All Questions
Tagged with reference-request nt.number-theory
1,409 questions
3
votes
2
answers
625
views
Intuition behind $\zeta(2) = \frac{\pi^2}{6}$ [closed]
In literature, there are many proofs of the well-known result $$\zeta(2) = \frac{\pi^2}{6}.$$
However, as far as I know, they do not offer an intuitive explanation of why this result should be true. ...
user60665
6
votes
2
answers
381
views
Lattice-cube minimal blocking sets
Let $C_d(n)$ be the lattice cube consisting of the $n^d$ points with
each of its $d$ coorindates in $\lbrace 1,2,\ldots,n \rbrace$.
Define a blocking set for a lattice cube to be a set of points
in ...
- 151k
4
votes
0
answers
115
views
Relations between an projective variety and galois cohomology
Let $f_1, \cdots, f_k$ be homogeneous polynomials over $\mathbb{Q}[x_0, \cdots, x_n]$. They define an projective variety $X$ over $\mathbb{P}^n(\mathbb{C})$, namely their set of zeros $$X = Z(f_1, \...
- 423
3
votes
2
answers
621
views
Who needs a symmetric upper asymptotic density on the integers?
The upper asymptotic density on $\mathbf Z$, viz. the function
$$
{\sf d}^\ast: \mathcal P(\mathbf Z) \to [0,1]: X \mapsto \limsup_{n \to \infty} \frac{|X \cap [1,n]|}{n},
$$
has a ''symmetric ...
- 10.5k
11
votes
2
answers
826
views
Sums of subsets of $\mathbb{Z}/n\mathbb{Z}$
I have encountered a problem that I suspect has been thoroughly studied but I have not been able to find references. Can anyone point me to a published reference dealing with this or a closely related ...
- 2,851
13
votes
2
answers
726
views
Special values of $\zeta$ outside the real line and the critical strip
The values of Riemann's function at the integers have been extensively studied. I was wondering, is there anything interesting known (or conjectured) to happen arithmetically outside the real line (...
- 17.6k
1
vote
1
answer
1k
views
When can we write fundamental units explicitly
Given a number field $K$, the Dirichlet Unit Theorem tells us about the structure of the unit group $O_K^\times$. However, the proofs do not seems to give any way to explicitly write out a set of ...
- 453
1
vote
2
answers
534
views
How to prove $\mathop {\lim }\limits_{x \to \infty } \sum\limits_{{f_x}(p) = 1} {\frac{1}{p}} = \ln 2$ for $p \le x$?
Let ${f_x}(m) = \sum\limits_{\left. p \right|m} {{f_x}(p)}$ be a
strongly additive function on positive integer number $m$, where $p$ is a prime number. Set
$${f_x}(p) = \left\{ {\begin{array}{*{20}{...
- 113
0
votes
1
answer
346
views
Upper bounds for solutions to a Pell-like equation
Let $N$ be a fixed positive integer that is not a square and $m$ be any nonzero integer. Let $x$ and $y$ be positive integers that solve $$x^2 - N y^2 = m^2$$ with $x + y$ minimal (in light of the ...
- 2,283
5
votes
2
answers
818
views
Blueprint of L-functions and need for introducing them ( Hasse-Weil L-functions )
Dear All,
This question may appear elementary to all the experts in number theory , but forgive me. I really wanted to know how did the $L$-functions came into existence, especially the Hasse-Weil L-...
8
votes
0
answers
161
views
Looking for an elliptic curve E st ${\large Ш}(\mathbb Q,E)$ cont. an element of order $p^2$ and certain other properties
I am looking for an elliptic curve $E$ with Weierstraß coefficients in $\mathbb{Q} $ so that for some prime $p$ the following conditions are satisfied:
(1) ${\large Ш}_{p^{\infty}}(\mathbb{Q},E)$ ...
- 1,805
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
4
votes
1
answer
421
views
Birch's conjecture from Representation Theory
Birch has a conjecture about which automorphic forms on $PGL(2)$ are the lifts from nonsplit $O(3)$. Temporarily ignore global issues, and focus on the local nonarchimedian picture. The automorphic ...
