Questions tagged [nt.number-theory]
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
15,952
questions
3
votes
0
answers
142
views
Algorithm for holonomic sequence
A sequence $(a_n)_{n\geq 0}$ of complex numbers is called polynomially recursive (P-recursive) or holonomic if there exists a number $r$ and rational functions $P_1(n), \ldots, P_r(n)$ such that $a_n =...
- 31
4
votes
1
answer
380
views
Symmetrizing with respect to Galois Group: Trace and Norm
In invariant theory the Reynold's Operator gives rise to an element invariant for that group.
For a Galois extension $K/F$ with $K=F[\alpha]$ the trace of $\alpha$ is an element of $K$. If $\alpha$ ...
- 2,543
6
votes
1
answer
712
views
Relation - Anabelian geometry and Tate conjecture
A lot has been said about the relation between Anabelian algebraic geometry and Mordell conjecture.
I would like to know what is the relation between Anabelian algebraic geometry and Tate ...
- 1,700
3
votes
1
answer
738
views
Gauss' Circle Problem at $\left ( \frac{1}{2}, \frac{1}{2} \right ) $
GCP (Gauss' Circle Problem) asks for a closed form for the number of square-lattice points inside a circle, centered at the origin, of radius $r$.
Let's denote by $N(r)$ the number of these points. ...
- 183
14
votes
1
answer
677
views
$\mathbb{Z}$-module structure of the subring generated by an algebraic number
Let $a$ and $b$ be algebraic numbers which are not necessarily algebraic integers. Is there some invariant that allows us to determine whether $\mathbb Z[a]$ and $\mathbb Z[b]$ are isomorphic as $\...
- 151
2
votes
2
answers
242
views
$n$-distant permutations more than not
Let $\mathfrak{S}_{2n}$ be the permutation group of the letters $[2n]=\{1,2,\dots,2n\}$. Call a permutation $\pi\in\mathfrak{S}_{2n}$ has an $n$-distant pair if there is some $j\in [2n-1]$ such that $\...
- 41.8k
9
votes
1
answer
518
views
Canonical models of Shimura varieties for GL2
Let $N \ge 4$ and let $Y_1(N)$ be the complex manifold $\Gamma_1(N) \backslash \mathcal{H}$, where $\Gamma_1(N) \subset \mathrm{SL}_2(\mathbf{Z})$ is the usual congruence subgroup and $\mathcal{H}$ ...
- 36.1k
5
votes
1
answer
270
views
Question about Fourier coefficients of a newform at primes
For $q:=e^{2\pi i z},$ let $f(z)=\sum_{n\ge 1}\lambda(n)n^{(k-1)/2}q^n$ be a normalized newform of type $(k,\chi)$ and level $N$. For any prime $p,$ we have
$$\lambda(p)=2\cos(\theta_p)\;\;\;\text{...
- 111
6
votes
0
answers
153
views
Large cubes in sum/difference sets
If $A$ is a subset of an abelian group with $|A+A|\leq K|A|$ then one can show that $A$ contains a large cube of size depending on $K$. Here a cube is a set of the form
$$C=\{a_0+\sum_{i=1}^d e_ia_i:...
- 131
3
votes
1
answer
454
views
Bound on the number of primitive divisors of the $n$th Fibonacci number
It is a result of Carmichael that for any integer $n > 12$, the Fibonacci number $F_n$ has at least one primitive divisor, that is, a prime factor $p$ such that $p$ does not divide any $F_m$ with $...
user40023
2
votes
0
answers
88
views
Example of action of an infinitely countable group that has important ergodic/statistical property?
I work in probability and I am looking for an important example of action of an amenable countable group in other areas of math for which the (pointwise) ergodic theorem is actually quite important. ...
- 21
6
votes
1
answer
483
views
Galois representation and weight one Hilbert modular form
Let $f$ be a primitive weight one Hilbert modular form for the totally real field $F$ (the weight of $f$ is parallel). Assume that $f$ is $N$-new, $p$-stable and cuspidal. Let $\rho:G_F \rightarrow \...
