Questions tagged [continued-fractions]

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continued fraction for logarithmic integral

Does the logarithmic integral function $\operatorname{li}(x)$ have the continued fraction expansion $$\operatorname{li}(x) = \cfrac{x}{\log x -1 -{}} \ \cfrac{1}{\log x -3 -{}} \ \cfrac{4}{\log x -...
Jesse Elliott's user avatar
1 vote
1 answer
152 views

Bounded, aperiodic irrationals with bounded, aperiodic sum

If $q = [q_0;q_1 \dots]$, say $q_i$ is the $i$-th partial quotient of $q$. My question is the following: Can one construct an explicit example of irrational $r,s > 0$ such that $\{ 1,r,s\}$ is $\...
Descartes Before the Horse's user avatar
1 vote
0 answers
178 views

Theoretical invariant distribution of discrete dynamical systems, including the Riemann Zeta map

Update on 3/10/2021: I added Example 5 in the Appendix. This generic example encompasses the Riemann Zeta dynamical system. A simple version of this post, targeted to engineers, machine learning ...
Vincent Granville's user avatar
2 votes
1 answer
514 views

About generalized continued fractions

Let us consider the sequences $(x_n), (a_n)$, starting with $n=0$ and $x_0\in ]0,1[$, defined by the following generalized Gaussian map: $$x_{n+1}=\frac{\lambda_n}{x_n^{\alpha_n}}-\Big\lfloor \frac{\...
Vincent Granville's user avatar
5 votes
1 answer
326 views

Irrationality of $e^{x/y}$

How to prove the following continued fraction of $e^{x/y}$ $${\displaystyle e^{x/y}=1+{\cfrac {2x}{2y-x+{\cfrac {x^{2}}{6y+{\cfrac {x^{2}}{10y+{\cfrac {x^{2}}{14y+{\cfrac {x^{2}}{18y+\ddots }}}}}}}}}}...
Sourangshu Ghosh's user avatar
6 votes
1 answer
638 views

Algebraic and rational parts of a real number

Let $\alpha$ be a positive real number. Does it make sense to define the closest rational to $\alpha$ as the number $R(\alpha)=\frac{p_1}{p_2}$ such that $p_1,p_2$ are positive co-prime integers ...
Vincent Granville's user avatar
8 votes
1 answer
714 views

An alternative to continued fraction and applications

This post is inspired by the Numberphile video 2.920050977316, advertising the paper A Prime-Representing Constant by Dylan Fridman, Juli Garbulsky, Bruno Glecer, James Grime and Massi Tron Florentin, ...
Sebastien Palcoux's user avatar
1 vote
0 answers
75 views

continued fractions and cusp non-excursions

Consider the modular surface $X:=\mathbb{H^2}/PSL_2(\mathbb{Z})$. Fix a width-of-cusp parameter $w, 0<w<<1$. Let $B_w$ be the cusp neighborhood of width $w$. (So $w=1$ corresponds to the ...
Kevin M Pilgrim's user avatar
17 votes
0 answers
702 views

Picture of Lambert's proof that $\pi$ is irrational?

With a suitably generous notion of "picture proof" or "proof without words" or "geometric proof," there do exist such proofs of the irrationality of square roots and even ...
Timothy Chow's user avatar
3 votes
0 answers
228 views

Asymptotic expansions for the continued fraction $[1,x,x^2,x^3,\cdots]$

The $n$-th convergent is defined as $$R_n(x) = \frac{P_n(x)}{Q_n(x)}=[1;x,x^2,\cdots,x^n]=1+\frac{1}{x+}\frac{1}{x^2+}\frac{1}{x^3+\cdots}\frac{1}{x^n}$$ where $P_n(x), Q(x)$ are polynomials ...
Vincent Granville's user avatar
2 votes
1 answer
114 views

Continued fractions, Chebyshev and non-homogenous approximation

In Khinchin's book, "Continued Fractions," he considers the question, given an irrational, $\alpha$, and a real number, $\beta$, how to find integral $x$ and $y$ such that $$\alpha x - y \...
Randall Fairman's user avatar
6 votes
0 answers
233 views

Continued fractions and class groups

Let $d$ be a positive integer. It is well-known (due to Lagrange) that the continued fraction of $\sqrt{d}$ is eventually periodic. Moreover, it is known that the equation $$\displaystyle x^2 - dy^2 = ...
Stanley Yao Xiao's user avatar
4 votes
0 answers
252 views

Why is Haven's discovery important?

