All Questions
Tagged with reference-request nt.number-theory
1,408 questions
4
votes
0
answers
252
views
Height pairings of Heegner points of nontrivial conductor
I am studying Gross's and Zagier's proof of the BSD conjecture for elliptic curves of rank $\leq 1.$ Their calculation essentially boils down to the following ingredients:
(1.) Finding a suitable ...
- 1,805
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
6
votes
0
answers
474
views
(In)finitely many natural numbers are not the sum or difference of two perfect powers
Are there infinitely many positive integers which are neither a sum nor a difference of
two perfect powers?
This question was proposed some years ago at KoMaL.
It's easy to see that the odd ...
- 3,153
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
1
answer
515
views
On comparing two almost injective divisor maps
Edit 2018.08.08 This answer https://mathoverflow.net/a/307881 will be updated to give recent information about S, especially a forthcoming preprint. End Edit 2018.08.08
In an introductory post on ...
- 13k
25
votes
8
answers
3k
views
Relatively concise English expositions of the proofs of the various Weil conjectures
Where can I find relatively concise (i.e. not excessively wordy and waxing poetic about history and intuitions and such, doesn't spend an eternity carefully developing various parts of the theory of ...
user97565
5
votes
0
answers
772
views
The Grimm Machine(s): A Collatz Conjecture Rival?
Edit 2018.08.08 This answer https://mathoverflow.net/a/307881 will be updated to give recent information about S, especially a forthcoming preprint. End Edit 2018.08.08
Just as the Collatz ...
- 13k
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
1
vote
3
answers
286
views
Reference book for primality testing [closed]
im searching for good reference to understand the primality testing idea
especially the Elliptic curves and primality for stirling numbers first and second ones , so can any one suggest to me good ...
- 199
1
vote
1
answer
231
views
An estimation of $p_n$
There seems to exist an asymptotic line
$$\displaystyle a+bx\sim \frac{x e^x}{p_{n}-x e^x}\; ,\;n=\lfloor e^x\rfloor\tag1$$
Which suggests an estimation
$$\displaystyle g(n)=\Big(1+\frac{1}{a+b\ln ...
- 862
10
votes
2
answers
4k
views
Reference book for Galois Representations
I am an undergrad. I have taken courses in algebraic number theory and have a basic idea about $p$-adic numbers. I have also read a little bit of infinite Galois theory. But I have no idea about ...
- 239
3
votes
1
answer
654
views
Order of vanishing of Artin $L$-functions at $s=1$
Let $E/F$ be a finite Galois extension of number fields with Galois group $G$. Let $S$ be a finite set of places of $F$ containing the infinite places. For $\chi$ an irreducible complex character of $...
- 1,661
4
votes
2
answers
928
views
Examples of Sets with Positive Upper Density
While reading the statement of Roth's theorem I started asking myself what are examples of sets of positive upper density? It's not hard to come up with a few:
Flip a coin with probability $\mathbb{...
- 22.8k
4
votes
1
answer
181
views
Asymptotics on the number of ways to pair off $\{1, 2, \dots, 2n\}$ into primes
Given $S = \{1, 2, \dots, 2n\}$, we can always pair off elements into $n$ pairs such that each sum to a prime. The proof of this fact is easy and follows from Bertrand's postulate.
Now, let $\gamma(n)...
- 141
13
votes
3
answers
1k
views
linear independence of $\sin(k \pi / m)$
I have tried searching the literature for a result like the following, but have not found anything.
For a positive integer $m$, is it known that
$$\{ \sin (k \pi / m): 1 \leq k \leq m/2, (k,m)=1 \}$$
...
- 133
6
votes
1
answer
739
views
Numbers divisible only by primes of the form 4k+1
Let $A(N)$ denote the number of positive integers $n\le N$ composed of prime numbers $p\equiv 1\pmod 4$ only. Is there an asymptotic formula for $A(N)$ (as $N$ tends to infinity)?
- 63
5
votes
1
answer
703
views
How to construct an abelian variety with CM by a given CM field?
Let $F$ be a totally real number field, and let $K$ be a quadratic extension of $F$ which cannot be embedded into $\mathbb{R}$.
Then $K$ is a so called CM field.
For instance, take $F = \mathbb{Q}(\...
- 11.3k
6
votes
2
answers
730
views
Definition of Hecke operators on orthogonal modular forms
In his paper Automorphic forms with singularities on Grassmannians, Borcherds poses Problem 16.5:
"Describe how the correspondence in this paper behaves under
the
action of Hecke operators."
