Questions tagged [transcendental-number-theory]
The transcendental-number-theory tag has no usage guidance.
163
questions
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=\{...
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 $\mathbb C(X)$?
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 ...
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 $A:=\sum_{n\ge0}\frac{\alpha^{2^n}}{2-\beta^{2^n}}$ ...
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}\...
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 $W(1)=\Omega$ ...
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 ...
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)=...
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 ...
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 $\log{n}$?
...
-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 ...
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 ...
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: $$\lim_{n\rightarrow\infty}\frac{1}{2^{n}}\sum_{m=0}^{\left\lfloor n\frac{\ln2}{\ln a}\right\rfloor }\binom{n}{m}=\begin{cases}
1 & \textrm{if }a=3\...
6
votes
0
answers
95
views
Counterexamples or reasonings about the transcendence of series involving the Möbius function, and polynomials in the denominator
This afternoon I tried to read and understand some sections of the paper Some Applications of Diophantine Approximation by R. Tijdeman procedding from Number Theory for the Millenium III, A K Peters (...
3
votes
1
answer
241
views
Power series equation with solution $1/e$ [closed]
As $e$ is transcendental, there is no polynomial equation with integer coefficients having $e$ as a root.
Are there classical equations of the form
$$\sum_{i=0}^{\infty} a_ix^i =1$$
that have $e$ ...
3
votes
0
answers
72
views
Algebraic independece of modular forms for Fricke group over $\mathbb C(q)$
Let $q=e^{\pi i \tau}$ for $\tau \in \mathbb H$ and
$$
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},
$$
a Fourier series of quasi-modular form ...
3
votes
1
answer
220
views
Linear independence of approximate square roots
From Galois theory, we know that $[\mathbb{Q}(\sqrt{2}, \sqrt{3}, \dots \sqrt{p_k}) : \mathbb{Q}] = 2^k$. Suppose I plug in rational approximations to the square roots, then of course the classical ...
1
vote
0
answers
568
views
Are either $\pi + e$ or $\pi e$ transcendental if we add or multiply digit-wise? [closed]
Since $x^2 - (e + \pi)x + e \pi = (x - \pi)(x - e)$ has transcendental roots, we know that the coefficients are not both rational, and not even algebraic (see comment by José).
My question is, can we ...
2
votes
0
answers
137
views
Compare my software's representation of exponential numbers and 0?
Suppose I have a real number
$$
x=\sum_{i=1}^n a_i e^{\lambda_i}
$$
where $a_i,\lambda_i$s are complex algebraic numbers.
Is there an algorithm to determine whether it is greater than 0 or less than ...
4
votes
1
answer
259
views
Does Banach-Mazur distance between regular polygons admit any structure that lends to approximation or exact results in particular situations?
Banach-Mazur distance between $P_5$ and $P_3$ is $d(P_5,P_3)=1+\frac{\sqrt5}2$ where $P_n$ is regular polygon in $n$ sides. Do closed form or approximate results exist (at least at special infinitely ...
1
vote
0
answers
151
views
Classifying transcendental functions for which the Hermite-Lindemann-Weierstrass theorem is true
The Hermite-Lindemann-Weierstrass theorem is the following statement regarding the exponential function $\exp : \mathbb{C} \rightarrow \mathbb{C}$:
Theorem (Hermite-Lindemann-Weierstrass): Let $\...
1
vote
1
answer
157
views
Is the Prouhet-Thue-Morse constant transcendental in any integer base $b>2$?
I have asked the following question on Math.SE some time ago and offered a bounty, yet received no answers nor comments, so I'm posting it here.
The Prouhet-Thue-Morse constant, defined as
$$
\tau =...
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 ...
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 ...
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 ...
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 ...
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^{-\...
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}{...
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 ...
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,
$$
P(q)=1-24\sum_{n=1}^{\infty}\sigma_1(n)q^n,
$$
$$
Q(q)=1+240\sum_{n=1}^{\infty}\sigma_3(n)q^n,
$$
$$
R(q)=1-504\...
5
votes
2
answers
286
views
Algebraic exponential values
Is there a non-zero real number $t$ for which there exist infinitely
many prime numbers $p$ with $p^{it}$ an algebraic integer?
I would even be surprised to find a real $t \neq 0$ with both $2^{it}$...
1
vote
0
answers
55
views
Equality of combinations of exponentials and logarithms
Suppose I have some combination of exponentials, logarithms, and arithmetic operations on rational numbers. For example, $e^{e^{r_1} + \log r_2} - r_3$. Under what conditions does an algorithm exist ...
4
votes
1
answer
438
views
Explicit numbers with square root cancellation in Weyl's exponential sum
I'm interested in examples of real numbers $\alpha$ where we have
$$\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 ...
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\...
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 ...
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|\...
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)\...
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 ...
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}(...
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 ...
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 ...
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 ...
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 ...
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.
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 $r=r_1+r_2-1$ ...
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 ...
-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 $f'$ ...
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},...
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 ...
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 ...