Today my attention was caught by one of those little stories that appear when you open a certain browser: an inmate achieved a number theoretic breakthrough It is about continued fractions and I would ...
Manfred Weis's user avatar
  • 12.6k
2 votes
1 answer
155 views

On a transcendental number defined as a variation involving the Lambert $W$ function in the nested square root representation of the golden ratio

Define the real number $\xi$ satisfying $$\xi=\sqrt{1+W\left(1+\sqrt{1+W(1+\ldots)}\right)}\tag{1}$$ where $W(x)$ denotes the main branch of the Lambert $W$ function, as reference I add that Wikipedia ...
user142929's user avatar
1 vote
0 answers
201 views

What is the nearest Ford circle for any point in $\mathbb R^2$

I want to draw Ford circles within a "distance Estimated system" (ray marching). Therefore, given a point $(x,y)$ from $\mathbb R^2$, I need the shortest distance to any circle with center $(p/q,1/2q^...
jukzi's user avatar
  • 111
3 votes
1 answer
332 views

Proof of continued fraction identity of subfactorial

This question is part of a wider conjecture I have formed with someone which has its roots in Raayoni et al. (2019). The subfactorial function can be written as $$!n=\frac{n!}e+\frac{(-1)^n}{n+2-\...
TheSimpliFire's user avatar
7 votes
0 answers
273 views

Possible Birkhoff spectra for irrational rotations

Let $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ be the unit circle (think of it as of the interval $[0,1)$ with endpoints identified). Assume that $\alpha$ is irrational and consider the rotation by $\alpha$, ...
Dominik Kwietniak's user avatar
3 votes
1 answer
267 views

Distribution of partial quotients of $\sqrt{d}$ with $X < d \leq 2X$

Recall that for a real number $\alpha$, it has a continued fraction expansion usually written as $$\displaystyle \alpha = [a_0; a_1, a_2, \cdots].$$ Moreover, $\alpha$ is rational if and only if its ...
Stanley Yao Xiao's user avatar
1 vote
0 answers
61 views

Maximal orders in Clifford algebras

Let $$ \mathcal{C}_n(R)=R\langle e_1,\ldots,e_n\rangle/(\{e_i^2+1\}, \{e_ie_j+e_je_i:i\neq j\}) $$ be the Clifford algebra for the negative definite quadratic form $-\sum_ix_i^2$ obtained by adjoining ...
yoyo's user avatar
  • 487
6 votes
1 answer
506 views

Are the coefficients of certain product of Rogers-Ramanujan Continued Fraction non-negative?

Let $$R(q) = \cfrac{q^{1/5}}{1 + \cfrac{q}{1 + \cfrac{q^{2}}{1 + \cfrac{q^{3}}{1 + \cdots}}}}$$ The following equality is famous: $$\cfrac{q^{1/5}}{R(q)} = \prod_{k>0} \cfrac{(1-q^{5k-2})(1-q^{...
TOM's user avatar
  • 427
7 votes
2 answers
513 views

Average number of iterations for the Euclidean algorithm to terminate

Let $N$ be a positive integer and $0 \leq s < N$. We try to divide $s$ into $N$ using the Euclidean algorithm: $N = q_1 s + r_1 $ $r = q_2 r_1 + r_2 $ $\vdots$ $r_{K-1} = q_{K-1} r_K$ If we ...
soupy's user avatar
  • 423
7 votes
2 answers
191 views

Evaluation of hypergeometric type continued fraction

Is there a (possibly hypergeometric-type) explicit evaluation of the continued fraction $$a-\dfrac{1.(c+d)}{2a-\dfrac{2.(2c+d)}{3a-\dfrac{3.(3c+d)}{4a-\ddots}}}$$ Even the special case $d=0$, $a=1$ ...
Henri Cohen's user avatar
  • 11.4k
5 votes
2 answers
744 views

Does using continued fractions work to give a homeomorphism $\mathbb{Q}^+ \rightarrow (\mathbb{Q}^+)^2$?

Let $\mathbb{Q}$ be the topological space of rational numbers (with topology induced by inclusion in the real line) and let $\mathbb{Q}^+$ be the set of positive ($x>0$) rationals. I'm looking for ...
stargazer's user avatar
3 votes
0 answers
152 views

Why does division parallelize but not continued fractions and is there an analog of multiplication to continued fractions?

All the basic arithmetic operations $\times,+,/,-$ can be parallelized. However continued fraction representation of a rational number is not parallelized. The process of Euclid's algorithm looks ...
Turbo's user avatar
  • 13.6k
9 votes
3 answers
1k views

Have new conjectures generated by the Ramanujan machine been proven?