- 1,046
7
votes
1
answer
432
views
Intuition behind centralizers of Langlands parameters
In the description of the Langlands correspondence for $\mathbb{Q}_p$, we consider admissible representations of $G(\mathbb{Q}_p)$ for $G$ a reductive group defined over $\mathbb{Q}_p$, and admissible ...
- 933
1
vote
2
answers
856
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Intuition behind the Riemann $\zeta$ functional equation
Let $\Lambda(s)=\pi^{-s/2} \Gamma\left(\frac{s}{2}\right)\zeta(s).$ Then $\Lambda\left(\frac{1}{2} + s\right) = \Lambda\left(\frac{1}{2} - s\right)$ constitutes the functional equation for the Riemann ...
- 655
1
vote
1
answer
237
views
Sparse subset of $\mathbb{N}$ with a summation property
For $A\subseteq \mathbb{N}$ and an integer $k\geq 1$ we set $S_A(k) = \{B\subseteq A: B\text{ is finite and } \sum_{b\in B} b = k\}.$
We say that a set $A\subseteq \mathbb{N}$ is sparse if $$\text{lim ...
- 45.5k
39
votes
2
answers
3k
views
How can one understand the Eisenstein series E2 in terms of automorphic representation?
The weight 2, level 1 Eisenstein series $E_2(z)$ is a non-holomorphic automorphic form. It is defined as the analytic continuation to $s = 0$ of the series
$$ E_2(z, s) = \sum_{\substack{m, n \in \...
- 497
1
vote
1
answer
395
views
What is the Euler product for double summations?
I know that the Euler product of a summation of multiplicative function is given by
$$\sum_nf(n)=\prod_p (1+f(p)+f(p^2)+....),$$
and if we have the Möbius function then it will be
$$\sum_n\mu (n)...
- 49
11
votes
0
answers
540
views
Polynomial mapping primes to primes
Consider a non constant polynomial $P\in\mathbb{Z}[X]$ sending prime numbers to prime numbers. I encountered on the web two different proofs that $P$ is the identity polynomial, one on mathoverflow ...
- 2,710
0
votes
0
answers
254
views
Estimating number of integer points of a region under a hyperbola
Let $X$, $T$, and $x_0$ be positive real numbers. Consider the region in $\mathbb{R}^2$
defined by
$$
xy \leq X, \ \ x_0 \leq x \leq x_0 + T, \ \ \frac{X}{x_0 + T} \leq y.
$$
Let $A$ be the area ...
- 3,595
7
votes
0
answers
658
views
High dimensional analogue of Ramanujan's pi formula
The question below comes to my mind when I am trying to explore something related to the formulas found by Jesus Guillera:
a)Generalized hypergeometric function
$${}_3 F_2\left(\begin{matrix}1/4&...
- 3,317
55
votes
4
answers
4k
views
An interesting integral expression for $\pi^n$?
I came on the following multiple integral while renormalizing elliptic multiple zeta values:
$$\int_0^1\cdots \int_0^1\int_1^\infty {{1}\over{t_n(t_{n-1}+t_n)\cdots (t_1+\cdots+t_n)}} dt_n\cdots dt_1.$...
- 1,277
4
votes
2
answers
1k
views
Riemann Hypothesis and Euler product
It is conjecture that under certain conditions a L-function satisfies RH.
Among these conditions there is the necessity for the L-function to have an Euler product. (Some L-functions with a functional ...
- 1,121
11
votes
4
answers
1k
views
A simple number theory confirmation
Suppose $a,b\in\Bbb N$ are odd coprime with $a,b>1$ then is it true that if all four of $$x_1a+x_2b,\mbox{ }x_2a-x_1b,\mbox{ }x_1\frac{(a+b)}2+x_2\frac{(a-b)}2,\mbox{ }x_2\frac{(a+b)}2-x_1\frac{(a-...
- 13.7k
6
votes
2
answers
365
views
Provoking involutions further
Let $\mathfrak{S}_n$ denote the permutation group, and $I_0(n)=\sum_{j\geq0}\binom{n}{2j}\frac{(2j)!}{2^jj!}$ stand for involutions see A000085 for more interpretations. There is also these numbers $...
