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

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Prime square offsets: Why is +7 more frequent than -7?

For a prime $p$, define $\delta(p)$ to be the smallest offset $d$ from which $p$ differs from a square: $p = r^2 \pm d$, for $d,r \in \mathbb{N}$. For example, \begin{eqnarray} \delta(151) & = &...
Joseph O'Rourke's user avatar
6 votes
1 answer
2k views

The connection between the Weil conjectures and Ramanujan's conjecture

I'm writing an essay about Ramanujan's conjecture and have some questions: 1 How is Ramanujan's conjecture connected with the Weil conjectures? 2 How could Ramanujan's conjecture be assumed true or ...
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5 votes
0 answers
124 views

Anyone got two Galois reps to compare?

I've got a new criterion for comparing Galois reps which are four dimensional if we know the kernel of the residual representation mod $5$ (or any large odd prime) and the Sato-Tate groups (should be ...
Watson Ladd's user avatar
  • 2,419
6 votes
1 answer
352 views

Semi-Simplicity of Mod-$\ell$ Galois Representations

I am currently studying mod-$\ell$ Galois representations of $CM$ elliptic curves. More precisely, let $\ell$ be a prime, $K$ a number field and $E/K$ a $CM$ elliptic curve, where all endomorphisms of ...
Rdrr's user avatar
  • 881
3 votes
1 answer
339 views

Squarefree values of polynomials at prime arguments

This is a reference request. Assume that $f_1,\ldots,f_r \in \mathbb{Z}[t]$ are non-zero linear polynomial. Letting $\mu$ be the M\"{o}bius function, is there any work on $$ \sum_{p\leq x} \prod_{i=...
Dr. Pi's user avatar
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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 =...
user108968's user avatar
4 votes
1 answer
379 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 $\alpha$ ...
P Vanchinathan's user avatar
6 votes
1 answer
710 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 ...
tttbase's user avatar
  • 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. ...
user3141592's user avatar
14 votes
1 answer
676 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 $\...
user108921's user avatar
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 $\...
T. Amdeberhan's user avatar
9 votes
1 answer
516 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 $\mathcal{H}$ ...
David Loeffler's user avatar
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{...
square-free's user avatar
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:...
Brandon Hanson's user avatar
3 votes
1 answer
453 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 $...
user avatar
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. ...
letta's user avatar
  • 21
6 votes
1 answer
482 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 \...
Adel BETINA's user avatar
  • 1,046
7 votes
1 answer
430 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 ...
Alexander's user avatar
  • 861
1 vote
2 answers
853 views

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 ...
Lucian's user avatar
  • 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 ...
Dominic van der Zypen's user avatar
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 \...
little dog's user avatar
1 vote
1 answer
393 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)...
Asmaa's user avatar
  • 49
11 votes
0 answers
537 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 ...
Ayman Moussa's user avatar
  • 2,575
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 ...
Johnny T.'s user avatar
  • 3,547
7 votes
0 answers
656 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&...
Y. Zhao's user avatar
  • 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.$...
Leila Schneps's user avatar
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 ...
Bertrand's user avatar
  • 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-...
Turbo's user avatar
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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 $...
T. Amdeberhan's user avatar
5 votes
1 answer
338 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 ...
Student88's user avatar
  • 337
7 votes
2 answers
296 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 $y \in \mathbb{Z}$, $...
trenta3's user avatar
  • 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 ...
Salvo Tringali's user avatar
9 votes
1 answer
844 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 $\...
Shubhodip Mondal's user avatar
8 votes
1 answer
706 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 ...
Seewoo Lee's user avatar
  • 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)=...
joro's user avatar
  • 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 ...
tiansong's user avatar
  • 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 <...
whl likes fish's user avatar
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?
liu's user avatar
  • 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 ...
SJY's user avatar
  • 579
1 vote
1 answer
521 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 ...
user3141592's user avatar
45 votes
5 answers
3k 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 ...
T. Amdeberhan's user avatar
8 votes
3 answers
837 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.
NTnewbie's user avatar
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 $\...
Paladin's user avatar
  • 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 $q$ ...
Stanley Yao Xiao's user avatar
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 ...
Paladin's user avatar
  • 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 ...
Vim's user avatar
  • 253
6 votes
1 answer
343 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 ...
Gerhard Paseman's user avatar
32 votes
1 answer
915 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 ...
Pedro Namtior's user avatar
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 $w >3$ ...
Mare's user avatar
  • 25.8k
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{...
Mare's user avatar
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