All Questions
Tagged with reference-request nt.number-theory
204 questions
8
votes
3
answers
2k
views
Numerical evaluation of the Petersson product of elliptic modular forms
It is known how to compute the Fourier expansion of elliptic modular forms using modular symbols, and it is known how to get numerical evaluations of $L$-functions of various type ; it's possible to ...
7
votes
1
answer
313
views
Prescribed values for the uniform density
Strauch & Tóth [1] Georges Grekos [3][4] showed that for any choice of upper and lower density, there is some subset of $\mathbb{N}$ with the chosen densities, provided the lower is no more than ...
- 9,114
7
votes
3
answers
826
views
Analytic equivalents for primes in arithmetic progressions
By way of context: it is known that the prime number theorem $\pi(x) \sim x/\log x$ is (nontrivially) equivalent to the statement that $\zeta(s)$ does not vanish on the line $\Re s=1$.
I would like ...
- 12.8k
7
votes
1
answer
1k
views
What would be the consequences of $\displaystyle{\lim\inf_{n\to\infty}p_{n+k}-p_{n}\sim k\log k}$?
The question is in the title: what would be the number theoretic consequences if we managed to establish the conjectured asymptotic equality $\displaystyle{\lim\inf_{n\to\infty}p_{n+k}-p_{n}\sim k\log ...
- 6,984
7
votes
1
answer
488
views
Ref. request: Additive probability measure on $\mathcal P({\bf N})$ supplies subset of $\mathbf R$ without Baire property
ZFC proves, among the other things, the existence of a (finitely) additive probability measure $\theta: \mathcal P(\mathbf N) \to \mathbf R$ on the power set of $\mathbf N$ such that $\theta(X) = 0$ ...
- 10.5k
6
votes
2
answers
804
views
Must Mersenne numbers be divisible by arbitrary large primes with exponent one?
Let $M_n$ denote the Mersenne numbers $M_n=2^n-1$.
As $n$ varies, must $M_n$ be divisible by arbitrary large prime $p$
with exponent one, i.e. $p \mid M_n, p^2 \nmid M_n$?
In other words, must the ...
- 25.4k
5
votes
0
answers
1k
views
Differential Galois number theory
Following https://math.stackexchange.com/questions/635659/can-the-error-term-involved-in-the-pnt-be-expressed-in-a-galois-theoretic-framew?noredirect=1#comment1341143_635659, I vainly tried to find ...
- 6,984
5
votes
1
answer
434
views
consecutive prime gaps and explicit bound
I am aware of the theorem that $p_{n+1}- p_n \leq n^{0.525}$ which is true for all sufficiently large numbers due to Baker, but if i want to make the implicit "for all sufficiently large numbers" ...
user95470
4
votes
1
answer
493
views
Counting number of points on a lattice in a hypercube
Suppose I have a lattice $\Lambda \in \mathbb{R}^n$. Let $X_i >0$ for $i=1,..,n$. I am interested in some references regarding counting number of points of $\Lambda$ inside $[-X_1, X_1] \times \...
- 3,625
4
votes
2
answers
593
views
Squares in Lucas sequences
Good night, everyone!
According to a celebrated result by J. H. Cohn, the only perfect squares in the Fibonacci sequence are $F_{0}=0$, $F_{1}=F_{2}=1$, and $F_{12}=144$. It is also known that the ...
- 87
4
votes
1
answer
700
views
Total sum of characters of the symmetric group $\frak{S}_n$
Let $\chi_{\mu}^{\lambda}$ denote a value of an irreducible character of the symmetric group $\frak{S}_n$, where $\mu, \lambda\vdash n$. When $\mu=(n)$, then it's known that
$$\sum_{\lambda\vdash n}\...
- 43.2k
60
votes
1
answer
6k
views
What were the main ideas and gaps in Yoichi Miyaoka's attempted proof (1988) of Fermat's Last Theorem?
Out of sheer curiosity I have been reading Stewert and Tall's "Algebraic Number Theory and Fermat's Last Theorem" (2001). As it contains various bits of history, I found out to my own shame that I was ...
- 7,127
47
votes
1
answer
3k
views
Which small finite simple groups are not yet known to be Galois groups over Q?
The subject line pretty much says it all. To expand just a little bit:
1) What is the smallest simple group that is not yet known to occur as a Galois group over $\mathbb{Q}$? (Variants: not known ...
- 65.4k
43
votes
3
answers
3k
views
Is this integral representation of $\zeta(2n+1)$ known?
Background: I'm an undergraduate at an institution with no researchers in analytic number theory, and no ties to the analytic number theory community. I believe I have found what is, as far as I can ...
- 383
42
votes
4
answers
4k
views
Are these fast convergent series for $\log(2)$, $\log(3)$ and $\log(5)$ already known and proven?
Now that some of the previously MSE formulae that I left here have been applied Dec.2023 to compute high precision record values ($10^{12}$ decimal digits) of trascendental constants $\Gamma(1/3)$ (Eq....
- 2,836
41
votes
2
answers
17k
views
Introductory text on Galois representations
Could someone please recommend a good introductory text on Galois representations? In particular, something that might help with reading Serre's "Abelian l-Adic Representations and Elliptic Curves" ...
34
votes
2
answers
3k
views
Shimura-Taniyama-Weil VS Grothendieck's dessins
When listening to the beautiful lectures by Gilles Schaeffer at
the SLC68, the following (perhaps crazy) question occurred to me:
did anyone attempt (succeed?) to combinatorially prove modularity of ...
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
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 ...