Recently the following preprint was published with new automatically generated conjectures on generalizes continuous fractions, e.g., for the Euler constant: Raayoni, Pisha, Manor, Mendlovic, Haviv, ...
Andreas Rüdinger's user avatar
18 votes
1 answer
1k views

Divergent Series & Continued Fraction (from Gauss' Mathematical Diary)

I've asked that question before on History of Science and Mathematics but haven't received an answer Does someone have a reference or further explanation on Gauß' entry from May 24, 1796 in his ...
Marcus's user avatar
  • 396
8 votes
2 answers
398 views

Riemann-Hilbert and orthogonal polynomials

Sorry for perhaps naive questions, I am not at all a specialist in the subject but I need it for my research. I know that there are close relations between Riemann-Hilbert problems and orthogonal ...
Henri Cohen's user avatar
  • 11.4k
34 votes
2 answers
1k views

Representations of $\zeta(3)$ as continued fractions involving cubic polynomials

$\zeta(3)$ has at least two well-known representations of the form $$\zeta(3)=\cfrac{k}{p(1) - \cfrac{1^6}{p(2)- \cfrac{2^6}{ p(3)- \cfrac{3^6}{p(4)-\ddots } }}},$$ where $k\in\mathbb Q$ and $p$ is a ...
Wolfgang's user avatar
  • 13.2k
5 votes
0 answers
107 views

"middle" partial denominator in continued fraction expansion of square roots

Suppose $d$ is a positive integer that is not a perfect square such that the negative Pell equation, $x^{2}-dy^{2}=-1$ has no solution. Then we know the minimal period of the continued fraction ...
user132145's user avatar
8 votes
1 answer
175 views

Limit of quotients of elements of special Fibonacci matrices

Let $F_n$ be the $n$-th Fibonacci number, started with $F_0=0,F_1=1$, and consider the matrices $$M_n=\pmatrix{F_{n+3} & F_{n+1} \\ F_{n+2} & F_{n}}.$$ Let $$\pmatrix{\alpha_n & \beta_n \\...
L.Remete's user avatar
1 vote
0 answers
90 views

Bound for truncation error of continued fraction for $E_1(z)$

Let $z \in \mathbb C \setminus(-\infty,0)$. It is known that $$E_1(z) = \cfrac{e^{-z}}{z+\cfrac{1}{1+\cfrac{1}{z+\cfrac{2}{1+\cfrac{2}{z+\cfrac{3}{1+\cdots}}}}}}.$$ For example, see http://functions....
Jesse Elliott's user avatar
1 vote
1 answer
117 views

Distinctness of quadratic surd continued fraction convergent ratio limit

In this question on math.stackexchange.com I have made two conjectures the first of which I have proved. The second has not been settled. I post it here to seek a proof. Given a quadratic surd $\sqrt ...
Hans's user avatar
  • 2,169
0 votes
0 answers
63 views

Reduction of a Jacobi-type continued fraction

I am trying to reduce the following Jacobi-like continued fraction(or J-fraction): $$f(z)=z+K_{n=1}^{\infty}\frac{R_{n}k^2}{z+Q_{n}}$$ where, $$R_{n}=n\left(n-\frac{1}{2}\right),\; Q_n=n\left(n+m+\...
Subho's user avatar
  • 121
6 votes
0 answers
503 views

Theory of Irrational Tangles?

According to one possible definition, an $n$-tangle $T$ is a subset $T \subseteq \Bbb{R}^2\times [0,1] =: X$ that is homeomorphic to a disjoint union $[0,1] \times n := [0,1] \amalg \ldots \amalg [0,1]...
Nicolas Schmidt's user avatar
6 votes
2 answers
398 views

Periods of the continued fraction expansions of Galois-conjugate quadratic-irrationals

Question: Given a quadratic irrational $x = a + b\sqrt{D}$ ($a,b \in \Bbb{Q}$, $D \in \Bbb{N}_{> 0}$ square-free) and its Galois conjugate $x' = a - b\sqrt{D}$, is it true that the continued ...
Nicolas Schmidt's user avatar
5 votes
0 answers
322 views

Transcendental Continued Fractions

Liouville famously showed the existence of a transcendental number by considering $\alpha =\sum\limits_{n=0}^\infty10^{-n!}$ and showing that it did not satisfy 'Liouville's criterion' $|\alpha-p/q|\...
Elie Ben-Shlomo's user avatar
0 votes
0 answers
80 views

Different characterizations of Liouville numbers

Usually, Liouville numbers are defined as follows: $x$ is Liouville if for ever $i\in\mathbb N$ there exist $n,m\in\mathbb Z$ such that \begin{equation} \left|x-\frac nm\right|<\frac1{m^i}. \end{...
Eduard Tetzlaff's user avatar
3 votes
1 answer
357 views

Matrix continued fractions

I am aware of the classical continued fraction in the field of real numbers, but recently I have come across the term matrix continued fraction and when I checked on the internet there are varieties ...
GA316's user avatar
  • 1,219
1 vote
2 answers
243 views

Does the set of Diophantine $m$-tuples has full measure?

