Prime numbers, Diophantine equations, analytic number theory, algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

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2
votes
0answers
55 views

Monte Carlo variant of Hilbert's Tenth Problem

Let $k \in \mathbb{N}$. Given an algorithm $\mathcal{A}$ which takes as argument a polynomial $P \in \mathbb{Z}[x_1,\dots,x_k]$ and either returns true or false, we say that $\mathcal{A}$ works for ...
-2
votes
0answers
60 views

College - Ecuation to solve in 3 variables (p,q,r) in [1,2] [on hold]

This is the image containing the ecuation
0
votes
0answers
101 views

Mellin transform on $\mathbb{Z}[\omega]$

I'm eager to ensure some facts which are elementary for many experts here. Let $\omega=\frac{-1+i\sqrt{3}}{2}$ be a complex cube root of unity. The Eisenstein integers $\mathbb{Z}[\omega]$ (a unique ...
0
votes
0answers
25 views

Projection formula for smooth representations of locally profinite groups

Let $G$ be a locally profinite (i.e., locally compact, Hausdorff, and totally disconnected) topological group, $H \le G$ a closed subgroup, $(\pi, V)$ a smooth representation of $G$, and $(\sigma, W)$ ...
3
votes
1answer
59 views

Lower bound of first moment of $L$-function on $\mathrm{GL}(3)$

Let $\pi$ be an automorphic form on $\mathrm{GL}(3,\mathbb{A}_{\mathbb{Q}})$. Do we know any case that \[\int_0^{T} \left|L(\frac{1}{2} + it, \pi)\right| dt \gg T\] holds unconditionally? I know the ...
5
votes
0answers
68 views

On the ratio of Gilbreath sequences

Definitions: let $n \in \mathbb{N}_{>0} \cup \{ \infty \}$ and let $E_n$ be the set of sequences $(d_i)_{i=1}^n$ such that $d_1=1$, $d_i$ is an even integer (for $i > 1$) and $0<d_i \le i$. ...
1
vote
0answers
105 views

is there an analogy between fractals and automorphic forms? [on hold]

Disclaimer: this question is rather vague and thus might not be suitable for this site. Still, as I already asked a similar question on MSE and got no feedback, I finally decided to take the plunge ...
-4
votes
0answers
75 views

an question about number theory [on hold]

Let $s_i=\frac{(q^n-1)...(q^n-q^{i-1})}{(q^{i-1})...(q^i-q^{i-1})}$, where $q$ is prime and $n$ is a positive integer. Now can anyone tell me this, $\lim_{n\mapsto \infty}\frac{\sum_{1\leq i\leq ...
1
vote
0answers
75 views

Clarifications on twisted forms

Suppose $F = F(\bar{k})$ is a finite algebraic group over a number field $k$. The absolute Galois group $\Gamma_k$ of $k$ acts on $F$ by group automorphisms via a homomorphism $\rho: \Gamma_k \to ...
6
votes
0answers
347 views

Number Theory over $\mathbb{F}_q [t]$, why is it important/interesting?

This is something I was curious about, but not really sure why. I was hoping maybe someone could give me a good explanation. I understand that there are similarities between $\mathbb{Z}$ and ...
3
votes
2answers
168 views

Inequality for an integral involving $ \exp $, $ \sin $ and $ \cos $

Let $ t > 0 $ and $ k \in \{ 0,1,2,\ldots \} $. Does the following inequality hold? $$ \int_{k + 1/2}^{k + 3/2} \frac{x \sin(2 \pi x)}{1 + 2 e^{2 \pi t} \cos(2 \pi x) + e^{4 \pi t}} ...
3
votes
0answers
159 views

A good book on adeles and ideles

Many results in number theory are stated either in a classical language or in an adelic one. I am often impressed of the efficiency and the satisfactory computational properties of the adelic setting, ...
0
votes
2answers
148 views