29
votes
5
answers
5k
views
Partial sums of multiplicative functions
It is well known that some statements about partial sums of multiplicative functions are extremely hard. For example, the Riemann hypothesis is equivalent to the assertion that $|\mu(1)+\mu(2)+\dots+\...
- 29k
27
votes
2
answers
2k
views
Etale site is useful - examples of using the small fppf site?
Edit: After the answers and comments, I'm hoping for a little bit of elaboration (in the comment to the answer below.) Also, question 2 was discussed here:
Points in sites (etale, fppf, ... )
There, ...
- 3,555
25
votes
1
answer
2k
views
The origin of Discrete `Liouville's theorem'
It is known that discrete Liouville's theorem for harmonic functions on $\mathbb{Z}^2$ was proved by Heilbronn (On discrete harmonic functions. - Proc. Camb. Philos. Soc. , 1949, 45, 194-206).
If ...
- 12.3k
24
votes
0
answers
1k
views
Exotic 4-spheres and the Tate-Shafarevich Group
The title is a talk given by Sir M. Atiyah in a conference with the following abstract:
I will explain a deep analogy between 4-dimensional smooth geometry (Donaldson theory)...
- 1,629
23
votes
1
answer
3k
views
A list of proofs of the Hasse–Minkowski theorem
I am currently doing a project in which I intend to include the most insightful possible proof of the Hasse–Minkowski theorem (also known as the Hasse principle for quadratic forms, among other names) ...
21
votes
2
answers
1k
views
Most squares in the first half-interval
It is well known that if $p$ is an odd prime, exactly one half of the numbers $1, \dots, p-1$ are squares in $\mathbb{F}_p$. What is less obvious is that among these $(p-1)/2$ squares, at least one ...
- 14.7k
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
19
votes
1
answer
2k
views
Legendre and sums of three squares
The Three-Squares-Theorem was proved by Gauss in his Disquisitiones, and this proof was studied carefully by various number theorists. Three years before Gauss, Legendre claimed
to have given a proof ...
- 32.6k
19
votes
1
answer
1k
views
Ehresmann's theorem over the $p$-adics
I am looking for a version of Ehresmann's theorem for analytic manifolds over the $p$-adic numbers $\mathbb{Q}_p$ or, more generally, local fields. I follow the conventions from Serre's book "Lie ...
- 21.3k
19
votes
1
answer
1k
views
Deligne's letter to Bhargava from March 2004
I am quite interested in moduli spaces for Rings and Ideals, a letter from Deligne to Bhargava is cited in Melanie Wood's thesis Moduli spaces for Rings and Ideals (pdf), studying the minimal free ...
- 461
18
votes
1
answer
562
views
Is special value of Epstein zeta function in 3 variables a period?
Kontsevich-Zagier's article "Periods" contains the following question
Is $\displaystyle \sum_{x,y,z \in \mathbb{Z}}' \frac{1}{(x^2+y^2+z^2)^2}$ an extended period?
($\sum'$ means we do not sum ...
- 809
18
votes
3
answers
6k
views
The multiplicative order of 2 modulo primes
Artin's Conjecture says that any positive integer, which is not a square, is a primitive root modulo infinitely many primes. Christopher Hooley gave in
Hooley, Christopher (1967). "On Artin's ...
- 25.5k
17
votes
1
answer
2k
views
Is there an analog of the Birch/Swinnerton-Dyer conjecture for abelian varieties in higher dimensions?
I am wondering if there is a multi-dimensional analog of the
Birch/Swinnerton-Dyer (BSD) conjecture.
The recent famous result inching toward resolution of that conjecture is:
Bhargava, Manjul, and ...
- 151k
17
votes
2
answers
1k
views
The GCD-matrix: generalizing a result of Smith?
Let $M$ be the $n\times n$ matrix, known as the GCD matrix, of entries $M_{ij}=\gcd(i,j)$. In the paper
H J S Smith, On the value of a certain arithmetical determinant, Proc. London Math. Soc. 7:208-...
- 43.2k
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
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'...
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
1k
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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
15
votes
1
answer
1k
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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
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
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
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
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
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
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
0
answers
644
views
Can there be arbitrarily many cubic fields unramified outside $\{p,\infty\}$?
Observe, trivially, that since quadratic fields correspond to rational integers modulo squares (viz. discriminants), there are (roughly about, but certainly at most) $2^{|S|+1}$ quadratic fields ...
- 13.8k
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)),$$
where the function $d_i$ counts the number ...
- 3,062
14
votes
4
answers
4k
views
Deligne's letter to Piatetskii-Shapiro from 1973
Could anyone point me to a place where I could find Deligne's letter to Piatetskii-Shapiro from 1973? It is cited for example in Berkovich's "Vanishing cycles for formal schemes II".
13
votes
2
answers
880
views
Arithmetic progressions modulo $p$ under the squaring map
I feel that the following problem should be known, but I'm not sure where to look for it.
Fix a real constant $\frac{1}{2} \ge \epsilon > 0$. For varying primes $p$, Let $A_p$ denote the set of ...
user631
13
votes
1
answer
760
views
Infinitely many integer solutions to $X^4+Y^4-18Z^4= -16$
We found infinitely many integer solutions to
$$X^4+Y^4-18Z^4= -16 \qquad (1)$$.
The interesting part in this diophantine equation is the sum of
the reciprocals of the degrees is $3/4 < 1$, which ...
- 25.4k
13
votes
1
answer
1k
views
Apéry's constant $\zeta(3)$ fastest convergent series
UPDATE Feb.02.2024
The series below, Eq.(3) for computing and Eq.(2) for verifying, were applied by Andrew Sun on Dec.22.2023 to get over $2\cdot10^{12}$ decimal digits and break the number of ...
- 2,836