All Questions
Tagged with reference-request nt.number-theory
1,408 questions
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
17
votes
2
answers
3k
views
Some unpublished notes of Hofstadter
I'm looking for some unpublished notes called "Eta Lore," which are apparently related to a talk Douglas Hofstadter first gave at the Stanford Math Club in 1963. I know these notes exist because they'...
17
votes
2
answers
2k
views
Is every odd positive integer of the form $P_{n+m}-P_n-P_m$?
I am looking for a comment, reference, remark, or proof of three conjectures as follows:
Conjecture 1: Let $x$ be an odd positive integer. Then there exist two integers $n, m \ge 2$ so that $$x=P_{n+...
- 1,776
17
votes
4
answers
10k
views
Prime/undecomposable matrices
Prime matrices as defined in the following paper Prime matrices P. F. RIVETT AND N. I. P. MACKINNON carry over many properties of factorization as in natural numbers to matrices over the field of ...
- 2,855
17
votes
1
answer
701
views
Combinatorics problem about sum of natural numbers
Following combinatorics problem is claimed to be an open problem in "The Princeton Companion to Mathematics" (pp. 6)
Let $a_1,a_2,a_3,...$ be a sequence of positive integers, and suppose that each $...
- 422
17
votes
1
answer
1k
views
Catalan's constant fast convergent series
NOTE. UPDATE 2 introduces proven series for Catalan's constant that is possibly the fastest currently known.
Working with some conjectured continued fractions that were published here, I have found ...
- 2,836
17
votes
1
answer
882
views
Reference Request: Conductors of Twists of Hyperelliptic Curves
It is my understanding that if I twist a hyperelliptic curve of genus 2 whose Jacobian has conductor $N$ by a prime $p$ with $p\nmid N$, that the conductor of the Jacobian of the twist is expected to ...
- 715
17
votes
1
answer
1k
views
Serre and Tate's conjectures on étale cohomology
In the appendix of Serre and Tate "Good Reduction of Abelian Varieties" [Annals of Mathematics 88 (1968), 492-517], the authors make the following conjectures.
Suppose that $X$ is a smooth proper ...
- 1,832
17
votes
0
answers
367
views
Average value of j-invariant at infinity
Let $\xi\in\mathbb{R}$ and consider the average value (with respect to hyperbolic length) of the $j$-invariant ($j(z)=q^{-1}+744+196884q+\ldots$, $q=e^{2\pi iz}$) along a geodesic aimed at $\xi$:
$$
\...
- 609
17
votes
0
answers
891
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
16
votes
1
answer
2k
views
On J. T. Condict's Senior Thesis on Odd Perfect Numbers
I am trying to locate a copy of J. T. Condict's senior thesis on odd perfect numbers:
J. Condict, On an odd perfect number's largest prime divisor, Senior Thesis, Middlebury College (1978).
I am ...
16
votes
3
answers
4k
views
Is it known that the Collatz-like sequence with 7n+1 diverges to infinity starting with 7?
In this question I was wondering if the $3$ in the Collatz conjecture is arbitrary, and when I wrote that question I tried to change to $7n+1$ starting with the seed number $7$, the sequence appears ...
- 541
16
votes
4
answers
1k
views
Reference for a linear algebra result
I asked the following question (https://math.stackexchange.com/questions/1487961/reference-for-every-finite-subgroup-of-operatornamegl-n-mathbbq-is-con) on math.stackexchange.com and received no ...
- 26.9k
16
votes
1
answer
4k
views
Order of magnitude of $\sum \frac{1}{\log{p}}$
Question: What is the order of magnitude of the following sum?
$$ \sum_{\substack{p<n\\\text{$p$ prime}}} \frac{1}{\log{p}} $$
Additional information: Since
$$ \sum_{\substack{p<n\\\text{...
- 3,004
16
votes
1
answer
1k
views
Has this strong number theoretic conjecture of Euler been proved, and where could I find such a proof?
Polya cites this work of Euler as an example of a conjecture which Euler considered impossible to doubt, and yet still needing a demonstration.
It is on pages 90-98 of "Induction and Analogy in ...
- 179
16
votes
1
answer
2k
views
Reference to a Don Zagier result and the congruent number problem
I was looking for a reference/explanation as to how Don Zagier managed to find the side lengths of a rational right triangle with area 157. There have been many literature references to the fact that ...
- 163
16
votes
1
answer
1k
views
On (a generalization of) the Gauss Circle Problem
Most (if not all) references I read about the Gauss Circle Problem that proves a bound below $O(R^{2/3})$ reduces the GCP to the Dirichlet Divisor Problem by the well known expression of $r_2(n)$, the ...
- 5,169
16
votes
1
answer
705
views
Connection between Bernoulli numbers and Riemann-Siegel theta function?