Equidistribution of rational points on an algebraic variety

Suppose that we have a variety $X \subset \mathbb{P}^{n}$ defined over $\mathbb{Q}$. Suppose we have $S$ many rational points on $X$ inside the box defined by $|x_i| \leq B_i$ for $i = 0, \cdots, n$, ...
2
votes
0answers
126 views

Tate's conjecture and symmetry of Hodge-Tate weights

I'm reading Bellaiche's notes on the Block-Kato conjecture (Hawaii summer school). Here is the link http://people.brandeis.edu/~jbellaic/BKHawaii5.pdf At page 10 he claims that an indirect ...
5
votes
0answers
74 views

The topology on the Robba ring

I've been reading Kedlaya's paper http://arxiv.org/abs/math/0208027 on finiteness of rigid cohomology and there's something I can't quite resolve in my understanding of the topology on the Robba ring. ...
1
vote
1answer
135 views

Is the Cassels-Tate pairing defined for elliptic curves over function fields?

The Cassels-Tate pairing is typically defined for elliptic curves (or abelian varieties) over number fields, but it seems like it should be defined for elliptic curves over function fields as well. ...
7
votes
1answer
228 views
+100

What is the decomposition group at $p$ in the Galois group unramified outside $\ell?$

Let $p\ne\ell$ be two prime numbers, and let $K_{\ell}$ be the maximal extension of $\mathbb Q$ ramified only at $\ell$ and $\infty$ (i.e. it is the fixed subfield of $\overline{\mathbb Q}$ by all the ...
0
votes
0answers
57 views

What is the proper Zariski-closed subset in these examples for Vojta's more general abc conjecture?

In A more general abc conjecture, p. 7 Paul Vojta conjectures: If $x_0,\ldots x_{n-1}$ are nonzero coprime integers satisfing $x_0 + \cdots x_{n-1}=0$ $$ \max\{|x_0|,\ldots |x_{n-1}|\} \le C ...
5
votes
0answers
77 views

Isotropy of Apollonian disk-packing

Is there any sense in which the "epsilon-tail" of an Apollonian disk-packing (by which I mean the union of the disks of radius less than epsilon) exhibits more and more statistical isotropy as epsilon ...
7
votes
2answers
827 views

Does the Gamma function preserve integers?

Does the Gamma function $\Gamma: \mathbb{C} \to \mathbb{C}$ preserve the Kummer ring $\mathbb{Z}[\exp(2\pi\imath/m)]$? And if not, then what about the Gaussian integers $\mathbb{Z}[\imath]$ or the ...
5
votes
1answer
315 views

Why do $12$ and $120$ occur very often in the denominators of $\zeta(-n)$ for odd $n$?

$\zeta(-n) = - \dfrac{B_{n+1}}{n+1}$ $\zeta(-2n) = 0$ $\zeta(-1) = - \dfrac{1}{12}$ $\zeta(-3) = \dfrac{1}{120}$ $\zeta(-5) = - \dfrac{1}{252}$ $\zeta(-7) = \dfrac{1}{240}$ $\zeta(-9) = - ...
1
vote
0answers
61 views

Definition of primitive divisor of a Lucas sequence

If $\{a_n\}_{n=1}^\infty$ is a sequence of integers, then the definition of primitive divisor of one of its terms is quite natural: Def. 1. A prime number $p$ is a primitive divisor of $a_n$ if $p ...
-3
votes
0answers
44 views

Binomial theorem [closed]

I have a problem with binomial theorom. What is the result of solving of inequality: (n 1) + (n 2) + (n 3) + ... (n n) > 32 Sorry for this notation. Thanks for answer.
3
votes
1answer
303 views

what would be the consequences on the distribution of primes of $\Lambda=\infty$?

