All Questions
Tagged with reference-request nt.number-theory
1,408 questions
3
votes
1
answer
233
views
Addition law for elliptic curves of the form $x^2y^2+a(x+y)+b=0$
Did anybody consider addition law for elliptic curves of the form $$x^2y^2+a(x+y)+b=0\,?$$ Does this form have any specific name?
- 12.3k
1
vote
0
answers
140
views
Primes of the form $2^{m_0}p_1^{m_1}\ldots p_r^{m_r}+1$
Is it known any example of a set of primes $\{p_1,\ldots,p_r\}$ with the following property: there are infinitely many $(m_0,\ldots,m_r)\in\mathbb N^{r+1}$ such that $2^{m_0}p_1^{m_1}\ldots p_r^{m_r}+...
- 509
1
vote
1
answer
114
views
Differences of consecutive ordered fractional parts
Let $r$ and $h$ be a real numbers and $n>0$. Write the fractional parts $\{k*r+h\}$, for $k = 1,2, . . . n$, in increasing order as $$ a_1 < a_2 < \cdots < a_n.$$ Let $D_n$ be the set of ...
- 3,443
5
votes
0
answers
974
views
$\sum_{n=1}^{\infty}\frac{1}{a_n}=\infty$ $\sum_{n=1}^{\infty}\frac{1}{b_n}=\infty$ but $\sum_{n=1}^{\infty}\frac{1}{a_n+b_n}=c, c\in R$ [closed]
The following question is inspired from: Defining the slowest divergent series.
Let $a_n$ and $b_n$ be two strictly increasing sequences of natural numbers,with $\sum_{n=1}^{\infty}\frac{1}{a_n}=\...
- 3,866
7
votes
2
answers
2k
views
Applications of periodic continued fractions
Some answers from Applications of finite continued fractions in fact are Applications of periodic continued fractions. I think that it should be separate question.
What can you add to the following ...
2
votes
1
answer
656
views
Reference Request - Sharp Estimates for a Logarithmic Sum
Can anybody suggest a good (e.g. "non-technical") introduction to estimating bounds for logarithmic sums of the form
$$\sum_{i=1}^{r}{{\alpha_i}{\log(q_i)}}$$
where the $$\alpha_i$$ are positive ...
10
votes
1
answer
3k
views
Reference request: number theory of Z[1/p]
Can anyone suggest a good place to read up on the number theoretic properties of and techniques for $\mathbb{Z}[1/p]$, (that is, rational numbers with only powers of a prime $p$ in the denominator)?
...
- 2,235
5
votes
0
answers
585
views
Bloch Kato Exponential as formal lie group exponential
Let $K$ be a $p$-adic field and $V$ a $p$-adic representation. In their paper on tamagawa numbers of motives, Bloch and Kato define an exponential map as the connecting homomorphism
$$DR(V) \...
- 3,555
5
votes
3
answers
2k
views
Modular forms reference
If f is a weight 2 newform on $\Gamma_1(N)$ then there exists an abelian variety Af whose endomorphism algebra is isomorphic to the field generated by the coefficients of f.
I've seen this proven in ...
user1073
2
votes
2
answers
345
views
Looking for a reference to a classical formula for the sum of the base-$b$ digits of an integer
Pick an integer $b \ge 2$, and for $n \in \mathbf N$ let $s_b(n)$ denote the sum of the base-$b$ digits of $n$. It is a nice exercise to prove that $$s_b(n) = (b-1) \sum_{i=1}^\infty \left\{\frac{n}{b^...
- 10.5k
0
votes
0
answers
694
views
"Descending cohomology, geometrically" by Mazur:
(Exist texts of that talk or related texts: http://ttv.mit.edu/collections/harris60/videos/13881-problem-session-barry-mazur ?) Article: http://www.math.harvard.edu/~mazur/papers/page37.pdf
- 10.8k
3
votes
0
answers
454
views
What is the relation between the length of period of simple continued fraction expansion of quadratic algebraic numbers $\sqrt{A}$ and the integer $A$
What is the relation between the length of period of simple continued fraction expansion of quadratic algebraic numbers and the integer
As we know,$\sqrt{2} = [1;2,2,2,2,…]$; while $\sqrt{14}= [3;1,2,...