I have come across a strange approximation for the Riemann-Siegel theta function involving the Bernoulli numbers - namely that
$$\frac{1}{2} \log \left| B_{2 n}\right|\approx \vartheta (2n)\ ,\quad n ...
- 1,903
16
votes
1
answer
2k
views
Reference for the `standard' Tate curve argument.
I'd like a reference (e.g. something published somewhere that I can cite in a paper) for the proof of the following:
Let $E$ be an elliptic curve over $\mathbb Q$ with minimal discriminant $\Delta$...
- 10.5k
16
votes
1
answer
357
views
Galois cohomology for non-Galois extensions
If $L/K$ is a Galois extension with group $G$ then we can consider $H^*(G;L^\times)$. This is useful in algebraic number theory, and there are many results about it.
Now let $L/K$ be a finite ...
- 56.9k
16
votes
0
answers
11k
views
Deligne's letter to Jean-Pierre Serre
I'm looking for another letter of Pierre Deligne, this time to Jean-Pierre Serre (from around 1974 I think), in which he proves that the Galois representation associated to a certain Hecke eigenform ...
15
votes
4
answers
3k
views
Collecting alternative proofs for the oddity of Catalan
Consider the ubiquitous Catalan numbers $C_n=\frac1{n+1}\binom{2n}n$. In this post, I am looking for your help in my attempt to collect alternative proofs of the following fact: $C_n$ is odd if and ...
- 43.2k
15
votes
4
answers
3k
views
No Tonelli or Fubini
Whenever we can interchange summation (perhaps due to Tonelli-Fubini), good things happen. Otherwise, one has to struggle evaluating double sums in just one way, because the alternative results in a ...
15
votes
3
answers
1k
views
Unit fraction, equally spaced denominators not integer
I've been looking at unit fractions, and found a paper by Erdős "Some properties of partial sums of the harmonic series" that proves a few things, and gives a reference for the following theorem:
$$\...
- 305
15
votes
5
answers
2k
views
Zeros of the derivative of Riemann's $\xi$-function
The Riemann xi function $\xi(s)$ is defined as
$$
\xi(s)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s).
$$
It is an entire function whose zeros are precisely those of $\zeta(s)$.
Since $\xi$ is real ...
- 11.1k
15
votes
2
answers
1k
views
Evaluating the integral $\int_{1}^{\infty}\frac{\{u\}}{u^{2}}\left(\log u\right)^{k}du.$
I am trying to find a formula for the following integral for non-negative integer $k$:
$$\int_1^{\infty}\frac{\{u\}}{u^{2}}\left(\log u\right)^{k}du.$$
My first thought was to use the formula $$\...
- 11.4k
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
15
votes
4
answers
575
views
Are all partial consecutive harmonic subsums distinct?
Let $b \gt a \geq 0$ be integers, and as elsewhere let $H_n$ be $\sum^n_{i=1} 1/i$. A partial consecutive harmonic subsum is a number $H(a,b)$ of the form $H_b - H_a$ (with $ H_0=0$). If $c=a$ and $...
- 13k
15
votes
3
answers
1k
views
Does anyone remember what happened to the experimental search for polynomial identities for $\pi$?
So a while back I was on the internet and had encountered a website containing an experimental search for identities for $\pi$. My memory was that the page belonged to either Jonathan Sondow or ...
- 2,689
15
votes
1
answer
1k
views
If the tensor product of two representations are crystalline, are the original representations crystalline?
Let $K$ be a finite extension of the $p$-adic numbers. Suppose that $V$ and $W$ are two (finite dimensional, $p$-adic) continuous representations of $G_K$. Suppose that $V \otimes W$ is crystalline. ...
user631
15
votes
4
answers
2k
views
Are any good strategies known for Erdos-Turan conjecture on additive bases of order two?
The following problem can become a bit of an obsession. I'm curious if there are any serious strategies for attacking it. The problem is a certain Erdos-Turan conjecture.
Let $ B \subseteq {\mathbb ...
- 7,067
15
votes
1
answer
943
views
BSD conjecture for rank 1 elliptic curves
Let $E/\mathbb{Q}$ be an elliptic curve. The weak Birch and Swinnerton-Dyer conjecture predicts that
$$\text{ord}_{s=1}L(E, s)=\text{rank} E(\mathbb{Q}).$$
Thanks to the work of Gross-Zagier and ...
- 253
15
votes
1
answer
1k
views
A p-adic analogue for a formula of Riemann?
This might be naive question but I was wondering whether a p-adic analogue of the following (shockingly) beautiful formula $$\zeta(s)\Gamma(s) = \int_0^\infty \frac{t^{s-1}}{e^t-1} dt$$ (vaild for $\...