It is widely believed that the quantity $\Lambda:=\lim\sup\dfrac{t_{n+1}-t_{n}}{2\pi/\log t_{n}}$, where $t_{n}$ is the imaginary part of the $n$-th non-trivial zero on the critical line of the ...
3
votes
0answers
119 views

Moduli interpretation of Eisenstein series

Let $N \geq 11$ be an integer and consider the basis of Eisenstein series for $M_2(\Gamma_0(N))$ described in Theorem $4.6.2$ of Diamond--Shurman's book. Pick and Eisenstein series $F$ in this basis. ...
1
vote
1answer
132 views

What is the ring $A_{\Gamma}$ in the Cohen-Lenstra Heuristics?

I understand the work in Cohen and Lenstra's paper that leads up to the heuristics themselves, where they count weighted averages of functions defined over isomorphism classes of $A$-modules, where ...
-1
votes
0answers
83 views

Squares sum problem [closed]

In how many ways can I add add differential elements of set of squares (1, 4, 9, 16...) to get sum less then n? If I have n = 25, I can add 1 + 4, 1 + 9, 1 + 16, 1 + 4 + 9 and so on... There is some ...
7
votes
0answers
131 views

Intersection of a ring class field of a quadratic field K with the cyclotomic extension of K

Let $K$ be a quadratic field. Let $f\in\mathbb{Z}_{\geq 1}$. Let $\mathcal{O}_f=\mathbf{Z}+f\mathcal{O}_K$ be the unique order of $K$ of index $f$ in $\mathcal{O}_K$. Let $H_f^{ring}$ denote the ring ...
4
votes
0answers
118 views

Moduli interpretation of Hecke operators on Shimura curves

In his book on Automorphic Forms, Shimura gives (chapter 9) definitions of the the Hecke operators for Shimura curves. One can give definitions of the Hecke and Atkin-Lehner operators in terms of the ...
6
votes
0answers
77 views

Weyl law for Maass forms with nontrivial character

The classical Weyl law for $\Gamma = \mathrm{SL}_2(\mathbb{Z})$ counts the number of Maass cusp forms on $\Gamma \backslash \mathbb{H}$ with Laplace eigenvalue less than $T$. This is originally due to ...
2
votes
3answers
257 views

Prime factors of the members of a certain recurrence

It is possible to prove elementarily that there are infinitely many primes that divide some element of the sequence $a_0 = k\ge 0$, $a_n = a_{n-1}^2+ 1$ for all $n\ge 1$ by showing that for all $m$, ...
1
vote
0answers
53 views

density of zeroes of Epstein zeta functions on vertical strips

There are many results (e.g. Davenport and Heilbronn) asserting that Epstein zeta functions in general have zeroes outside the line $\mathrm{Re\ } s = n/4$. Is there any result about the density of ...
3
votes
4answers
386 views

solutions to special diophantine equations [closed]

Let $0\le x,y,z,u,v,w\le n$ be integer numbers obeying \begin{align*} x^2+y^2+z^2=&u^2+v^2+w^2\\ x+y+v=&u+w+z\\ x\neq& w \end{align*} (Please note that the second equality is ...
-9
votes
0answers
79 views

Find all numbers $x$ satisfying $x^3 \equiv 1 \pmod A$ [closed]

Given any two numbers $M,N \in \{1,2,\ldots,10^{18}\}$, find all numbers $x$ lying between $M$ and $N$ satisfying $$x^3\equiv1 \pmod A,$$ where $A$ can be any number $A \in \{1,2,\ldots,10^{5}\}$ . ...
0
votes
0answers
51 views

For which recurrence relations is it decidable whether a formal power series has a maximal zero coefficient?

In this MSE question, I asked whether one can prove that a generating function has infinitely many coefficients equal to zero. The answer given (and accepted) to that rather broad question was “No”. ...
0
votes
0answers
64 views

Are the first ten zeros of this Dedekind zeta function non-simple?

