f(n)$ giving all primes" or sometimes "infinitely many primes", where $f$ is a classical function (I mean ...
- 1,980
11
votes
2
answers
2k
views
Good books on Dirichlet's class number formula
I refrained from asking the technical questions; maybe everyone didn't like my attitude. At least, help me finding good books.
Can anyone suggest a good book that gives a complete reference to "...
Trust God
0
votes
0
answers
115
views
What is known about the lower bound for the integers $n$ for which $n$ minus the first $k$ odd primes are $k$ composite numbers?
Question edited in view of the comments below
By Yamada's paper we can conclude that if $n>e^{e^{36}}$ be an even number then it can always be written as the sum of a prime and a semi-prime.
My ...
user57432
2
votes
0
answers
76
views
Is there an estimate available for a sum of the form $\sum_{\mathbf{x} \equiv \mathbf{a} (H) } \mu^2(x_1 x_2)$
I am interested in a sum of the shape
$$
\sum_{ \substack{ 1 \leq x_1, x_2 \leq B\\
\mathbf{x} \equiv \mathbf{a} (H) } } \mu^2(x_1 x_2).
$$
I figured it must have been considered before, but I have ...
- 3,625