- 2,029
15
votes
1
answer
954
views
Funktorialität in der Theorie der automorphen Formen
In 2010 Langlands wrote an article with the title Funktorialität in der Theorie der automorphen Formen: Ihre Entdeckung und ihre Ziele. On the IAS website, he says that
This note ... was written ...
- 18.7k
15
votes
1
answer
969
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
15
votes
0
answers
673
views
Exposition of Drinfeld's proof of function field Langlands for GL(2)
I know, or think I know, the vague outline of the proof: the Galois-to-automorphic direction is "classical," i.e. follows from converse theorems due to Grothendieck et al., and for the ...
- 311
15
votes
0
answers
591
views
For how many primes does an elliptic curve over a totally imaginary field have supersingular reduction?
An elliptic curve over a finite field, $k$, of characteristic p is called supersingular if it has no $p$-torsion over $k^{\mathrm{alg}}$, or equivalently, if $\mathrm{End}(E)$ is an order in a ...
- 605
15
votes
0
answers
476
views
Any references on zeta-function like sums of inverse determinants over lattices of matrices?
I'm sorry for the title, it was little difficult to phrase..
Let us consider a matrix lattice $L\subset M_n(\mathbb{C})$. By this I mean a discrete additive group in $M_n(\mathbb{C})$. Let us ...
- 201
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:
$$\...
- 43.2k
14
votes
4
answers
4k
views
Deligne's letter to Piatetskii-Shapiro from 1973
Could anyone point me to a place where I could find Deligne's letter to Piatetskii-Shapiro from 1973? It is cited for example in Berkovich's "Vanishing cycles for formal schemes II".
14
votes
3
answers
2k
views
Norms in quadratic fields
This should be well-known, but I can't find a reference (or a proof, or a counter-example...). Let $d$ be a positive square-free integer. Suppose that there is no element in the ring of integers of $\...
- 38k
14
votes
4
answers
2k
views
Partitions-sum of divisors identity
A few years ago I first read about the marvelous Euler identity:
$\sum_{n\in\mathbb{N}}p(n)z^n=\prod_{k\geq1}\frac{1}{1-z^k}$,
where $p(n)$ is the number of partitions of $n$ ($p(0)=1$ by convention)...
- 2,479
14
votes
6
answers
10k
views
Frobenius number for three numbers
Given integers $a,b,c$ such that $\gcd(a,b,c) = 1$, it is well known that there exists only a finite set of numbers $n$ such that $n$ is not expressible as $ax+by+cz$ for non negative integers $x$,$y$,...
- 3,463
14
votes
4
answers
3k
views
Jacobi's theorem on sums of two squares (reference request)
One of Jacobi's theorems states that the number of representations of a positive integer $n$ as a sum of two squares of integers equals
$$4(d_1(n)-d_3(n)),$$
where the function $d_i$ counts the number ...
- 3,062
14
votes
3
answers
1k
views
How many sequences of rational squares are there, all of whose differences are also rational squares?
After commenting on a
question
of Joseph O'Rourke's, I thought it interesting that a number theory result (artihmetic progressions of rational squares cannot be arbitrarily long) had applications to ...
- 13k
14
votes
1
answer
749
views
reference for: no finite set of positive (integer) binary quadratic forms represents all primes
This recent question asks for a set of forms (binary quadratic) representing all primes.
Set of quadratic forms that represents all primes
When the question was asked on MSE last month
https://math....
- 25.7k
14
votes
1
answer
495
views
powered partition function generator: 1/2 of them are zeros?
Ramanujan delivered his famous congruences
$$p(5n+4)\equiv_50, \qquad p(7n+5)\equiv_70, \qquad p(11n+6)\equiv_{11}0$$
for the integer partitions with generating function $F(x)=\prod_{k=0}^{\infty}\...
- 43.2k
14
votes
4
answers
3k
views
Fourier decay rate of Cantor measures
For $0<\theta<\frac{1}{2}$, denote by $C_\theta$ the Cantor set with dissection ratio $\theta$, i.e. the Cantor set obtained from dissection parttern $(\theta, 1-2\theta,\theta)$. It is known ...
- 981
14
votes
1
answer
1k
views
Transcendence of $\Gamma(1/3), \Gamma(1/4)$
This is a re-post from MSE as I did not get even a single comment there.
Wikipedia mentions that the transcendence of $\Gamma(1/3), \Gamma(1/4)$ was proved by G. V. Chudnovsky. Does anyone have a ...
- 2,249
14
votes
2
answers
858
views
References for particular topics related to Langlands
I have never really concentrated on Langlands, which explains my poor level of understanding of it. But I have read quite a few introductory papers related to Langlands, and to the circle of ideas ...
