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Questions tagged [transcendental-number-theory]

5
votes
0answers
100 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
579 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
78 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
324 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
260 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\...
6
votes
2answers
223 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 $2^{it}$...
0
votes
0answers
40 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 ...
0
votes
0answers
25 views

Additive and linear independence of squares of logarithms

Since Alan Baker work, a lot is known about linear combinations of logarithms, and more. However, squares of logarithms can be useful and attractive too. Let $\ Q\ :=\ \{q_1\ \ldots\ q_n\}\ $ be ...
4
votes
1answer
225 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
345 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\...
2
votes
0answers
72 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
163 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
98 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
316 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
128 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
748 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
150 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
571 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
221 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
443 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
474 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
126 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
236 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
73 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
209 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
285 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
545 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
340 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
779 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 ...
2
votes
0answers
103 views

Inhomogeneous approximation mod p

I am interested in the following discrete analog of Kronecker's inhomogeneous approximation theorem. Given $x_1,\ldots,x_n$ distinct residue classes modulo a prime $p$, and further residue classes $...
5
votes
1answer
290 views

Linear independence of p-adic logarithms (analog of Baker's theorem)

We have the following theorem of Baker: Theorem 1. Let $\alpha_1, \ldots, \alpha_m \in \mathbb{C}$ be algebraic numbers $\neq 0, 1$ such that $\log \alpha_1, \ldots, \log \alpha_m$ are linearly ...
18
votes
2answers
720 views

Could there be a special-function counterexample to Schanuel's conjecture?

It is not too hard to show that if Schanuel's conjecture is true, then the only algebraic numbers admitting a "closed-form expression" (as defined precisely in this paper) involving $e$, $\pi$, and ...
-1
votes
1answer
300 views

rational power transcendental [closed]

A common example that rational power transcendental number is rational is $2^{\log_2 3}$.The irrationality of $2^{e}$ and $2^{\pi}$ is still unknown. Gelfond-Schneider theorem gives the answers only ...
7
votes
2answers
279 views

Effective Lindemann–Weierstrass theorem

The Lindemann–Weierstrass theorem states that if $\alpha_1, ..., \alpha_n$ are algebraic numbers which are linearly independent over the rational numbers $ℚ$, then $e^{\alpha_1}, ..., e^{\alpha_n}$ ...
5
votes
2answers
638 views

A transcendence question involving the exponential function

Let $(z_n)$ be a sequence of complex numbers satisfying $|z_n|\to +\infty$ and such that $\{e^{z_n}\mid n \in \mathbb{N}\}$ is infinite. Is it always true that $\{(z_n,e^{z_n})\mid n \in\mathbb{N}\}$ ...
2
votes
0answers
270 views

Automorphism group of $\mathbf{C}$ over $\overline{\mathbf{Q}}$

Choose an embedding $\overline{\mathbf{Q}}\rightarrow \mathbf{C}$ from an algebraic closure of the field of rationals to the field of complex numbers. Question 1: Is it true that $\mathbf{C}$ is ...
2
votes
0answers
87 views

Transcendence of a $q$-series

Let $q\ge 2$ be an integer. Fourier's proof of irrationality of $e$ adapts to prove the irrationality of $$\Psi_q=\sum_{n\ge0}\frac1{\prod_{k=0}^{n-1}(q^n-q^k)}$$ Is this number knwon to be ...
30
votes
3answers
1k views

Unexpected applications of transcendental number theory?

In the last pages of "Equations Différentielles à points singuliers réguliers", Deligne provides a proof, attributed to Brieskorn, of the so-called local monodromy theorem (on the quasi-unipotence of ...
26
votes
0answers
598 views

Is the Flajolet-Martin constant irrational? Is it transcendental?

Facebook has a new tool to estimate the average path length between you and any other person on Facebook. An interesting aspect of their method is the use of the Flajolet-Martin algorithm. In the ...
2
votes
0answers
154 views

Transcendency of certain integrals

Let $p$ be an even monic polynomial with rational coefficients of degree at least $4$. Can the integral $$\int_{-\infty}^\infty e^{-p(x)}dx$$ be an algebraic number? Is anything known about ...
4
votes
1answer
252 views

Transcendence of a ratio of p-adic logarithms

Let $p, \ell_1, \ell_2$ be distinct prime numbers, and $x_1, x_2 \in \overline{\mathbf{Q}}^\times$. If $$ \frac{\log_p x_1}{\log_p \ell_1} = \frac{\log_p x_2}{\log_p \ell_2}, $$ does it follow that ...
5
votes
0answers
114 views

On existence of rapid Arithmetic geometric procedure?

We know that $\pi$ can be computed by Arithmetic Geometric mean using Gauss-Legendre procedure which does provide fastest convergence rate as well with a guarantee of $2^n$ bits of $\pi$ at $n$th ...
5
votes
1answer
432 views

Transcendence of $\Gamma(1/3), \Gamma(1/4)$

This is a re-post from MSE as I did not get even a single comment there. Wikipedia mentions that the transcendence of $\Gamma(1/3), \Gamma(1/4)$ was proved by G. V. Chudnovsky. Does anyone have a ...
32
votes
2answers
1k views

Can we find two positive integers $n$ and $m$ ($n,m>1$) such that $n^\pi = m$? [duplicate]

I came across this apparent random question in some math questions website. At first, I thought it was easy to show that there are no non-trivial integer solutions to this equation, but then I ...
12
votes
2answers
599 views

For what real $t$ is $\{n^t : n \geq 1\}$ linearly independent over $\mathbb{Q}$?

It's straightforward that $t$ must be irrational. I have googled many variations of this question and browsed through some books on transcendental number theory. There is much that is said about when ...
11
votes
1answer
445 views

Transcendence of products of certain real algebraic numbers

Let \begin{equation} z := \prod_p p^{1/p^2}, \end{equation} where the product is over all prime numbers $p$, and we always take the positive real root. Is $z$ transcendental or algebraic, or (as I ...
5
votes
1answer
212 views

transcendence of beta values

(1) Can anybody suggest a readable reference for Schneider's theorem that the number $$ \beta(a, b)=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)} $$ is transcendental for $a, b \in \mathbb{Q}$ such that none ...
3
votes
0answers
163 views

What is the relation between the length of period of simple continued fraction expansion of quadratic algebraic numbers $\sqrt{A}$ and the integer $A$

What is the relation between the length of period of simple continued fraction expansion of quadratic algebraic numbers and the integer As we know,$\sqrt{2} = [1;2,2,2,2,…]$; while $\sqrt{14}= [3;1,2,...
8
votes
5answers
1k views

Small values of a polynomial evaluated at roots of unity

The MO answer https://mathoverflow.net/a/98176/11926 notes the following: Let $\gamma$ be an algebraic number that is not a root of unity. Then Baker's theorem implies that there is a constant $C(\...