# Tagged Questions

**8**

votes

**2**answers

320 views

### An algorithm for Poincare recurrence time

Define the function $[0,+\infty) \rightarrow R$:
$$ f = \cos (t) + \cos (\sqrt{2} t) + \cos (\sqrt{3} t) + \cos (\sqrt{5} t ) . $$
I want a number $t $ bigger than $10^7$ such that
$$ f(t) > 4 ...

**6**

votes

**1**answer

405 views

### Multiplicity one theorem

I am reading Dorian Goldfeld's book Automorphic forms and L functions for the groups GL(n,R) ...

**2**

votes

**1**answer

352 views

### A property of e?

Define $f(n) = \lfloor {ne}\rfloor$ if $n$ is odd and $f(n) = \lfloor {n/e}\rfloor$ if $n$ is even. Is the set $\{n, f(n), f(f(n)),\dots\}$ bounded for every $n$?
Computer sampling suggests that ...

**1**

vote

**0**answers

48 views

### Asymptotics on number of bounded prime gaps [duplicate]

It's been over 2 years since the groundbreaking paper by Yitang Zhang in which he has shown that infinitely many prime pairs are by some constant $H$, with $H\leq 70000000$. Over the course of the ...

**0**

votes

**1**answer

88 views

### Weil group of a local field, small notational problem

In Bushnell and Henniart, The Local Langlands conjecture for GL(2), there is a proposition on p. 184 in which they prove the following:
Let $F$ be a non-archimedean local field, $\mathcal W_F$ its ...

**2**

votes

**1**answer

336 views

### A question on the bounds of the $n$-th composite $c_n$

While trying to prove the inequality $$c_{p_n-m}+c_{m-n}>p_n+2$$ I tried the bounds of $c_n$ (denotes the $n$-th composite number) given in this paper to prove that the sum $c_{p_n-m}+c_{m-n}$ ...

**5**

votes

**1**answer

225 views

### Are all complex zeros of $\dfrac{\zeta'}{\zeta}(s) \pm \dfrac{\zeta'}{\zeta}(1-s)$ on the critical line $\Re(s)=\frac12$?

Numerical evidence suggests that all complex zeros (real ones exist as well) of:
$$\frac{\zeta'}{\zeta}(s) \pm \frac{\zeta'}{\zeta}(1-s)$$
reside on the critical line with $\Re(s)=\frac12$.
I made ...

**17**

votes

**1**answer

405 views

### Completely multiplicative functions with values in $\{-1,1\}$

This question is from Eric Saias and myself:
Let $A$ be the set of abscissas of convergence of Dirichlet Series $\sum_{n\ge 1} \frac{f(n)}{n^s}$
where $f(n)$ is completely multiplicative and $f(n) \in ...

**4**

votes

**0**answers

256 views

### Are there always at least *five* divisions?

@JosephO'Rourke asked a question about a Collatz like function related to primes:
$f(n) =
\begin{cases}
n^2 & \text{if} \;n \;\text{is prime} \\
\lfloor n/2 \rfloor & \text{if} \;n ...

**24**

votes

**1**answer

1k views

### Is there an arbitrarily long arithmetic progression whose members are palindromes?

I suspect there isn't for some simple reason, but I could not find anything on it and if the opposite would hold in a more general sense, then that would solve this question.

**1**

vote

**2**answers

155 views

### Roots of not-necessarily reciprocal polynomials

Consider (irreducible) monic polynomials with integer coefficients which satisfy some fixed linear condition $L$ (for example, the coefficient of $x$ is $1,$ or something more complicated). The ...

**3**

votes

**2**answers

290 views

### roots of reciprocal polynomials

Consider the set $S = \{x \in \mathbb{R} \left| f(x) = 0\right. \},$ where $f$ is a reciprocal monic irreducible polynomial with integer coefficients (reciprocal means that the sequence of ...