- 2,429
0
votes
0
answers
266
views
Completing a dyadic sum
Suppose I knew the behaviour of a given sum in every other interval, for example:
$$
\sum_{\substack{0\leq a \leq x\\ a\equiv 1 (k)}} \sum_{x/(a+k/2)< b \leq x/a} f(b) \sim g(x),
$$
for any $x>1$...
- 3,799
4
votes
0
answers
104
views
Asymptotics of the dimensions of the plus and minus space of Atkin-Lehner
Let $S_2^{+}(p)$ be the space of newforms of level $p$ that have Atkin-Lehner eigenvalue +1, and $S_2^{-}(p)$ be the space that have Atkin-Lehner eigenvalue $-1$. What is known about the asymptotics ...
- 2,429
5
votes
0
answers
145
views
Character sums over a sumset
Suppose that $p$ is a prime, $A$ is a subset of $\mathbb F_p$, and $P$ is a polynomial over $\mathbb F_p$ of degree $d$. Using Weil's bound, it is not difficult to show that
$$ \left| \sum_{a,b\in A}...
- 23k
6
votes
2
answers
754
views
ASCII prime plots and prime-rich quadratic polynomials
This is a series of questions inspired by the MathOverflow question
Find the least prime so that p-1 has two factors greater than $m$ and $n$ posted by Aaron Sterling.
I suggested plotting primes by ...
- 13k
5
votes
1
answer
819
views
Is this known alternating sum for Euler's constant?
This probably is known, but Wolfram Alpha doesn't recognize it
and couldn't find it in Mathworld (there is something close,
but using floor).
We have
$\lim_{s \to 1} (\zeta(s)-1/(s-1)) = \gamma$
...
- 25.4k
11
votes
1
answer
408
views
Integers with a large prime divisor in short intervals
For an integer $n$, denote by $P^+(n)$ the largest prime divisor of $n$. Then we have the following:
There exists some $c>0$, such that for all $x$ sufficiently large the number of integers $n\in[...
6
votes
1
answer
680
views
Counting number of points in a lattice with bounded sup norm
Let $\Lambda$ be a lattice in $\mathbb{R}^n$. For $\bar{x} \in \mathbb{R}^n$, let
$\| \bar{x} \| = max_{1 \leq i \leq n} \{ |x_i| \}$, i.e. the sup norm. Let $\lambda_1, ..., \lambda_n$
be a ...
- 579
4
votes
5
answers
688
views
Nonlinear equations in integers
Linear patterns in subset of the integers (for example, primes) such as arithmetical progressions is a hot topic in mathematics. Recently, much progress has been made in this area. For example,
the ...
- 71
3
votes
1
answer
425
views
An electronic copy of Vishik's work on $p$-adic $L$-functions for modular forms
This question is very simple.
Would someone be so nice as to send me an electronic copy of M. M. Vishik, Non-Archimedean measures connected with Dirichlet series, Mat. Sb. (N.S.), 1976, Volume 99(...
- 10.9k
0
votes
1
answer
107
views
Variation on the definition of the uniform distribution mod 1 [closed]
A sequence $x_{n}$ is said to be uniformly distributed mod 1 if $\forall a,b$ with $0\leq a<b<1$,
$$\lim_{n\rightarrow \infty}\frac{1}{n}|\lbrace j=1,...,n :\lbrace x_{j}\rbrace\in [a,b]\rbrace|...
3
votes
1
answer
1k
views
Two Dirichlet's series related to the Divisor Summatory Function and to the Riemann's zeta-function
Considering the $\textit{Divisor Summatory Function}$, $D(n)$, defined as
$$
D(n) = \sum_{k=1}^{n}d(k) ,
$$
where
$$
d(n) = \sum_{k|n}^{n}1.
$$
One can observe the following pattern in the values of $...