- 3,094
6
votes
1
answer
1k
views
reference for p-local and p-complete integers
Can anyone suggest a good thorough reference for $p$-localization and $p$-completion of the integers? I'm an algebraic topologist who's found himself washed up without any intuition.
In particular, ...
- 499
6
votes
2
answers
806
views
Still more generalized Dirichlet Theorem
Dirichlet proved a classical theorem about approximating irrational real numbers with rational numbers, saying that for any irrational real number $\alpha$, you can find infinitely many rational ...
- 315
6
votes
0
answers
291
views
Legendre polynomials and formal groups
Let $P_n(x)$ be Legendre polynomials:
$$\frac{1}{\sqrt{1-2tx+t^2}}=\sum\limits_{n=0}^{\infty}P_n(x)t^n.$$
Usual arguments from the theory of formal groups allow to
prove that for any $n$
$$P_n(x)=Q_n(...
- 12.3k
3
votes
2
answers
876
views
Asymptotics for the number of partitions of $n$ into odd prime parts
Hello!
I am interested in the asymptotic behavior of the function $p_o(n)$ defined as the number of partitions of $n$ into odd prime parts A099773 - http://oeis.org/A099773 .
I couldn't find any ...
- 3,463
1
vote
1
answer
193
views
L-function of twist
I'd like to ask the following easy question, since I can't find a reference.
Let $X/\mathbb{Q}$ a smooth projective variety. How does one express the $L$-function of the twist, $L(H^i(X)(r), s)$ in ...
- 3,555
5
votes
1
answer
655
views
A theorem of Stickelberger on the number of prime ideals in a decomposition
Suppose that $p$ is unramified in a number field $K$ of degree $n$. Apparently, Stickelberger proved that $\big( \frac{Disc(K)}{p}\big) = (-1)^{n - g}$, where $g$ is the number of prime ideal factors ...
- 7,347
3
votes
2
answers
639
views
Minimum sum among fixed length factors of a number
I think this question is famous, but I searched a lot in the internet with many keywords and unfortunately I did not find any related problem. I will appreciate any helpful answers and any guidance ...
- 4,784
1
vote
1
answer
347
views
Reference on elements of finite order in principal congruence subgroups of symplectic groups
We should start with the definition of the symplectic group for an arbitrary ring $R$.
The symplectic group $Sp(g,R)$ is the subgroup of $SL(2g,R)$ such that all elements satisfy $M=J_g^t M J_g$ with $...
- 85
4
votes
0
answers
332
views
Diophantine equations over cyclotomic fields
Let $\mathbb{Q}^{\text{ab}}$ be the compositum of all finite abelian extensions of $\mathbb{Q}$. Explicitly, $\mathbb{Q}^{\text{ab}}$ is the field obtained from $\mathbb{Q}$ by adjoining all roots of ...
- 11.3k
7
votes
1
answer
918
views
Integral expression for zeta(2)
By computing the sum of all Bernoulli numbers via Borel summation (I learned this technique from Varadarajan's excellent book Euler through time. A new look at old themes, 2006) I found that $$\sum ...
- 32.6k
2
votes
0
answers
412
views
Conjectures on fractions where each digit appears once in numerator and denominator
This is a highly redacted version of a question that was asked before. Please see Criteria of considering relevance of the question to the domain of research topics for details.
Some numerical ...
- 335
4
votes
1
answer
234
views
Inequality due to Siegel (assumptions) and upper bounds on number field discriminants
In Siegel's 1969 paper, Abschätzung von Einheiten, on page 73, he states the inequality
$$\log\sqrt d\le n-1+{n\over 2}\log\pi+r_2\log 2\qquad (*)$$
and compares with the bound due to Minkowski that
$$...