**13**

votes

**2**answers

361 views

### Elliptic curves and supercuspidal representations of conductor $p^2$

Let $E$ be an elliptic curve defined over $\mathbf{Q}$. Let $p \geq 5$ be a prime of additive reduction for $E$.
Let $f$ be the newform associated to $E$, and let $\pi$ be the irreducible admissible ...

**0**

votes

**0**answers

65 views

### Explicit twisted Padé approximants

This is a follow-up of Twisted Padé approximants
Let $z\in\mathbb Z_p$ with $v_p(z)>0$. One puts $f_z(x)=(1+z)^x$ for all $x\in\mathbb Z_p$. I try to determine the twisted Pade approximants ...

**1**

vote

**0**answers

92 views

### What are the differences between p-adic Whittaker functions and archimedean Whittaker functions? [closed]

What are the differences between p-adic Whittaker functions and archimedean Whittaker functions? Are there some references about the differences? Thank you very much.

**3**

votes

**1**answer

189 views

### Explicit description of rings of Witt vectors

I have some basic questions on the rings of Witt vectors. The first example one looks at is $W(\mathbb{F}_{p})= \mathbb{Z}_{p}$. Is it known if $W(\mathbb{F}_{p}[x]/(x^{n})) = ...

**5**

votes

**2**answers

262 views

### What is a “generalized zeta function”?

Out of procrastination I computed $$\sum_{k=1}^\infty k^{-k^2}\sim 1.06255080549625593786944593879.$$
The inverse symbolic calculator identified this number as "From generalized Zeta function". I do ...

**3**

votes

**3**answers

383 views

### Are There Infinitly Many $n$ Which $a\times n!+1$ Be Composite?

$ax+1$ is a linear polynomial with integral coefficients.
Are there infinitly many $n$ which $a\times n!+1$ be composite?
As I know this problem is true for polynomials with degree greater that 1, ...

**1**

vote

**0**answers

50 views

### Distribution of smooth values of polynomials

Let $\xi$ be a positive parameter. We say a positive integer $n$ is $\xi$-smooth, or friable, if for all primes $p$ dividing $n$, we have $p \leq \xi$. Let $T(X, \xi)$ denote the set of $\xi$-smooth ...

**2**

votes

**1**answer

193 views

### Siegel-Walfisz for the Möbius function

I am working through the proof of the Bombieri-Vinogradov theorem in Analytic Number Theory (Iwaniec, Kowalski). My problem is that on page 424, it is said that $\mu(m)$ satisfies $D_f(x;q,a)\ll ...

**13**

votes

**1**answer

452 views

### A Collatz-like function that bifurcates on primes

This is likely piling one mystery on another, but ...
I was exploring a function $f(n): \mathbb{N} \mapsto \mathbb{N}$ defined as follows:
$$
f(n) =
\begin{cases}
n^2 & \text{if} \;n \;\text{is ...

**0**

votes

**1**answer

100 views

### Twisted Padé approximants

Let $f$ be a continuous function defined on $\mathbb Z_p$. By Mahler theorem, there exists a sequence $(\gamma_k)_{k\in\mathbb N}$ of $\mathbb Z_p$ such that for every $z\in\mathbb Z_p$
...

**7**

votes

**1**answer

764 views

### A Question about Palindromic Numbers and System of Arithmetic Progression

Based from Harminc and Sotak's result, www.fq.math.ca/Scanned/36-3/harminc.pdf
We know that under certain condition, an arithmetic progression can contain an infinitely many palindromes.
My question ...

**1**

vote

**1**answer

251 views

### Sums of two squares: positive lower density? [duplicate]

This question was (indirectly) raised in this post.
A set $A\subseteq \mathbb{N}$ has positive lower density if $$\text{lim inf}_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n} > 0.$$
Does the set ...