- 534
5
votes
0
answers
504
views
An explicit formula for $\zeta(2m+1)$ with good convergence
The question: Is the following formula known?
$$\zeta(2m+1)=\frac{(-1)^m 2^{4m+2}\pi^{2m}}{2^{2m}-1} \sum\limits_{k=1}^m \frac{(2^{2k}-1)b_{2k}}{2^{2k}(2k)!}\cdot \sum\limits_{v=k}^m \frac{(2^{2v-2k+...
- 293
2
votes
0
answers
188
views
Solution of the Diophantine equation $x^4+y^4+z^4=2t^4$ are well-known? [closed]
Are solutions of the Diophantine equation $x^4+y^4+z^4=2t^4$ well-known?
I give a solution:
$x=m^2-n^2, y=m^2-2mn, z=n^2-2mn, t=m^2+n^2-mn$
- 477
11
votes
3
answers
745
views
Counting points on lattices
I expect that the following is a standard problem from analytic number theory, but I don't know where exactly to look for an answer.
Let f: ℤr→ H be a surjective homomorphism into a ...
- 527
1
vote
2
answers
375
views
binomial/factorial identity mod p
In trying to determine the spectrum of a well-known ergodic transformation, I came up with the following useful (for me) result.
Let $p$ be a prime and $a$ a positive integer. Then for $M$ a positive ...
- 4,679
9
votes
1
answer
639
views
Borel's Paris Lectures
I am trying to read Harish-Chandra's book on automorphic forms on Semisimple Lie groups, and he keeps referring to Borel's Paris lecture notes. Does anyone have an online version of these notes or ...
- 105
6
votes
1
answer
368
views
Bounds re Asymptotic Formula for the Sum of Largest Prime Factors
I have a reference request related to the result :
$\sum_{n=2}^{x} P(n)$ ~ $\frac{\pi^2}{12}\frac{x^{2}}{log(x)}$ as $x \rightarrow \infty$
where $P(n)$ is the largest prime factor of the positive ...
- 71
7
votes
2
answers
1k
views
questions on Néron-Tate canonical height
I have three questions regarding height pairings:
In [Serre, Lectures on the Mordell-Weil theorem], p. 85 f., it is stated that the following function is a local height function:
"Let $V/R$ be a ...
user19475
6
votes
2
answers
310
views
Conjectured congruence for the Apery numbers
Numerical evidence for the first hundred Apery numbers
$$A_n=\sum\limits_{k=0}^n\binom{n}{k}^2\binom{n+k}{k}^2$$
suggests the following congruence relation
$$A_n\equiv 0\; (\mathrm{mod}\; 5),\;\;\...
- 16.5k
6
votes
1
answer
788
views
Exponential sums over finite fields with even characteristic
I am looking for an elementary evaluation (if one exists) of the exponential sum
$$
G_r(a,b) = \sum_{x \in \mathbb{F}_{2^r}} \psi(ax^2 + bx),
$$
where $a,b \in \mathbb{F}_{2^r}^*$ are both units, $\...
- 197
11
votes
1
answer
891
views
The maximum of the preimage of [1,x] through Euler's totient function
A friend of mine and I have shown the following:
"For each $x \geq 1$ let $m := m(x)$ be the greatest positive integer such that $\varphi(m) \leq x$, where $\varphi$ is the Euler's totient function.
...
user40023
0
votes
0
answers
98
views
Eigenvalues of a sequence of matrices involving the divisor function
Let $A_{n,k},k=1,\ldots,n$ be a sequence of $n\times n$ upper triangular matrices where $A_{n,1}=I_n$ and $A_{n,k},\quad 2\leq k\leq n$ be a regularly shifted and scaled matrix, with $P_{n,k}$ an $n\...
- 10.4k
5
votes
1
answer
1k
views
Verlagerung made "explicit"
Suppose given a finite extension $L/K$ of number fields. I would like to develop a better intuition for the Verlagerung giving an embedding $Ver : Gal(K^{ab}/K) \to Gal(L^{ab}/L)$.
For example, let $...
- 2,029
9
votes
1
answer
570
views
The Dissertation of F. J. van der Linden
Does anyone have access to the 1984 dissertation of Franciscus Jozef van der Linden under Hendrik Lenstra? It is called Euclidean Rings with two infinite primes. The theory is that this has the ...