- 1,049
1
vote
1
answer
393
views
The Dirichlet series of the Hasse–Weil L-function
I have the following question:
Is there is a paper claiming that the Dirichlet series of the Hasse–Weil $L$-function (associated with an elliptic curve over rationals) is of finite order.
Thank you in ...
- 1,197
6
votes
1
answer
121
views
Name or references for minimal $N$ such that $\left(\frac{a}{b}\right)_n = \left(\frac{a}{b'}\right)_n$ whenever $b \equiv b' \bmod (N)$
Let $\left( \dfrac{a}{b} \right)_n$ denote the nth power residue symbol, a generalization of the Legendre symbol. I have recently seen it quoted that there is a minimal ideal $N$ (minimal by ideal ...
- 1,570
1
vote
1
answer
316
views
Solution or Reference Request for a Closed Form of the Sum
I have been working for quite a while on finding a closed formula for the Legendre Symbol. Inspite of my best efforts I can't come anything better with a formula for the symbol $\left(\dfrac{q}{p}\...
user57432
2
votes
1
answer
91
views
Reference or a short argument that a certain subset generates the ring of p-typical symmetric functions under plethysm
Let p be a prime, and let $\Lambda_p$ be the subring of the ring of symmetric functions $\Lambda$ (over $\mathbb{Z}$) such that $$x \in \Lambda_p$$ iff there is an $i \in \mathbb{N}$ such that $p^ix \...
- 101
2
votes
1
answer
604
views
2-dimensional galois representations
Class field theory gives us a framework in which we can understand one-dimensional galois representations. I'd like to learn about the galois representations attached to modular forms, but I'm having ...
- 3,555
3
votes
0
answers
338
views
The hypertriangular function of $n$
I'm looking for papers or recent results on the hypertriangular function of $n$:
$$H_t(n)= \displaystyle\sum\limits_{k=1}^{n} k^k$$
This is A001923 in the OEIS.
I don't have much experience with ...
- 181
0
votes
1
answer
246
views
Does the modified Szpiro conjecture require minimal model?
The modified Szpiro conjecture is described in
Wikipedia
and here and here.
The modified Szpiro conjecture states that: given $\varepsilon > 0$, there exists a constant $C(\varepsilon)$ such that ...
- 25.4k
0
votes
1
answer
237
views
Congruences for generalized Franel numbers
Let us define generalized Franel numbers $f^{(m)}_n$ through recurrence relations:
$f^{(1)}_n=1$ for all $n$, and $$f^{(m)}_n=\sum\limits_{k=0}^n\binom{n}{k}^3f^{(m-1)}_k.$$ In fact $$f^{(m)}_n=\sum\...
- 16.5k
2
votes
1
answer
339
views
Finite Flat Group Schemes for Modular Forms of Higher Weight
Let $f$ be a newform (normalized, cuspidal) of weight $k \ge 2$. Then for a prime $\ell$ there is an $\ell$-adic Galois representation associated to $f$. If $k=2$, this comes from an abelian variety, ...
- 15.4k
6
votes
0
answers
408
views
Asymptotic formula for restricted partition function
Let $p(n)$ be the partition function. Hardy and Ramanujan - and Uspensky, independently proved the asymptotic formula
$$(1) \quad p(n) \sim \frac1{4\sqrt{3}} \frac{e^{c_0\sqrt{n}}}{n} \text{ as } n \...
user40023
3
votes
2
answers
737
views
What's known about complete split primes in Q(E[p])?
Let E be an elliptic curve over $\mathbf{Q}$, and p be a prime of good reduction for E such that the Galois representation $\bar\rho_p$ of $\mathbf{Q}$ on the p-torsion of E surjects onto Aut(E[p]). ...
- 2,406
3
votes
1
answer
389
views
Galois deformations with Panchiskin condition
Let $L/\mathbf{Q}_p$ be a finite extension and we consider a fixed $L$-linear representation $V$ of the absolute Galois group $G:=\operatorname{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p)$. Assume that ...