**4**

votes

**3**answers

358 views

### genus 2 Siegel theta series of 3-dimensional lattices

Let $(V,f)$ be a $3$-dimensional positive definite quadratic space over $\mathbf Q$.
Let $G(V)$ be a set of representatives of the isometry classes of maximal integral lattices on $V$.
To an ...

**1**

vote

**0**answers

66 views

### Representations of Hamilton's real/complex quaternions algebra

A lot of works and questions deal with classifying representations of a simple central algebra of given dimension over a non-archimedean field, for instance here.
But do we know precisely such a ...

**3**

votes

**2**answers

378 views

### Sums of sets of lower density 0

We say that a set $A\subseteq \mathbb{N}$ has lower density 0 if $$\text{lim inf}_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n} = 0.$$
Given $A,B\subseteq \mathbb{N}$ we set $A+B = \{a+b: a\in A, b\in ...

**3**

votes

**1**answer

126 views

### Simultaneous approximation by rationals with relatively prime numerators

The following seems hard to me (or perhaps just not true), but perhaps I am mistaken. It is known that given irrational numbers $x_1$ and $x_2$, there are infinitely many simultaneous rational ...

**3**

votes

**2**answers

255 views

### n torsion groups of quadratic twists of elliptic curves

If $E$ is an elliptic curve over a number field $K$ and $E^{F}$ is a quadratic twist of $E$. Then it is stated in ``Ranks of twists of elliptic curves and Hilbert’s tenth problem" due to Mazur and ...

**13**

votes

**3**answers

474 views

### Which irrationals yield bounded sets of iterates?

For $r > 0$, define $f(n) = \lfloor {nr}\rfloor$ if $n$ is odd and $f(n) = \lfloor {n/r}\rfloor$ if $n$ is even. For which real numbers $r$ is the set $\{n,f(n), f(f(n)),\dots\}$ bounded for every ...

**1**

vote

**4**answers

605 views

### Distribution of composite numbers

I have moved this question to math.stackexchange.com. People who are interested in this question can discuss at :http://math.stackexchange.com/questions/1272431/distribution-of-composite-numbers
...

**2**

votes

**0**answers

131 views

### Mod 2 modular forms in levels 5 and 25--how to account for this Hecke isomorphism?

The space $P1$ of my earlier question 203755 "Two spaces attached to mod 2 level 9 modular forms...", is essentially the space of mod 2 level 3 modular forms. That such a space should appear inside ...

**3**

votes

**0**answers

276 views

### How is $ \sum_{x \in X(\mathbb{F}_q)} \dots $ a generalization of cardinality?

This quarter Maxim Kontsevich is offering a course on exponential integral. There is not much in the way of notes, but is one page with mysterious comments.
Let $X$ be an algebraic variety over ...

**3**

votes

**0**answers

116 views

### Farey Fractions Estimate Equivalent to the Prime Number Theorem?

Wikipedia's article on Farey Fractions points to an article of Jerome Franel that some averages are equivalent to the Riemann hypothesis.
Let $F_n$ be the $n$-th Farey sequence, then the number of ...

**5**

votes

**2**answers

290 views

### A (likely) positivity property of the Lerch zeta-function

The problem is to show that $\Re L(b/2,1/2,p+1)>0$ for all real $b\ne0$ and all real $p>-1$, where
$$L(\lambda,c,s):=\sum_{k=0}^\infty\frac{\exp(2\pi i\lambda k)}{(k+c)^s}$$
is the Lerch ...

**2**

votes

**1**answer

253 views

### Powers modulo a fixed integer

We say that a set $A\subseteq \mathbb{N}$ has positive measure if $$\text{lim inf}_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n} > 0.$$
For $b\in\mathbb{N}$ with $b>1$ we consider the sets $$S_b ...

**0**

votes

**0**answers

84 views

### Concentration of large prime factors of polynomials

For each positive integer $n$, let $P(n)$ denote the largest prime factor of $n$ (and for completeness, define $P(1) = 1$, say). Let $f(x) \in \mathbb{Z}[x]$ be an irreducible polynomial of degree at ...

