All Questions
Tagged with reference-request nt.number-theory
1,408 questions
-6
votes
1
answer
488
views
Automorphisms of partitions [closed]
I would like to know whether the notion of automorphism of the set of partitions of a positive integer $n$ has been considered so far or not. To make things clearer, I say that a partition of $n$ in $...
- 6,984
1
vote
1
answer
398
views
Is the Cassels-Tate pairing defined for elliptic curves over function fields?
The Cassels-Tate pairing is typically defined for elliptic curves (or abelian varieties) over number fields, but it seems like it should be defined for elliptic curves over function fields as well. ...
user60591
3
votes
2
answers
1k
views
On figurate numbers
Do you know a text where I can find a definition of polygonal number that is both geometrically and operationally sound?
I've basically seen two ways in which this topic is approached in the ...
- 8,842
4
votes
0
answers
164
views
Two variants of the Littlewood-Offord theorem
I found two different looking things being called the Littlewood-Offord theorem,
If $\vec{a} \in \mathbb{R}^k \setminus 0$ and $t \in \mathbb{R}$ then there are $O(\frac{2^k}{\sqrt{k}})$ points $x \...
- 2,246
1
vote
1
answer
325
views
Error term for prime harmonic
What is known about the asymptotic behavior of
$$
f(x)=\sum_{p\le x}\frac1p-\log\log x-B_1?
$$
Of course by Mertens we know that $f(x)=o(1),$ but has more been proved (in terms of $O$ or $\Omega_\pm$,...
- 9,114
-1
votes
2
answers
2k
views
Any grammar for the language $L =a^p$, $p$ is prime number of $\mathbb{N}$
Any grammar for the language
$$L =a^p,\text{ $p$ is prime and }p\in \mathbb{N}?$$
Is such a grammar related to any question of number theory like RH or the conjecture of twin primes?
- 3,104
4
votes
1
answer
181
views
Asymptotics on the number of ways to pair off $\{1, 2, \dots, 2n\}$ into primes
Given $S = \{1, 2, \dots, 2n\}$, we can always pair off elements into $n$ pairs such that each sum to a prime. The proof of this fact is easy and follows from Bertrand's postulate.
Now, let $\gamma(n)...
- 141
11
votes
1
answer
782
views
Atkin--Lehner operators in Hida theory
Let $p$ be a prime, and $F$ a $p$-adic Hida family of ordinary modular forms (of some tame level $N \ge 1$). I'd like to know whether, for $q$ a prime factor of $N$, the actions of the Atkin--Lehner ...
- 37k
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
13
votes
1
answer
1k
views
Reference for: CM Hilbert Modular forms arise from Hecke characters
For classical modular forms, the correspondence between the form having CM by an imaginary quadratic field $K$ and it being induced from a Hecke character on $K$ is well-known. (Ribet's paper is a ...
- 659
1
vote
1
answer
255
views
Exponential sum estimates similar to the one for $\sum_p (\log p) e(p \alpha)$, but for different sequences
Obtaining a non-trivial estimate for $\sum_p (\log p) e(p \alpha)$ over the minor arcs is one of the estimates required for obtaining the ternary Goldbach for $n$ sufficiently large via the circle ...
- 579
2
votes
2
answers
338
views
Weak form of Brocard's conjecture
I ask this out of curiosity, motivated by a question asked by one of my students.
The Brocard's conjecture claims that there exist at least four prime numbers between $p_{i}^2$ and $p_{i+1}^2$, where ...
- 66.3k
2
votes
1
answer
169
views
Higher dimensional analogs of logarithmic density
For a set $A\subseteq \mathbb{N}$ its lower/upper asymptotic/logarithmic densities are given by
\begin{align*}
\underline{d}(A)=\liminf_{N\to\infty} \frac{|A\cap [1,N]|}{N},\\
\bar{d}(A)=\limsup_{N\to\...
