Questions tagged [transcendental-number-theory]
The transcendental-number-theory tag has no usage guidance.
109
questions
2
votes
0answers
93 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
0answers
131 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 ...
3
votes
1answer
319 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)=...
1
vote
0answers
122 views
A question about rationality, irrationality or transcendence of definite integral [closed]
Forgive me for the following fundamental question. But I think I require the accuracy of an expert.
Consider an integral of the form:
$$\int_a^b f(x)dx,$$
where $f(x)$ is analytic and real valued for ...
6
votes
1answer
166 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
0answers
84 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
1answer
222 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 ...
4
votes
0answers
121 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
1answer
95 views
Any ideas for the following limit of partial sums of binomial coefficients?
Let $a$ be an odd integer $≥3$. It appears that: $$\lim_{n\rightarrow\infty}\frac{1}{2^{n}}\sum_{m=0}^{\left\lfloor n\frac{\ln2}{\ln a}\right\rfloor }\binom{n}{m}=\begin{cases}
1 & \textrm{if }a=3\...
5
votes
0answers
76 views
Counterexamples or reasonings about the transcendence of series involving the Möbius function, and polynomials in the denominator
This afternoon I tried to read and understand some sections of the paper Some Applications of Diophantine Approximation by R. Tijdeman procedding from Number Theory for the Millenium III, A K Peters (...
3
votes
1answer
191 views
Power series equation with solution $1/e$ [closed]
As $e$ is transcendental, there is no polynomial equation with integer coefficients having $e$ as a root.
Are there classical equations of the form
$$\sum_{i=0}^{\infty} a_ix^i =1$$
that have $e$ ...
3
votes
0answers
53 views
Algebraic independece of modular forms for Fricke group over $\mathbb C(q)$
Let $q=e^{\pi i \tau}$ for $\tau \in \mathbb H$ and
$$
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},
$$
a Fourier series of quasi-modular form ...
3
votes
1answer
147 views
Linear independence of approximate square roots
From Galois theory, we know that $[\mathbb{Q}(\sqrt{2}, \sqrt{3}, \dots \sqrt{p_k}) : \mathbb{Q}] = 2^k$. Suppose I plug in rational approximations to the square roots, then of course the classical ...
1
vote
0answers
524 views
Are either $\pi + e$ or $\pi e$ transcendental if we add or multiply digit-wise? [closed]
Since $x^2 - (e + \pi)x + e \pi = (x - \pi)(x - e)$ has transcendental roots, we know that the coefficients are not both rational, and not even algebraic (see comment by José).
My question is, can we ...
2
votes
0answers
131 views
Compare my software's representation of exponential numbers and 0?
Suppose I have a real number
$$
x=\sum_{i=1}^n a_i e^{\lambda_i}
$$
where $a_i,\lambda_i$s are complex algebraic numbers.
Is there an algorithm to determine whether it is greater than 0 or less than ...
4
votes
1answer
207 views
Does Banach-Mazur distance between regular polygons admit any structure that lends to approximation or exact results in particular situations?
Banach-Mazur distance between $P_5$ and $P_3$ is $d(P_5,P_3)=1+\frac{\sqrt5}2$ where $P_n$ is regular polygon in $n$ sides. Do closed form or approximate results exist (at least at special infinitely ...
1
vote
0answers
82 views
Classifying transcendental functions for which the Hermite-Lindemann-Weierstrass theorem is true
The Hermite-Lindemann-Weierstrass theorem is the following statement regarding the exponential function $\exp : \mathbb{C} \rightarrow \mathbb{C}$:
Theorem (Hermite-Lindemann-Weierstrass): Let $\...
1
vote
1answer
101 views
Is the Prouhet-Thue-Morse constant transcendental in any integer base $b>2$?
I have asked the following question on Math.SE some time ago and offered a bounty, yet received no answers nor comments, so I'm posting it here.
The Prouhet-Thue-Morse constant, defined as
$$
\tau =...
0
votes
0answers
195 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
1answer
565 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 ...
2
votes
1answer
406 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
0answers
206 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 ...
11
votes
1answer
717 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
0answers
86 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
1answer
395 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 ...
7
votes
1answer
323 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,
$$
P(q)=1-24\sum_{n=1}^{\infty}\sigma_1(n)q^n,
$$
$$
Q(q)=1+240\sum_{n=1}^{\infty}\sigma_3(n)q^n,
$$
$$
R(q)=1-504\...
5
votes
2answers
259 views
Algebraic exponential values
Is there a non-zero real number $t$ for which there exist infinitely
many prime numbers $p$ with $p^{it}$ an algebraic integer?
I would even be surprised to find a real $t \neq 0$ with both $2^{it}$...
1
vote
0answers
49 views
Equality of combinations of exponentials and logarithms
Suppose I have some combination of exponentials, logarithms, and arithmetic operations on rational numbers. For example, $e^{e^{r_1} + \log r_2} - r_3$. Under what conditions does an algorithm exist ...
4
votes
1answer
277 views
Explicit numbers with square root cancellation in Weyl's exponential sum
I'm interested in examples of real numbers $\alpha$ where we have
$$\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
0answers
378 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
0answers
105 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
0answers
221 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
0answers
109 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
1answer
342 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
0answers
143 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}(...
7
votes
1answer
857 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 ...
36
votes
1answer
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 ...
-1
votes
1answer
158 views
Is there a fixed positive integer $n $ for which :$\log^n x+e^{nx}=1$ with $0<x <1$? [closed]
I have tried to know if there is a fixed positive integer $n $ and $x$ is a real number such that $0<x<1$
for which :$\log^n x+e^{nx}=1$ but i can't succeed ,
Then my question here is :
...
3
votes
1answer
823 views
Chudnovsky algorithm and Pi precision
What are the precision/ number of correct Pi digits after N iterations of Chudnovsky algorithm. Looking for a formula (rather than a table) and reference.
3
votes
2answers
227 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 ...
2
votes
1answer
506 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
1answer
516 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
0answers
134 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
0answers
275 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
1answer
74 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
1answer
217 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
2answers
373 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
0answers
585 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 ...
-4
votes
1answer
374 views
Is there a fixed integer $n$ for which the difference :$\pi^n-\ e ^n$ is integer number? [closed]
I'm interested knowing more about nature of $\pi$ and $\ e$ since they are independent algebraically.
In this question I'm interested to know if there exist a integer $n$ for which the difference $\...
4
votes
2answers
1k views
Is it possible to know if $\log(\pi)$ is irrational or not since the $\log$ function is the inverse of the $\exp$ function?
I'm interested in knowing more about the question if $f(\pi)$ is rational or not, where $f$ is some well-known function. For example, $\cos(\pi) =-1$ is rational, while ${e}^{\pi}$ is irrational as ...
