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588 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 $\{(z_n,e^{z_n})\mid n \in\mathbb{N}\}$ ...
188 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 ...
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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 ...
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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 ...
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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 ...
117 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 ...
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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 ...
101 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 ...
255 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 ...
950 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 not integer solutions to this equation, but then i realized that the ...
522 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 ...
419 views

Transcendence of products of certain real algebraic numbers

Let $$z := \prod_p p^{1/p^2},$$ 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 ...
135 views

Solution to system of polynomial equations

Suppose $A$ is a set of polynomials:$$P_1(x,y_1,\dots,y_n)=0,$$ $$P_2(x,y_1,\dots,y_n)=0,$$ $$\vdots$$ $$P_k(x,y_1,\dots,y_n)=0$$ is a system of equations with coefficients over $\mathbb{Z}$, and ...
204 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 ...
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Lower bound on the irrationality measure of $\pi$

There seems to be a lot of work on the upper bound for the irrationality measure of $\pi$, but I could not find anything on a lower bound except the general $\mu(\pi)\geq2$. Looks like it is the best ...
700 views

Conjecture on irrational algebraic numbers

Conjecture: For every irrational algebraic number $q$ and natural number $b$, the representation of $q$ on base $b$ contains all the digits $[0,\dots,b-1]$. Questions: Has this conjecture been ...
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Strongest known version of Baker's theorem

The article I have checked for Baker's theorem is Waldschmidt's. But the article and the citations therein are from the time of '88. Question: What is the the strongest known lower bound for Baker'...