- 1,658
1
vote
2
answers
245
views
Looking for a reference for a paper by Mordell
On page 384 of the book "Number Theory:Volume 1:tools and Diophantine Equations" by Henri Cohen there is reference to the fact that: "It has been proved by Schinzel, Mordell nd successors that such an ...
- 2,143
3
votes
1
answer
247
views
4-th order diophantine equation
I met in many places the equation
$(a^4-b^4)(c^4-d^4)=\square$
It is well known that this was investigated by Euler.
But I was unable to find the general solution of this equation. Could you please ...
- 185
8
votes
1
answer
190
views
Number of orbits of $\mathrm{SL}_2(\mathcal O_A)$ on $\mathbf P^1(A)$ when $A$ is a quaternion algebra
This is a reference request.
Let $A$ be an anisotropic quaternion algebra over $\mathbf Q$. Let $\mathcal O_A$ be a maximal order in $A$. Then $\mathrm{SL}_2(A)$ acts transitively on the right on ...
- 1,980
3
votes
0
answers
177
views
Looking for an appropriate reference(s) for two conjectures on odd perfect numbers
(I apologize in advance if this question is unsuitable for MO. If so, please let me know and I will migrate it to MSE.)
Let $\sigma(M)$ be the sum of the divisors of the positive integer $M$. For ...
5
votes
2
answers
471
views
Kronecker theorems on linear forms.
Dickson's History of the Theory of Numbers, vol II p. 94 refers to some theorems of Kronecker on linear forms:
...find integers $w$ and $w^\prime$
such that $aw+a^\prime w^\prime$ takes
a value ...
- 11.1k
0
votes
1
answer
598
views
Reference for a lemma on étale maps
The Stacks Project has the following really nice Lemma concerning étale maps of rings:
Let $A\rightarrow B$ be a finitely presented, étale morphism of rings. Then there exists a presentation
$$ B\...
- 1,838
1
vote
0
answers
405
views
Generalized Ramanujan's identity with hyperbolic cotangent
Three weeks ago I derived an identity, which generalizes Ramanujan's identity with hyperbolic cotangent. Don't you know is it original or not?
$$
\sum_{n=1}^\infty \frac{\coth(\pi n)}{n^3}=\frac{7\pi^...
- 237
2
votes
1
answer
540
views
Is there an english translation of Delignes "La conjecture de Weil pour les surfaces K3."?
The title is pretty self explanatory: I'm looking for an english translation of Delignes inventiones paper "La conjecture de Weil pour les surfaces K3."
Anyone know if such a thing exists?
Thanks!
- 2,824
6
votes
0
answers
426
views
Conjectured new primality test for Mersenne numbers
How to prove that this conjecture about a new primality test for Mersenne numbers is true ?
Definition: Let $M_{q}=2^{q}-1 , S_{0} = 3^{2} + 1/3^{2} , \ and: \ S_{i+1} = S_{i}^{2}-2 \pmod{M_{q}}$
...
- 171
9
votes
1
answer
1k
views
Modern Proof of the Theorem of the Base
I am looking for a modern proof of the so-called "Theorem of the Base"--that the Neron-Severi rank of a smooth projective variety is finite. One can prove this for varieties over $\mathbb{C}$ easily ...
- 23k
2
votes
0
answers
276
views
Identity with Ramanujan's generalized continued fraction
Let $F(x,q)=\sum_{n\geq 0}x^n\dfrac{q^{n^2}}{(q)_n}$, where $(q)_n=(1-q)(1-q^2)\dots(1-q^n)$. Then:
$$H(x,q)=\frac{F(-xq,q)}{F(-x,q)}=\dfrac{1}{1-\dfrac{qx}{1-\dfrac{q^2x}{1-\dots}}}$$ is the ...
- 301
5
votes
1
answer
605
views
Who is attributed with the conjecture that every multiply-perfect number greater than $1$ is even?
I know that Descartes is considered to be the first to ask whether or not odd perfect numbers exist ($n$ such that $\sigma(n)=2n$, where $\sigma(n)$ is the sum of divisors of $n$), and he also ...