Since ...
user94539
2
votes
1
answer
169
views
Higher dimensional analogs of logarithmic density
For a set $A\subseteq \mathbb{N}$ its lower/upper asymptotic/logarithmic densities are given by
\begin{align*}
\underline{d}(A)=\liminf_{N\to\infty} \frac{|A\cap [1,N]|}{N},\\
\bar{d}(A)=\limsup_{N\to\...
- 1,658
14
votes
4
answers
3k
views
Jacobi's theorem on sums of two squares (reference request)
One of Jacobi's theorems states that the number of representations of a positive integer $n$ as a sum of two squares of integers equals
$$4(d_1(n)-d_3(n)),$$
where the function $d_i$ counts the number ...
- 3,062
1
vote
0
answers
165
views
Reference request: Bounding exponential sum $\sum_{x \in [0,X]} \Lambda(x) e(\beta_d x^d + \ldots + \beta_1 x )$
Let $1 \leq i \leq d$, $q \in \mathbb{N}$, and $0 \leq a_{i} < q$. Let
$$
\mathfrak{M}^{(i)}_{a_{ i}, q} (C) =\{ \beta_{i} \in [0,1) : | \beta_{i} - a_{i}/q | \leq (\log X)^{C} X^{-i} \} .
$$
We ...
- 3,625
2
votes
2
answers
2k
views
A multidimensional version of Hensel's lemma? (for more than one polynomial)
The classical Hensel's lemma is stated as follows: Let $f(x) \in \mathbb{Z}_p[x]$ and $a \in \mathbb{Z}_p$ satisfy
$$
|f(a)|_p < | f'(a) |_p^2.
$$
Then there is a unique $\alpha \in \mathbb{Z}_p$...
- 3,625
12
votes
3
answers
7k
views
Learning roadmap for algebraic number theory
I have read some elementary number theory from David Burton's text and I know groups and rings from Herstein's book Topics in Algebra and some field theory from different sources online. I am ...
1
vote
0
answers
70
views
A non-surjective coboundary map induced by a central extension
Let $k$ be a number field and
$$ 1\to A \to B \to C \to 1$$ be a central extension of finite groups over $\mathcal{O}_k$ (the ring of integers of $k$), with $B$ non-commutative. Consider the induced ...
- 11
4
votes
1
answer
325
views
Irreducible monic polynomials
I am looking for criteria for the irreducibility of monic polynomials with constant term $\pm1$ over $\mathbb Q$. Eisenstein's criterion clearly doesn't apply here.
For instance, for the family of ...
- 2,135
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
2
votes
1
answer
442
views
Want more details about the image of a Maass form in the AIM press release concerning LMFDB
Actually I came upon this through MO a couple of days ago: in here
(http://aimath.org/aimnews/lmfdb/) there is a mesmerizing image
The caption reads
A Maass form, one of the 20 different types of ...
- 18.4k
17
votes
1
answer
703
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
6
votes
2
answers
1k
views
Divergent Series as a topic of research
About a year ago, while studying real analysis, I got very much interested in divergent series. I discussed possible research topics related to divergent series with my teachers but couldn't find any. ...
- 422
2
votes
0
answers
149
views
$f(x)$-th largest number of prime factors
Given a sufficiently well-behaved function $1\le f(x)\le x$ and a multiset $S=\{\omega(n): 1\le n\le x\}$, what can be said about the asymptotics of the $f(x)$-th largest member of $S$? In other words,...
- 9,114
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
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
5
votes
0
answers
504
views
An explicit formula for $\zeta(2m+1)$ with good convergence
The question: Is the following formula known?
$$\zeta(2m+1)=\frac{(-1)^m 2^{4m+2}\pi^{2m}}{2^{2m}-1} \sum\limits_{k=1}^m \frac{(2^{2k}-1)b_{2k}}{2^{2k}(2k)!}\cdot \sum\limits_{v=k}^m \frac{(2^{2v-2k+...
- 293
5
votes
0
answers
143
views
Reference request: "effective'' semistable reduction
I am looking for the origin of the following idea: suppose $m$ and $n$ are relatively prime integers $\geq 3$. Let $E$ be an elliptic curve over a number field $K$. Let $L/K$ be a finite extension ...
- 51
3
votes
1
answer
425
views
An electronic copy of Vishik's work on $p$-adic $L$-functions for modular forms
This question is very simple.
Would someone be so nice as to send me an electronic copy of M. M. Vishik, Non-Archimedean measures connected with Dirichlet series, Mat. Sb. (N.S.), 1976, Volume 99(...