**11**

votes

**0**answers

567 views

### Meaningful review of Moriwaki's “Arakelov Geometry”

I have been asked to write a mathscinet review for Atsushi Moriwaki's Arakelov Geometry
book:
http://www.ams.org/bookstore-getitem/item=mmono-244
I could do the review the standard way in a day or ...

**0**

votes

**1**answer

167 views

### Is the following conjecture equivalent to the Second Hardy-Littlewood Conjecture? [closed]

Let $y$ be an arbitrary positive real number such that $y\ge 2$. Then if we can prove that, $$\lim_{x\to\infty}\dfrac{\pi(x)+\pi(y)}{\pi(x+y)}=1$$will it imply that for all sufficiently large $x$ ...

**0**

votes

**1**answer

143 views

### Redundancy of the Cantor Enumeration of the Rationals

What is the cardinality of the set of values corresponding to the first $n$ rationals generated in Cantor's enumeration scheme for proofing of their countability?
Edit:
following the suggestion of ...

**7**

votes

**0**answers

98 views

### primality and square freeness of the partition function

Divisibility properties of the partition function $p(n)$ seem to have been studied for the last three hundred years (most recently, Ken Ono has been quite active). However, I assume it is open that ...

**1**

vote

**1**answer

238 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 ...

**9**

votes

**1**answer

280 views

### Higher Fano varieties and Tsen's theorem

The rational connectivity of (complex) Fano manifolds ($c_1(T_X) > 0$) is one of the major, and surely most memorable achievements of Mori's bend-and-break method. To this day, despite intensive ...

**8**

votes

**1**answer

317 views

### Orthonormal bases of R^3 with components lying in the golden field

Greg Egan proved an interesting theorem about unit vectors in $\mathbb{R}^3$ whose components actually lie in the 'golden field' $\mathbb{Q}[\sqrt{5}]$. He found it in our studies of twin ...

**1**

vote

**0**answers

89 views

### Divisibility in $F3[t]$

Let $(P_n)_n$ be the sequence of polynomials on $\mathbb F_3$ defined by
$$P_n=\prod_{\substack{h\in\mathbb F_3[t]\\\deg\,h=n\\h\text{ monic}}}h\qquad (P_0=1).$$
For every $r\in\mathbb N^*$, ...

**9**

votes

**3**answers

349 views

### Computing minimal polynomials using LLL

I would like to use the Lenstra–Lenstra–Lovász lattice basis reduction algorithm (LLL algorithm) to compute the minimal polynomial of a (real) algebraic number $\alpha$ from a decimal approximation ...

**9**

votes

**2**answers

597 views

### What are the current trends in class field theory?

Being far from an expert in the subject I was wondering if people can hint towards a modern exposition of the developments in the last 10 years ? Or if not then suggest some sub-subjects in CFT that ...

**1**

vote

**0**answers

49 views

### Big Omega result about number of totally positive integers with fixed trace

There is much literature on the study of $N_a$, the number of totally positive integers with fixed trace $a$ in a totally real field.
That number has a natural geometric approximation $G_a$, and we ...

**0**

votes

**0**answers

96 views

### Generalized arithmetic progressions contained in Bohr sets

Recall that generalized arithmetic progression of dimension $d$ is by definition a set of the form $P = P_1+\dots+P_d$, where $P_j = \{lp_j\ \mid \ |l|\leq l_j\}\subset \mathbb Z$ is an ordinary ...

**7**

votes

**1**answer

314 views

### Distribution of the sequence $\bigl (\frac{\phi(n)}{n}\bigr )_{n=1}^{\infty}$ in $[0,1]$

Maybe this is a well-know problem.
What do we know about distribution of the sequence $\bigl (\frac{\phi(n)}{n}\bigr )_{n=1}^{\infty}$ in $[0,1]$? (Where $\phi$ is the Euler's totient function).
In ...