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

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61 views

Equations over $\mathbb{Z}[[T]]$ vs. equations over $\mathbb{Z}_p$

This question might be deemed totally unanswerable, unless there is an obvious counterexample. Answers to either effect would be welcome. Question. Let $X$ be a finite-type scheme over ...
5
votes
1answer
220 views

Parity of primes [duplicate]

While working on a completely different (combinatorial) problem, I ran a simple program to calculate the parity of the first ~50000 primes (number of 1s in their binary representation modulo 2). The ...
2
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0answers
119 views

What are the minimal degrees of the real and imaginary part of an algebraic complex number? [on hold]

Let $z=a+bi\in\mathbb C$ with $b\ne0$ be an algebraic complex number of minimal degree $n$. It is obvious that $a=\dfrac {z+\bar{z}}2$ and $b=\dfrac {z-\bar{z}}{2i}$ are also algebraic. For $n=3$, it ...
3
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1answer
103 views

When are all sums of the elements of a set different?

Consider a set $S = \{x_1, \dots, x_n\} \subset \mathbb{Q}\setminus\{0\}$ and assume that for any $I, J \subset [n]$ with $I \neq J$ we have that \begin{equation} \sum_{i \in I} x_i \neq \sum_{j \in ...
5
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1answer
179 views

What does the sum of the reciprocals of all the highly composite numbers converge to?

I've calculated the sum of the reciprocals of all the $156$ first highly composite numbers up to $10^{18}$: $\sum \dfrac{1}{HCC(n)} = \dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{6} + \dfrac{1}{12} + ...
-2
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0answers
44 views

Proving how many divisors of a prime factorization (including 1 and n) there are [on hold]

I'm trying to figure out this problem but I'm not sure where to start. Could anyone explain to me the question a bit more in depth or give a few hints? The problem is, Let n in Z+ with prime ...
3
votes
2answers
275 views

Intuition behind $\zeta(2) = \frac{\pi^2}{6}$ [on hold]

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. ...
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2answers
282 views

Splitting integers 1, 2, 3, … n to avoid least possible sum

For each positive integer n, partition the integers 1, 2, 3, … 2n into two sets of n integers each. Let g(n) be the least integer such that there is such a partition in neither of whose parts there is ...
1
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1answer
70 views

On the Saito Kurokawa representation

I know Saito-Kurokawa(SK) representation is the famous non-tempered representation of $SO(5)$. But since the tempered or non-tempered terms are concerned with local phenomenon, I am wondering that ...
2
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1answer
140 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 ...
2
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0answers
54 views

Invariant Theory over finite adeles

Classical invariant theory, among the other things, classifies polynomial functions over a vector space $V$ endowed with a quadratic form $Q$ which are invariant under the action of $SO(V,Q)$. I am ...
8
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1answer
165 views

A curious Gauss-Sum type identity

Let $q=e^{2\pi i/m}$, $a\in\mathbb{R}$ and $1\leq j\leq m-1$. I would like to prove that: $$(a-1)\sum_{n=0}^{m-1} q^n\frac{\prod_{k=0}^{j-2} (q^{n+k+1}-a)}{\prod_{k=0}^{j} (aq^{n+k}-1)}=0.$$ For ...
3
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0answers
135 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 ...
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0answers
268 views

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

This is the image containing the ecuation
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0answers
142 views

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

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 ...
2
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2answers
102 views
+100

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
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1answer
83 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 ...
6
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0answers
136 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
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0answers
115 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 ...
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0answers
82 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
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0answers
82 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 ...
7
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0answers
366 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
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2answers
178 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
1answer
282 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, ...
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2answers
153 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
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0answers
129 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
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0answers
78 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
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1answer
176 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
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1answer
334 views

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 ...
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0answers
58 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
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0answers
80 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
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2answers
835 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
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1answer
323 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) = - ...
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0answers
65 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
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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
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1answer
319 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
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0answers
126 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
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1answer
137 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 ...
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0answers
84 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 ...
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135 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 ...
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0answers
121 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
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0answers
80 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
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3answers
261 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 $m$, ...
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0answers
55 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
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4answers
392 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 ...
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0answers
52 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”. ...
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0answers
65 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
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1answer
295 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
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0answers
230 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
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354 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 $$ \binom{2n}{n} \equiv 2\pmod p ? $$ ...