- 63
2
votes
3
answers
339
views
Gauss sums over multiplicative subgroups
Hello,
Is anyone here aware of a well-motivated exposition of the Bourgain-Glibichuk-Konyagin estimate for exponential sums (or Gauss sums) over multiplicative subgroups? If any of you has a write-up ...
- 8,842
9
votes
0
answers
414
views
In which orders can the numbers of prime factors of consecutive integers be?
Let $\omega(m)$ be the number of distinct prime divisors of a positive integer $m>1$. I am interested in the relative orders in which the numbers $\omega(n+1),...,\omega(n+k)$ can occur.
Given a ...
- 2,719
0
votes
1
answer
121
views
Dirichlet series without order term
is there a name in use for Dirichlet series without the order term, analogously to Laurent or Puiseux polynomials? Is there work known about such expressions?
$D(s) = \sum_{0<n<N}a_n/n^s$
The ...
- 155
4
votes
1
answer
505
views
Number of divisors of a sum / ABC conjecture equivalent statement
I have two related questions that I can't seem to find any literature on:
1) What can be said about $\tau(a+b)$ knowing $\tau(a)$ and $\tau(b)$ (where $\tau(n)$ is the number of positive divisors of $...
- 2,235
3
votes
1
answer
398
views
Primes in short intervals with a preassigned frobenius
Edited after mistake in the first version.
It is known since Selberg that under the Riemann Hypothesis, given an $\epsilon>0$, there is a prime between $x$ and $x+O(x^\epsilon)$ for all $x$ in a ...
- 26k
3
votes
0
answers
196
views
Counting number of points in a lattice with bounded length
I am interested in counting number of lattices using the following theorem.
The following is Theorem IV (page 412) in Chapter VIII of "An introduction to the geometry of numbers (second printing, ...
- 579
1
vote
1
answer
323
views
Koch's "Extendible functions"
For more than a year now, I have been looking for a copy of the following CICMA Concordia preprint :
Author : Helmut Koch
Title : Extendible functions
Preprint, CICMA Concordia University Department ...
- 18.7k
2
votes
0
answers
116
views
Name of a difference of continuants
I am getting ready to publish the manuscript
http://arxiv.org/pdf/1408.4631v2.pdf
and I am trying to do due diligence on a quantity I study before it gets published. (This is cross-posted from Name/...
- 1,521
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
7
votes
0
answers
462
views
Looking for a paper of Hartshorne
In a famous paper
Hartshorne - Varieties of small codimension,
Hartshorne formulates a conjecture, which roughly says that varieties of small codimension in projective space are complete ...
- 21.3k
2
votes
0
answers
398
views
Counting factors: is this approach in the literature on multiperfect numbers?
Does the following approach (or something near it) exist in the number theory
literature?
I will provide some motivation for $\omega(p^n - 1)$ as $n \rightarrow \infty$
and for this question. First, ...
- 2,158
7
votes
2
answers
785
views
Inverse map for partition transform
Let $(a_n)$, $n\in\mathbb{N}$, be a sequence of complex numbers, then formally one has
(1)
$$\prod_{1}^{\infty}\left(1-a_nx^n\right)^{-1}=1+\sum_{1}^{\infty}\left(\sum_{j_1+2j_2+\cdots +nj_n=n}a_1^{...
- 2,480
6
votes
1
answer
621
views
Lorentzian characterization of genus
Suppose we take the "even" indefinite lattice from page 50 in Serre A Course in Arithmetic (1973)
$$ U \; = \;
\left( \begin{array}{cc}
0 & 1 \\\
1 & 0
\end{array}
\right),$$
...
- 25.7k
3
votes
0
answers
680
views
Birch/Swinnerton-Dyer "Notes on Elliptic Curves II"
I would like to know if any of you know if there is a more general treatment to what Birch and Swinnerton-Dyer did in "Notes on Elliptic Curves II" (http://www.ams.org/mathscinet-getitem?mr=179168).
...
- 521