- 25.7k
1
vote
0
answers
148
views
Estimating the sum of Dirichlet character $\sum_{0 \leq x < q} \chi(F(x))$ where $F(x)$ is a polynomial
Let $q \in \mathbb{N}$ and $\chi$ a Dirichlet character mod $q$. Let $F(x)$ be a polynomial with integer coefficients. I was wondering if a bound for the following sum was available or not:
$$
\sum_{0 ...
- 3,625
13
votes
0
answers
523
views
Euler Subgroups and Automorphic L-functions
Recently, I have read about the Whittaker expansion for $\mathrm{GL}_n$ and was struck by the utility of the mirabolic subgroup, $\mathrm{P}_n\subset \mathrm{GL}_n$ of matrices with bottom row $(0\; 0 ...
- 2,516
9
votes
3
answers
980
views
$\omega(p^n - 1)$ as $n \rightarrow \infty$
Although I am also interested in the number of distinct prime factors (not counting
multiplicity), today I use $\omega(m)$ to denote the number of (positive) prime
factors (with multiplicity) of the ...
- 2,158
5
votes
3
answers
3k
views
Effective way of finding generators on the curve and the rank conjecture
Hello everyone,
I have never heard of a polynomial time running algorithm that finds the generators of elliptic curves efficiently. I do know that Nagell-Lutz theorem is useful in computing the ...
20
votes
1
answer
787
views
Representation theory of reductive groups in characteristic $p$ as a limit of the theories in characteristic $0$
This question is out of plain curiosity. The first sentence of Deligne's
Les corps locaux de caractéristique $p$, limites de corps locaux de caractéristique $0$ (1984) reads (in rough translation) as ...
- 18.7k
3
votes
2
answers
1k
views
The arithmetic of higher genus curves
Genus 0 curves are well understood in number theory. There is also are rich theory a bunch of conjectures about the arithmetic of elliptic curves.
This leads me to the question, what we know about ...
- 2,810
3
votes
2
answers
328
views
equivalence of quadratic forms over finitely generated fields
Over number fields, two quadratic forms are equivalent iff they have the same dimension, signature, discriminant and Hasse invariant.
How is the situation like over finitely generated fields?
user19475
9
votes
0
answers
230
views
Clozel's unpublished manuscript
I'm looking for Clozel's unpublished manuscript
L. Clozel, Modular properties of automorphic representations I: Applications of
the Selberg trace formula (1993)
cited in Urban's Eigenvarieties ...
- 3,799
1
vote
6
answers
1k
views
List of structure theorems for vector valued Siegel modular forms (esp. of genus 2)
What are Siegel modular forms?
We start with defining their common domains $\mathbb{H}_g$ as the set of symmetric $g \times g$ matrices with positive definite imaginary parts.
The symplectic group $...
1
vote
0
answers
463
views
A question on (odd) perfect numbers
I have asked this question in MSE a few weeks back, but did not receive any responses. I have cross-posted it to MO, hoping that it is appropriate for this site.
Let $\sigma(x)$ be the (classical) ...
2
votes
0
answers
154
views
Equi-distribution of the parity of partitions
The integer partition function $p(n)$ has a generating function given by
$$\frac1{(q)_{\infty}}=\sum_{n=0}^{\infty}p(n)q^n$$
with $(q)_{\infty}=\prod_{m=1}^{\infty}(1-q^m)$. The long-standing problem ...
- 43.2k
3
votes
1
answer
447
views
A number array related to colored necklaces and the primes
I stumbled upon entry OEIS-A208535 on the enumeration of certain kinds of colored necklaces and noticed that the integers for the odd prime rows of the table there seem to be given by the Moreau ...
- 10.5k
9
votes
1
answer
1k
views
Class groups of orders
In Cox's book "Primes of the form $x^2 + ny^2$", he proves that in a quadratic imaginary field $K$, if $\mathcal O$ is an order of conductor $f \in \mathbb Z$, we have that the class group $\mathrm{Cl}...
- 647