We say that an $m$-tuple $\omega=(\omega_1,\ldots,\omega_m)$ satisfies the Diophantine condition of order $\nu \geq 0$ if there is a constant $C>0$ such that for all natural $q$ and integer $p_1,\...
demolishka's user avatar
11 votes
0 answers
398 views

Relation between a continued fraction and partitions

I am interested in the continued fraction $$\sum\limits_k {{z^{{2^k} - 1}}} = \frac{1}{{1 - \frac{{{T_0}z}}{{1 - \frac{{{T_1}z}}{{1 - \frac{{{T_2}z}}{{1 -{ \ddots }}}}}}}}}.$$ OEIS A104977 states ...
Johann Cigler's user avatar
1 vote
1 answer
185 views

Simultaneous Diophantine Condition and Growth Rate of Convergents Denominators

Let $\omega=(\omega_1,\ldots,\omega_{m})$ be an $m$-tuple of real numbers. Let $|\omega|_{m}:=\sup\limits_{1 \leq j \leq m}|\omega_j|_{1}$ be a metric on flat torus $\mathbb{T}^{m}=\mathbb{R}^{m}/\...
demolishka's user avatar
0 votes
0 answers
215 views

whether the quotient of continued fraction of algebraic irrational number is bounded or not is similar or equivalent to Collatz conjecture?

I vaguely recall that whether the quotient of continued fraction of algebraic irrational number is bounded or not is similar or equivalent to Collatz conjecture, could any one give the reference? or ...
XL _At_Here_There's user avatar
10 votes
1 answer
226 views

Distribution of good diophantine approximations

Let $\langle x \rangle: \mathbb{R} \to (-1/2,1/2]$ be the periodic function with period $1$ which is $x$ for $x \in (-1/2,1/2]$. Is there some function $D(a,b)$ of real numbers $a<b$ such that, for ...
David E Speyer's user avatar
3 votes
1 answer
390 views

Continued Fraction of Random Variables

So this is my first post in mathoverflow. I posted this problem in Mathstack, an I've also put a bounty on it, but did not get any response. If anyone can at least point out a reference on this ...
MAN-MADE's user avatar
  • 135
6 votes
2 answers
2k views

Distribution of distances of successive zeros of $f(x)={\rm cos}(x)+{\rm cos}(\alpha x)+{\rm cos}(\beta x)$

Let $\alpha$ and $\beta$ be incommensurate real numbers. Consider the function $f(x)={\rm cos}(x)+{\rm cos}(\alpha x)+{\rm cos}(\beta x)$ and its positive zeros $x_k(\alpha,\beta)$. Fix $\alpha$ and ...
Heis 's user avatar
  • 161
14 votes
0 answers
344 views

Quasiperiodic continued fractions

Is anything known about continued fractions in which the sequence of integers is quasiperiodic? Quasiperiodic is meant here in the sense of 1D quasicrystals. For example, draw an irrationally-sloped ...
F Flicker's user avatar
  • 141
6 votes
3 answers
588 views

Irrationality of generalized continued fractions

An infinite simple continued fraction $$\frac{1}{b_1 + \frac{1}{b_2 + \frac{1}{b_3+\dots}}} (b_i\in\mathbb Z)$$ is irrational. Now for a generalized continued fraction: $$\frac{a_1}{b_1 + \frac{a_2}...
bhbr's user avatar
  • 181
2 votes
0 answers
150 views

Has anybody studied continued fractions in function spaces?

For the text below, define $f^\infty(x) = \lim_{n\to\infty} f^n(x)$ where $f^n = \underbrace{f \circ \ldots \circ f}_{n}$. Usually 'continued fraction' means continued fraction in $\mathbb{R}$. For ...
edom's user avatar
  • 29
9 votes
1 answer
353 views

Some nice functional equations for $q$-continued fractions

Given $\large q=e^{2\pi i \tau}$. Define, $$\alpha(\tau) = \sqrt2\,q^{1/8}\prod_{n=1}^\infty\frac{ (1-q^{4n-1})(1-q^{4n-3})}{(1-q^{4n-2})(1-q^{4n-2})}$$ $$\beta(\tau) = q^{1/5}\prod_{n=1}^\infty\frac{ ...
Tito Piezas III's user avatar
7 votes
0 answers
166 views

The Heine $q$-continued fraction

Let $q=e^{2\pi i \tau}$. The Heine continued fraction is $$H_2(\tau)=\frac1{q^{1/24}}\frac{\eta(2\tau)}{\eta(\tau)} =1+\cfrac{q}{1-q+\cfrac{q^3-q^2}{1+\cfrac{q^5-q^3}{1+\cfrac{q^7-q^4}{1+\ddots}}}}$$ ...
Tito Piezas III's user avatar