- 10.9k
6
votes
1
answer
380
views
Applications of Level Lowering
What are some applications/consequences of level lowering of Galois representations? I understand the application of Ribet's theorem in the proof of Fermat's last theorem but I am wondering what other ...
- 1,629
8
votes
2
answers
537
views
Famous results about the value of a given limit assuming it exists
Chebyshev got famous showing that if the limit $l:=\lim_{x\to\infty}\frac{\pi(x)}{x/\log x}$ exists, then necessarily $l=1$, constituting a major breakthrough towards a proof of the famous prime ...
- 6,984
1
vote
0
answers
82
views
Reference request: Structure of $H^1({{\mathbf{Q}}_{q}},{{{{E}}_{{p}^{\infty}}}})$
I need reference on the structure of ${H}^{1}({{\mathbf{Q}}_{q}},{{E}_{{{p}^{\infty}}}})$, in particular when:
(1.) $q=p$
and/or
(2.) $E$ has multiplicative reduction at $q$.
Here, $E$ is an ...
- 1,805
0
votes
1
answer
980
views
How to calculate $N_{L/k}$(roots of unity)?
Suppose that $L/k$ is a Galois extension of number fields and that $G$
is the corresponding Galois group. Further, for $\frak p$ a prime ideal
of $\cal O$$_L$, let $K=L^{G(\frak p)}$, where [$L$ : $K$...
- 197
5
votes
2
answers
342
views
Minimum length of a convex lattice polygon containing k lattice points?
Let $f(k)$ denote the minimum length of a convex lattice polygon containing exactly $k$ lattice points (including lattice points on the boundary).
It is not too hard to show that $k = \frac{1}{4\pi} ...
- 191
9
votes
1
answer
646
views
Have the explicit Poisson-type formulas of Guinand and Meyer been observed before?
In a recent paper of Meyer Measures with locally finite support and spectrum PNAS vol. 113 no. 12:3152–3158 (behind a paywall, but see also these seminar notes) some new explicit Poisson-type formulas ...
- 2,901
13
votes
2
answers
2k
views
Has it been proved that odd perfect numbers cannot be triangular?
(Note: This question has been cross-posted from MSE.)
Euclid and Euler proved that every even perfect number is of the form $m = \frac{{M_p}\left(M_p + 1\right)}{2}$ where $M_p = 2^p - 1$ is a prime ...
7
votes
2
answers
708
views
Eisenstein Series on Siegel Space
I am looking for any reference dealing explicitly with Eisenstein series on Siegel space (the simplest case of $\rm{SP}_4$ is fine). Anything would be welcome, but in particular I'm interested in the ...
- 2,824
5
votes
2
answers
403
views
Is Howe's construction of tame supercuspidal representations independent of additive character?
Let $F$ be a $p$-adic field.
In "Tamely ramified supercuspidal representations of $Gl_n$" (Am. J. Math 73 (1977)), Howe constructs a supercuspidal representation $\pi_{\psi}$ of $GL_n(F)$ from the ...
- 1,453
1
vote
1
answer
258
views
Is it proved that for every integer $p>0$ there exists an integer $k>0$ such that every integer $n>0$ can be expressed as $j_1^p+\dots+j_k^p$?
It has been shown, by elementary methods, that every positive integer can be expressed as the sum of $4$ squares. This type of result has been proven for many different powers $p$, for example, when $...
- 1,247
8
votes
1
answer
431
views
Reference request: seminar report of Serre from late 60s on possibility of Galois representations attached to modular forms?
See here for a comment of Matt Emerton.
There are also various seminar reports of Serre, e.g. his report on mod p modular forms, but also his report from the late 60s on the possibility of Galois ...
- 83
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
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}}}$$ is the ...
- 301
3
votes
1
answer
171
views
Does positive relative density imply asymptotic additive basis behaviour?
First definitions: let $A, B \ \subset \mathbb{Z_{>0}}$ and $1\in A, 1\in B$. We define the relative density of $A$ with respect to $B$ to be $$rel(A, B) = \inf_n \frac{|A \cap [1,n]|}{| B \cap [1,...
19
votes
1
answer
2k
views
Has anything ever been done with the set $\{1,2,3,4,\ldots\}$ equipped with the operation $a \oplus b = a+b-1$ and the usual notion of multiplication?
Definition. $$\mathbb{J} = \{1,2,3,\ldots\}.$$
We can refer to the elements of $\mathbb{J}$ as "joiners."
The product of joiners is inherited from $\mathbb{Z}$.
The sum of joiners will be ...
- 3,793