- 205
2
votes
1
answer
942
views
Proof of the Friedlander–Iwaniec theorem
Does anybody know where I could find the proof of the Friedlander–Iwaniec theorem. The link that I find when I search for it is http://www.pnas.org/content/94/4/1054.full.pdf+html, but this seems more ...
- 1,890
7
votes
1
answer
856
views
Continued fraction representation of Zeta
A question at math.SE is asking for references. The fraction is quite nice! Check it out and post some references if you know of any.
I found this at arxiv, but it doesn't apply to Zeta.
- 1,043
4
votes
0
answers
559
views
Explicit description/calculation of norm group of ideles of characteristic $p$ global field
I posted the same question earlier in stack exchange,
(https://math.stackexchange.com/questions/1130391/algebraic-proof-of-2nd-inequality-of-global-class-field)
thinking it is most definitely not a ...
- 111
4
votes
1
answer
1k
views
Books on the Hardy-Littlewood circle method
Are there any good books providing an introduction to the Hardy-Littlewood method that do not require much of a background in complex analysis?
- 1,890
4
votes
2
answers
657
views
Intersection of Hilbert class fields of imaginary quadratic fields
In this question Hilbert class field of Quadratic fields it is mentioned that if $d\equiv 1 \mod 4$ then the Hilbert class field of $\mathbb{Q}(\sqrt{-d})$ contains $\mathbb{Q}(i,\sqrt{d})$.
Could ...
- 1,905
6
votes
2
answers
642
views
Is the square of the covering radius of an integral lattice/quadratic form always rational?
This is one of many observations from Pete L. Clark's questions on "Euclidean" quadratic forms. I sent Pete many positive integral forms that obeyed his condition. In turn, his condition turns out to ...
- 25.7k
2
votes
1
answer
514
views
Heegner points on elliptic curves
I want to know about Heegner point computations for a CM elliptic curve. What is the best reference book/paper for reading?
- 151
4
votes
0
answers
176
views
Are there any results about this higher degree Titchmarsh divisor problem?
Does there exist an asymptotic formula for $\sum_{p\le x}\tau(p−1)^n$ ? Here $n$ is an arbitrary positive integer and $\tau$ is the divisor function. The case of $n=1$ was done by Linnik, but when $n$ ...
- 41
9
votes
0
answers
3k
views
"Must read "papers on analytic number theory
Question: What would be some must-read
papers for an aspiring analytic number
theorist? In other words, what are the papers that any analytic number theorist would have read? (Background: Someone ...
5
votes
1
answer
324
views
Symmetry type of non-cohomological automorphic forms
By Katz-Sarnak philosophy a family of $L$-functions would have a symmetry type which would reflect the statistics of $L$-functions, such as low lying zeros and moments. Shin-Templier's paper on Sato-...
- 809
2
votes
0
answers
237
views
On the cardinality of the set of right-truncatable primes
We say that the (base ten) prime number $p=a_{n}a_{n-1}a_{n-2}\cdots a_{1}a_{0}$ is right-truncatable if all of the following numbers are prime:
\begin{eqnarray*}a_{n},\\a_{n}a_{n-1},\\ a_{n}a_{n-1}...
- 8,842
0
votes
1
answer
211
views
Question about sign change of Hecke eigenvalues
I want to write a survey on the subject 'Sign changes for coefficients of symmetric power $L$-functions'. So, I browse the Web and I got some papers. I read it and I gave special interest to the paper ...
- 1,257
6
votes
1
answer
1k
views
Must the $j$-invariant of an elliptic curve with an isogeny be integral?
Let $K$ be a quadratic field, and $E/K$ a non-CM elliptic curve with a $K$-rational $p$-isogeny, for $p$ a prime. I would like to say the following:
For large enough $p$, the $j$-invariant $j(E)$ ...
- 2,827
10
votes
2
answers
897
views
What are some consequences of the Mumford-Tate conjecture?
