Questions tagged [transcendental-number-theory]

Filter by
Sorted by
Tagged with
1 vote
1 answer
148 views

Hybrid numeration system on $[0,1]^2$

Let $X_0,X_1\in [0,1]$ and $b_1,b_2>0$ be integers. We are going to create a numeration system for vectors $(X_0,X_1)$, the base being the vector $(b_1,b_2)$, as follows. Recursively define $X_k=\{...
Vincent Granville's user avatar
0 votes
0 answers
117 views

Transcendence of Euler series

Is the Euler series $\sum_{n\ge0}n!X^n\in\mathbb C[[X]]$ transcendental over $\mathbb C(X)$?
joaopa's user avatar
  • 3,655
1 vote
0 answers
66 views

Transcendence detection from algebraic constructions

I have a simple image or intuition in my mind that I can't seem to shake off, so I thought I'd seek help here. Suppose we don't know if $\alpha \in \mathbb{C} - \mathbb{Q}$ is algebraic or ...
user161812's user avatar
2 votes
0 answers
104 views

Transcendence and Mahler's method

Let $\alpha,\beta\in\overline{\mathbb Q}$ be such that $0<|a|<1$ and $|\beta|=1$. One assumes that $\beta$ is not a root of unity. Is $A:=\sum_{n\ge0}\frac{\alpha^{2^n}}{2-\beta^{2^n}}$ ...
joaopa's user avatar
  • 3,655
4 votes
1 answer
432 views

About $\pi$, $e$ and transcendence

This is mostly curiosity on my part. I assume experts would have some up-to-date info. Are $1$, $\pi$ and $e$ linearly independent over $\mathbb{Q}$? Does the set $\{ m\pi+ne;\;\;m,n\in\mathbb{Z}\...
Liviu Nicolaescu's user avatar
3 votes
0 answers
190 views

Irrationality or transcendence of $i^{i\Omega}$ and $2^\Omega$, with $\Omega=W(1)$ and $W(x)$ being the main branch of Lambert $W$ function

In this post we denote the main (or principal) branch of the Lambert $W$ function as $W(x)$, I add as reference that Wikipedia has the article Lambert $W$ function. The particular value $W(1)=\Omega$ ...
user142929's user avatar
3 votes
0 answers
166 views

What is known about the irrationality of ratios and products of logarithms of integers?

Let $a,b,c,d$ be positive natural numbers such that $\{a,b\} \neq \{c,d\}$ and such that none are perfect powers. Is it true that $$\frac{\log a \log b}{\log c \log d} \notin \mathbb{Q} ?$$ The ...
Mark Lewko's user avatar
  • 11.7k
6 votes
1 answer
588 views

Arithmetic-geometric mean for rationals?

Let $\operatorname{AGM}(x,y)$ be the arithmetic-geometric mean of $x$ and $y$. Given an error $\varepsilon>0$, a bound $b\in\mathbb R_+$ and a function $f:\mathbb R\rightarrow\mathbb R$ with $f(x)=...
VS.'s user avatar
  • 1,816
13 votes
2 answers
712 views

Lindemann theorem for Artin-Hasse exponential

Though the Lindemann--Weierstrass theorem is not known in the $p$-adic settings, its "Lindemann" part -- the transcendence of $\exp(a)$ for algebraic $a$ with $0<|a|_p<p^{-1/(p-1)}$ -- was shown ...
Wadim Zudilin's user avatar
1 vote
0 answers
95 views

Efficiently computing the digits of irrational number

Is there irrational real number $C$, defined by algebraic numbers and elementary functions such that the $n$-th digit in base $b$ in the fractional part is computable in time polynomial in $\log{n}$? ...
joro's user avatar
  • 24.2k
-6 votes
1 answer
326 views

Numerical evidence that $\pi$ is not normal in base two [closed]

Confusion is possible, but we got numerical evidence against popular belief about the normality of $\pi$ in base two. According to wikipedia a real number is said to be simply normal in an integer ...
joro's user avatar
  • 24.2k
7 votes
1 answer
437 views

Algorithm for comparing $x + y \cdot \log(z)$ for $x,y,z$ rational?

I'm playing with some methods of comparing two real numbers of the form $x + y \log(z)$, where $x,y,z$ are rational numbers and $z$ is positive. There are various estimates on irrationality measures ...
Marty's user avatar
  • 13.1k
0 votes
1 answer
173 views

Any ideas for the following limit of partial sums of binomial coefficients?