- 41.8k
5
votes
1
answer
341
views
Mumford-Tate conjecture cases with small $l$-adic monodromy groups
My question concerns the Mumford-Tate conjecture for abelian varieties over number fields.
Most proven cases (that I am familiar with) show that the l-adic monodromy group is as large as it can ...
- 337
7
votes
2
answers
297
views
On the solutions of $f(x) = y^k$ with $f \in \mathbb{Z}[x]$, $k \in \mathbb{N}$
I was wondering if it is true that the set of integer solutions of the equation
$$ f(x) = y^k $$
is finite, where $f$ is an irreducible integer polynomial of degree $d \ge 2$ and $y \in \mathbb{Z}$, $...
- 109
18
votes
5
answers
2k
views
An elementary, short proof that the group of units of the ring of integers of a number field is finitely generated
Dirichlet's unit theorem states that (i) the group of units, $\mathscr{U}_K$, of the ring of integers of a number field $K$ is finitely generated, and (ii) the rank of $\mathscr{U}_K$ is equal to $r_1 ...
- 9,636
9
votes
1
answer
849
views
Newform of a cuspidal Automorphic Representation
I was going through these notes https://www.dpmms.cam.ac.uk/~ty245/2008_AGR_Fall/2008_agr_week1.pdf . There, Theorem 9.2 states that: If $\pi ^{\infty}$ is a cuspidal automorphic representation of $\...
- 493
8
votes
1
answer
707
views
Hecke operator which changes character
In This MO question, Werner said that Hecke operator "changes" characters. I'm looking for any explicit theory of this kind, about Hecke operator with characters.
Actually, there are somewhat ...
- 1,911
1
vote
1
answer
82
views
Complexity of quadratic polynomials isomorphism
Two polynomials $f,g$ are isomorphic iff $f(x_1,\ldots x_n)=g(\pi(x_1, \ldots x_n))$ for a permutation $\pi$.
$f,g$ are equivalent if there exists invertible linear transormation
$A$ such that $f(X)=...
- 24.2k
1
vote
0
answers
182
views
Can we choose $h(\mathcal{O})$ elliptic curves such that they have same trace and endomorphism ring $\mathcal{O}$ but distinct j-invariants?
Let $\mathbb{F}_q$ be a finite field of characteristic $p$ with $q=p^a$ elements. Let $E$ be an ordinary elliptic curve defined over $\mathbb{F}_q$. We know that $\text{End}(E)\otimes\mathbb{Q}$ is ...
- 139
5
votes
2
answers
336
views
Methods to decide whether two positive definite ternary quadratic forms are in the same spinor genus?
Are there any effective methods to decide whether or not two positive definite ternary quadratic forms are in the same spinor genus?
For example, the following three forms are in the same genus
<...
2
votes
2
answers
546
views
What is the physical interpretation of the Riemann Hypothesis? [closed]
Some propositions in math can be modeled as a physical system. Has anyone done this for RH?
- 55
1
vote
0
answers
70
views
Packing the box $[0,X^{\delta_1}] \times [0,X^{\delta_2}] \times [0,X^{\delta_3}] \subseteq \mathbb{R}^3$ with cubes
Let $0 < \delta_1 \leq \delta_2 \leq \delta_3 \leq 1$, and consider the box $B := [0,X^{\delta_1}] \times [0,X^{\delta_2}] \times [0,X^{\delta_3}] \subseteq \mathbb{R}^3$. Let $X > 3$ say. Is it ...
- 579
1
vote
1
answer
522
views
Proof claimed of Gauss' Circle Problem
I just wanted to ask wether this problem has already been proved or not.
I know that there are 2 other posts that deal with exactly the same question, but I decided to ask it again, since they are ...
- 183
45
votes
5
answers
4k
views
Fibonacci series captures Euler $e=2.718\dots$
The Fibonacci recurrence $F_n=F_{n-1}+F_{n-2}$ allows values for all indices $n\in\mathbb{Z}$. There is an almost endless list of properties of these numbers in all sorts of ways. The below question ...
- 41.8k
8
votes
3
answers
842
views
Asymptotic formula for sums of four squares?