Let $A/K$ be an abelian variety over a number field. On the one hand we have the singular cohomology group
$$V := H^1(A(\mathbb{C}),\mathbb{Q})$$
with respect to some fixed embedding $K \subset \...
- 2,827
1
vote
2
answers
191
views
Reference request for Frobenius numbers
The Frobenius number of a set of coprime integers is the largest number that not can be written as the sum of integer multiples of numbers in that set.
I'm looking for a general reference on ...
- 563
2
votes
0
answers
100
views
Quasi-algebraically closed fields reference request
I am looking for a road to understanding what quasi-algebraically closed fields are with the ultimate goal of understanding the paper by Lang 1952.
My current background is the first 6 chapters from ...
- 87
5
votes
1
answer
992
views
On a sum involving Euler totient function
Let
$$S_a(N)=\sum_{n\le N}\frac{\varphi(an)}{n^2}.$$
The usual machinery gives an asymptotic formula
$$S_a(N)=\frac1{\zeta(2)}\cdot\frac{a^2}{\varphi_+(a)}\log N+C(a)+O(N^{-1+\varepsilon}a^{1+\...
- 12.3k
14
votes
0
answers
644
views
Can there be arbitrarily many cubic fields unramified outside $\{p,\infty\}$?
Observe, trivially, that since quadratic fields correspond to rational integers modulo squares (viz. discriminants), there are (roughly about, but certainly at most) $2^{|S|+1}$ quadratic fields ...
- 13.8k
6
votes
3
answers
555
views
Source for embedding multiplicative group of an algebraic closure of a finite field?
It's easy to embed the (cyclic) multiplicative group of a finite field into the multiplicative group of $\mathbb{C}$ (or other algebraically closed field of characteristic 0): assign to a generator of ...
- 52.9k
9
votes
1
answer
1k
views
Where can I find a modern write-up of Heegner's solution of Gauss' class number 1 problem?
In a recent MO question someone mentioned Heegner's solution of the Gauss "class number 1" problem which takes the following form:
When the class number of an imaginary quadratic form is 1 an ...
- 621
7
votes
2
answers
595
views
Representation theory of Discrete Subgroups of Lie groups
My question is the following. Which representations of $Sp(2g, \mathbb Z)$ are extendable to representations of $Sp(2g, \mathbb C)$ or $Sp(2g, \mathbb R)$. Is there a general theory and a good ...
- 327
8
votes
0
answers
328
views
Asymptotics of A261668
In Uniform Approach to Double Shuffle and Duality Relations of Various q-Analogs of Multiple Zeta Values via Rota-Baxter Algebras, Proposition 10.8, Jianqiang Zhao mentiones the sequence:
$$a_n=\sum_{...
- 10.7k
1
vote
0
answers
165
views
Reference request: Bounding exponential sum $\sum_{x \in [0,X]} \Lambda(x) e(\beta_d x^d + \ldots + \beta_1 x )$
Let $1 \leq i \leq d$, $q \in \mathbb{N}$, and $0 \leq a_{i} < q$. Let
$$
\mathfrak{M}^{(i)}_{a_{ i}, q} (C) =\{ \beta_{i} \in [0,1) : | \beta_{i} - a_{i}/q | \leq (\log X)^{C} X^{-i} \} .
$$
We ...
- 3,625
6
votes
0
answers
261
views
Local character expansion for discrete series representations of $GL_n(F)$
I'm interested about what, if anything, is known about the local character expansion of discrete-series representations of $GL_n(F)$, where $F$ is a $p$-adic field.
First, some notation: let $G$ be a ...
- 1,453
4
votes
0
answers
96
views
Reference to an elementary lemma (implying existence and uniqueness of the base-$b$ representation)
I'm looking for a reference to the following elementary result (or to a generalization of it).
Lemma. Let $(H, +, \preceq)$ be an (additive) partially ordered commutative monoid such that
$$x+z ...
- 10.5k