Let $a$ be an odd integer $≥3$. It appears that: $$\lim_{n\rightarrow\infty}\frac{1}{2^{n}}\sum_{m=0}^{\left\lfloor n\frac{\ln2}{\ln a}\right\rfloor }\binom{n}{m}=\begin{cases} 1 & \textrm{if }a=3\...
MCS's user avatar
  • 1,256
6 votes
0 answers
95 views

Counterexamples or reasonings about the transcendence of series involving the Möbius function, and polynomials in the denominator

This afternoon I tried to read and understand some sections of the paper Some Applications of Diophantine Approximation by R. Tijdeman procedding from Number Theory for the Millenium III, A K Peters (...
user142929's user avatar
3 votes
1 answer
241 views

Power series equation with solution $1/e$ [closed]

As $e$ is transcendental, there is no polynomial equation with integer coefficients having $e$ as a root. Are there classical equations of the form $$\sum_{i=0}^{\infty} a_ix^i =1$$ that have $e$ ...
user2718's user avatar
3 votes
0 answers
72 views

Algebraic independece of modular forms for Fricke group over $\mathbb C(q)$

Let $q=e^{\pi i \tau}$ for $\tau \in \mathbb H$ and $$ P_2(q)=\frac{P(q)+2P(q^2)}{3}, \quad Q_2(q)=\frac{Q(q)+4Q(q^2)}{5}, \\ R_2(q)=\frac{R(q)+8R(q^2)}{9}, $$ a Fourier series of quasi-modular form ...
LWW's user avatar
  • 653
3 votes
1 answer
220 views

Linear independence of approximate square roots

From Galois theory, we know that $[\mathbb{Q}(\sqrt{2}, \sqrt{3}, \dots \sqrt{p_k}) : \mathbb{Q}] = 2^k$. Suppose I plug in rational approximations to the square roots, then of course the classical ...
Nikhil's user avatar
  • 263
1 vote
0 answers
568 views

Are either $\pi + e$ or $\pi e$ transcendental if we add or multiply digit-wise? [closed]

Since $x^2 - (e + \pi)x + e \pi = (x - \pi)(x - e)$ has transcendental roots, we know that the coefficients are not both rational, and not even algebraic (see comment by José). My question is, can we ...
user avatar
2 votes
0 answers
137 views

Compare my software's representation of exponential numbers and 0?

Suppose I have a real number $$ x=\sum_{i=1}^n a_i e^{\lambda_i} $$ where $a_i,\lambda_i$s are complex algebraic numbers. Is there an algorithm to determine whether it is greater than 0 or less than ...
gondolf's user avatar
  • 1,487
4 votes
1 answer
259 views

Does Banach-Mazur distance between regular polygons admit any structure that lends to approximation or exact results in particular situations?

Banach-Mazur distance between $P_5$ and $P_3$ is $d(P_5,P_3)=1+\frac{\sqrt5}2$ where $P_n$ is regular polygon in $n$ sides. Do closed form or approximate results exist (at least at special infinitely ...
VS.'s user avatar
  • 1,816
1 vote
0 answers
151 views

Classifying transcendental functions for which the Hermite-Lindemann-Weierstrass theorem is true

The Hermite-Lindemann-Weierstrass theorem is the following statement regarding the exponential function $\exp : \mathbb{C} \rightarrow \mathbb{C}$: Theorem (Hermite-Lindemann-Weierstrass): Let $\...
Stanley Yao Xiao's user avatar
1 vote
1 answer
157 views

Is the Prouhet-Thue-Morse constant transcendental in any integer base $b>2$?

I have asked the following question on Math.SE some time ago and offered a bounty, yet received no answers nor comments, so I'm posting it here. The Prouhet-Thue-Morse constant, defined as $$ \tau =...
Klangen's user avatar
  • 1,943
1 vote
0 answers
233 views

Transcendental functions generating almost integers

Informally speaking, an "almost integer" is a real number very close to an integer. There are some known ways to construct such examples in a systematic way. One is through the use of certain ...
Almost user's user avatar
7 votes
1 answer
632 views

Digits in an algebraic irrational number

I am trying to solve a problem and I got a conditional result related to normality of algebraic irrational numbers (Borel conjecture). I know that by using Ridout theorem or Schmidt subspace theorem ...
Jean's user avatar
  • 515
1 vote
1 answer
660 views

Closest area of research to Transcendental Number Theory or/and Geometry of Numbers?