This question asks about the zeros of the zeta function of the number field with defining polynomial: ...
0
votes
1answer
292 views

The periodic architecture underlying the natural numbers [closed]

EDIT In the original version of this post I did not include a well specified mathematical question, and learning by failing, I realize there should have been. Closing or deleting the question is the ...
8
votes
0answers
223 views

Zero's in the decimal representation of powers of 3

This looked like an easy exercise, when a friend of mine asked me if I know a way to prove that the decimal representation of $3^k$ always contains a zero for $k\ge k_0$, but the more I think about ...
14
votes
0answers
349 views

Does $\binom{2n}{n} \equiv 2 \pmod p$ ever hold?

Well, the title does not tell the whole story; the complete question is: Are there any primes of the form $p=2n(n-1)+1$, with integer $n\ge 1$, such that $$ \binom{2n}{n} \equiv 2\pmod p ? $$ ...
3
votes
0answers
106 views

Fourier expansions of newforms at width-1 cusps

Let $f_E$ be the newform attached to the Elliptic Curve $E$ with cremona label $\textbf{100a1}$ and let $\alpha = \left[\begin{matrix} 1&0 \\ 10&1 \end{matrix}\right] \in SL_2(\mathbb{Z})$. ...
1
vote
1answer
109 views

Factorisation of local quaternionic zeta functions

Vignéras, in her Arithmetics of quaternion algebras, begin chapter II.4 recalling that we know the number of integer ideals of fixed norm of a quaternion algebra $H$ over a local field $K$, ramified ...
0
votes
1answer
214 views

Conjecture on prime numbers

Given a prime $p$, let $a_n=pn+n-1$. I have noticed that $\forall{p}\exists{n}\in[2,p]:a_n\in\mathbb{P}$. For example: $p=7,a_3=23,a_4=31,a_6=47$. What is this conjecture called, and has it been ...
-3
votes
1answer
158 views

Andrica's and Legendre's Conjectures [closed]

My question is, which of these two conjectures is stronger, Andrica's or Legendre's? Could proving one prove the other? If the upper bound for the prime gap above any given natural number $n$ were to ...
0
votes
0answers
42 views

At least a cubic form in a hypercube which is equivalent to a reduced form which is not monic

In my research appeared a question, but I am not too familiar to the theory of reduced forms (sorry if the question is easy, but I was not able to figure out it): Let ...
7
votes
0answers
166 views

Reciprocal polynomials with roots off the unit circle

A polynomial is called reciprocal if its coefficients read forwards are the same as those read backward - there is an obvious translation of reciprocal polynomials into cosine polynomials (since a ...
6
votes
1answer
116 views

Cuspidal modular forms - toroidal or minimal compactification?

Let $Y$ be a Siegel variety and let $X$ be a toroidal compactification of $Y$. For any tuple of integers $\underline k$ we have the usual sheaf $\omega^{\underline k}$. The space of modular forms of ...
0
votes
0answers
73 views

Conjectured alternate form for vanishing of $\Re\zeta(1/2+it)$ except at zeros

Heavily based on Agno's question. Define: $$ \chi(s)=\pi^{-(\frac{s}{2})} \Gamma(\frac{s}{2}) $$ Agno conjectured: only for $\sigma=\frac12$, $\Re(\chi(s)) = \Re(\zeta(s)) =0$ is always true, ...
1
vote
0answers
136 views

on the Rankin-Selberg L-function

Let $n,m$ be two different positive integers. I heard that for cuspidal tempered automorphic representations $\pi_{n}$ and $\pi_m$ of $GL_n$ and $GL_m$, the Rankin-Selberg L-function $L(s,\pi_n ...
-1
votes
1answer
140 views

Longest sequence of sum of distinct squares [closed]

I want to find longest sequence of distinct squares that $\alpha_{_1}$ + ... + $\alpha_{_n}$ is given number. In particular I want to find largest square in that sequence. I've tried use Lagrange's ...
3
votes
0answers
274 views

polynomials with roots on the unit circle

Suppose $P(x) \in \mathbb{Z}[x]$ is irreducible, and such that at least one of its roots has modulus $1.$ Is there anything we can say about the reduction of $P(x)$ modulo primes? Do these have some ...