Does there exist asymptotic formula for ways to write n as sum of four squares? Or can this be proved impossible? I can only find reference for sums of five squares.
- 83
7
votes
2
answers
437
views
Generalization of Legendre`s conjecture
Legendre`s conjecture states that there is always a prime between $n^2$ and $(n+1)^2$ for every natural $n$.
It is natural to create following generalization:
Is it true that for every $\...
- 131
2
votes
1
answer
128
views
Density of "simultaneous squares"
Let $(u,v)$ be a pair of non-zero integers. We say that $(u,v)$ is a pair of simultaneous squares if for all primes $p$ dividing $u$, we have $\left(\frac{v}{p}\right) = 1$ and for all primes $q$ ...
- 25.5k
4
votes
2
answers
680
views
Is there always at least one prime in intervals of this form?
Take some 4 consecutive primes $p_n,p_{n+1},p_{n+2},p_{n+3}$ where $p_n \geq 5$.
Now form two products: $p_n \cdot p_{n+3}$ and $p_{n+1} \cdot p_{n+2}$.
Is there always at least one prime in the ...
- 131
15
votes
2
answers
638
views
Is the following series consisting of equally distributed $\pm 1$ bounded?
Apologise in advance if this problem isn't research-level (I'm quite certain it isn't). It's just I found it quite intriguing because it turned out to be much more subtle than it appeared at my first ...
- 253
6
votes
1
answer
345
views
Friable Numbers In Short Intervals: Density Estimates?
I am hoping for explicit numerical estimates like the following sample (with made up numbers, though it might be true): for every $n \gt 10^6$ and every $b$ with $b^2 \lt n \lt b^3$, the number of ...
- 13k
32
votes
1
answer
916
views
Strange convergence of Euler's series for $\zeta(2)$
Using Maple to compare $\pi^2$ and the partial sums of $6\sum_{n=0}^{\infty}\frac{1}{n^2}$ I have noticed something that appears strange.
For instance, let $S_{k}=6\sum_{n=0}^{k}\frac{1}{n^2}$ be ...
- 403
2
votes
1
answer
308
views
Counting certain tuples
The following counts Cohen-Macaulay modules in a certain Gorenstein algebra. I search for a closed formula, see also Elementary interpretation of a homological result .
Let $n \geq 4$ and $w >3$ ...
- 26.1k
3
votes
1
answer
267
views
Elementary interpretation of a homological result
Translating a homological/representation theoretic result into elementary things, I obtained the following (in case I made no mistake):
Let $n \geq 4$ and $w >3$ and let $w$ be an unit in $\mathbb{...
- 26.1k
2
votes
2
answers
259
views
Prove a family of series having integer coefficients
I encountered a certain family of infinite series in some work, which is given by
$$F_r(x)=\frac1{2^r}\sum_{k=0}^r\binom{r}k\frac1{1+x(2k-r)^2}.$$
I've convincing date to believe the following is true,...
- 41.8k
17
votes
1
answer
978
views
Theta functions, re-expressed
Recall the classical $\theta(q):=\prod_{k=1}^{\infty}(1-q^k)$ and
define the sequences $a_n$ and $b_n$ by
$$\frac{\theta^3(q)}{\theta(q^3)}=\sum_{n=0}^{\infty}a_nq^n \qquad \text{and} \qquad
F(q):=\...
- 41.8k
11
votes
1
answer
346
views
Is there an asymptotic formula that describes the correlation of multiplicative inverses in Farey sequences?
Added 14/04/17: So I am placing a bounty on this question because I am very interested in knowing about strategies for calculating the asymptotic behaviour of this sum. I have calculated $S(X)$ for $X&...
- 2,470
2
votes
1
answer
131
views
Combinatorial proof for bicolored graphs
A bicolored graph is a graph each vertex of which has been assigned one of two colors such that each edge connects vertices of different colors. A bipartite graph is a graph $G$ which admits such a ...
- 41.8k
13
votes
3
answers
696
views
Arithmetic problem for bicolored graphs
A bicolored graph is a graph each vertex of which has been assigned one of two colors such that each edge connects vertices of different colors. A bipartite graph is a graph $G$ which admits such a ...
- 41.8k