I am highly interested in doing research in either of 1- Transcendental Number Theory and Algebraic Independence; 2- Diophantine Approximation and Geometry of Numbers. There is no person working ...
user avatar
9 votes
0 answers
293 views

Constructing an infinite chain of subsets of 'hyper' algebraic numbers?

This question is cross posted from MSE. Let $F$ be a subset of $\mathbb{R}$ and let $S_F$ denote the set of values which satisfy some generalized polynomial whose exponents and coefficients are ...
Mason's user avatar
  • 191
12 votes
1 answer
989 views

Is the p-adic Lindemann-Weierstrass Conjecture still open?

The p-adic Lindemann-Weierstrass Conjecture: Let $\alpha_{1},\ldots,\alpha_{N}\in\overline{\mathbb{Q}_{p}}$ be distinct $p$-adic algebraic numbers satisfying $\left|\alpha_{n}\right|_{p}<p^{-\...
MCS's user avatar
  • 1,256
1 vote
0 answers
100 views

unknown sequences of rational numbers with sum of a transcendental number [closed]

Starting with a transcendental number like $e$, it is known that we can write it as a sum of infinitely many rational numbers of the form : $e= \sum \limits _{n=0}^{\infty }{\dfrac {1}{n!}}={\frac {1}{...
MasM's user avatar
  • 289
5 votes
1 answer
451 views

Number defined by a recursive binary sequence

In a math column in Scientific American many years ago, I encountered a peculiar binary sequence I describe below. Unfortunately I can't find a reference on this, so I would be grateful for any ...
Dominic van der Zypen's user avatar
9 votes
1 answer
600 views

Algebraic independence of $P,Q,R$ or $E_2,E_4,E_6$ over $\mathbb C(z)$

Let $P,Q,R$ be the Fourier series of the Eisenstein series $E_2,E_4,E_6$, that is, $$ P(q)=1-24\sum_{n=1}^{\infty}\sigma_1(n)q^n, $$ $$ Q(q)=1+240\sum_{n=1}^{\infty}\sigma_3(n)q^n, $$ $$ R(q)=1-504\...
LWW's user avatar
  • 653
5 votes
2 answers
286 views

Algebraic exponential values

Is there a non-zero real number $t$ for which there exist infinitely many prime numbers $p$ with $p^{it}$ an algebraic integer? I would even be surprised to find a real $t \neq 0$ with both $2^{it}$...
Pablo's user avatar
  • 11.2k
1 vote
0 answers
55 views

Equality of combinations of exponentials and logarithms

Suppose I have some combination of exponentials, logarithms, and arithmetic operations on rational numbers. For example, $e^{e^{r_1} + \log r_2} - r_3$. Under what conditions does an algorithm exist ...
pavpanchekha's user avatar
  • 1,461
4 votes
1 answer
438 views

Explicit numbers with square root cancellation in Weyl's exponential sum

I'm interested in examples of real numbers $\alpha$ where we have $$\left| \sum_{n=1}^N \mathrm e(\alpha n) \right| \ll N^{1/2} $$ or perhaps with the weaker estimate with the right side replaced ...
Anonymous's user avatar
5 votes
0 answers
432 views

Why does Faltings' Siegel lemma imply Siegel lemma?

Recall the Siegel lemma: Let $A = (a_{ij})$ be an $N \times M$ matrix with rational integer coefficients. Put $a = \max_{i,j} |a_{ij}|$. Then, if $N < M$, the equation $Ax = 0$ has a solution $x\in\...
joaopa's user avatar
  • 3,655
3 votes
0 answers
159 views

Closed form of a Fredholm number

It is known that the so called Fredholm number $\sum _{n=0} ^{\infty} \beta^{2^n}$ is transcendental for any algebraic $\beta$ with $0<|\beta|<1$. However, it may be the case, that a closed form ...
FusRoDah's user avatar
  • 3,680
5 votes
0 answers
323 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
1 vote
0 answers
123 views

Siegel lemma with one contrainst

Les $A=(a_{ij})_{\substack{1\le i\le m\\1\le j\le n}}$ ($n>m$) be a matrix of integers entries. Can one determine a "small" (depending on the size of entries of $A$) solution $(x_1,x_2,\cdots,x_n)\...
joaopa's user avatar
  • 3,655
4 votes
1 answer
381 views

Existence of normal number except random numbers

For normality, see https://en.wikipedia.org/wiki/Normal_number. For random number/sequence, see https://en.wikipedia.org/wiki/Algorithmically_random_sequence. Now, is there any number that is normal ...
XL _At_Here_There's user avatar
1 vote
0 answers
161 views

Is there any irrational algebraic number among the set? [closed]

Suppose $S$ is set of numbers such that every number in it expands in decimal digits,every digit is 0 or 1,and $\lim_{n\rightarrow\infty}\frac{C_{n}(0)}{n}=\frac{1}{2}$ where ${C_{n}(0)}$ and ${C_{n}(...
XL _At_Here_There's user avatar
8 votes
1 answer
1k views

Why is the Euler-Mascheroni constant not a Liouville number?

Let $\gamma$ be the Euler-Mascheroni constant. Why is $\gamma$ not a Liouville number? Are there any upper bounds for the irrationality measure of $\gamma$ known? Any pointers to the literature are ...
Christoph Mark's user avatar
38 votes
1 answer
3k views

Is 0.24681012141618202224... transcendental?

Is a number whose infinite decimal part is the sequence of even numbers, transcendental? How about a number whose infinite decimal part is the odd numbers? Would the odds be more difficult to prove ...
user avatar
3 votes
2 answers
233 views

Integer programming of free energy

Given $k$, $f(x) = e^{x_1}+e^{x_1+x_2}+\cdots+e^{x_1+x_2+\cdots+x_k},$ $x=(x_1,x_2,\ldots,x_k),$ where $x_i \in \{0,1\}$. We want to compute: $\inf_{x \neq y}|f(x)-f(y)|$ or a lower bound of $\inf_{x ...
Jun Li's user avatar
  • 79
1 vote
1 answer
655 views

Could the complex zeros of Riemann zeta function be of the form $ s=0.5+ik$ with $k$ a positive integer? [closed]

I have checked in Andrew Odlyzko, Tables of zeros of the Riemann zeta function, to know if there is an example of zeros of Riemann zeta function with integer imaginary parts, but I don't see that. I ...
zeraoulia rafik's user avatar
6 votes
1 answer
590 views

Transcendence of $e^{\frac{\pi^2}{12 \log 2}}$

Is it known whether $e^{\frac{\pi^2}{12 \log 2}}$ is transcendental or algebraic? This number showed up in this other question.
user405683's user avatar
3 votes
0 answers
141 views

$\mathbb{Z}$-linear independence of arguments of units in non-CM number fields

Playing a little bit with Groessencharacters a stumbled on the following question: Let $K$ be a non CM number field with $r_1$ real embeddings and $2r_2>0$ complex embeddings. Set $r=r_1+r_2-1$ ...
Hugo Chapdelaine's user avatar
8 votes
0 answers
385 views

Transcendence of the $p$-adic number $\sum_{n\ge0}a^{2^n}$

Let $p$ a prime number and $a\in\overline{\mathbb Q}\subset\mathbb C_p$ be an algebraic $p$-adic number such that $|a|_p<1$. Is the number $s_a=\sum_{n\ge0}a^{2^n}$ transcendental over $\mathbb ...
joaopa's user avatar
  • 3,655
-1 votes
1 answer
81 views

Preservation of algebraically dependence for derivative [closed]

It is well-known (see Allouche monography for example) that if $f$ is an algebraic function over $K(X)$ then $f'$ is also algebraic. I wonder whether $f$ and $g$ are algebraically dependent, then $f'$ ...
joaopa's user avatar
  • 3,655
5 votes
1 answer
233 views

Algebraic independence of $a^a$ and $b^b$ for algebraic irrationals $a,b$

Let $a,b$ be algebraic irrationals. Are there conjectures or unconditional results about the algebraic independence of $a^a$ and $b^b$? Probably Schanuel's conjecture is related, maybe only $\log{a},...
joro's user avatar
  • 24.2k
0 votes
3 answers
756 views

algebraic independence of exponential functions

Let $a_1,\ldots,a_n$ be $\mathbb Q$-linearly independant algebraic numbers. Are the functions $e^{az},\ldots,e^{a_nz}$ algebraically independent functions (over $\mathbb C(z)$ or $\mathbb Q(z)$)? I ...
joaopa's user avatar
  • 3,655
3 votes
0 answers
635 views

Doubts on a paper by Lang in Transcendental Number Theory

... namely, in the seventh page of his Transcendental numbers & diophantine approximations. I do hope that somebody in the site has read this very paper recently... Here is a screenshot of the ...
Jamai-Con's user avatar