Questions tagged [transcendental-number-theory]
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163
questions
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views
Does algebraic independence of logarithms conjecture imply L-W?
Assume that algebraic independence of logarithms conjecture is true: Let $a_1,\cdots,a_n$ be algebraic numbers such that $\log(a_1),\cdots,\log(a_n)$ are $\mathbb Q$-linearly independant. Then, $\log(...
6
votes
0
answers
216
views
Proof $\pi$ is transcendental without symmetric function theory
This is a crosspost of my question from MSE from 3 weeks ago which was bountied but has received no response.
For an algebra assignment, I was asked to do a literature review and write up a proof of ...
2
votes
0
answers
237
views
"Visually" constructible numbers
Let $S \subseteq \mathbb{E}^2$ ($\mathbb{E}^2$ denotes the Euclidean plane), an $S$-line is a straight line passing through $2$ different points of $S$ and an $S$-circle is a circle centered at a ...
21
votes
2
answers
2k
views
Is the integral of $e^{-x^2}$ from $0$ to $1$ known to be irrational?
Is it known whether $$\int_0^1 e^{-x^2} \, dx$$ is irrational? It is well-known that $\int_0^\infty e^{-x^2} \, dx=\frac{\sqrt{\pi}}{2}$ which is irrational, but what about the prior integral? Also, I ...
3
votes
0
answers
79
views
Exponential of Liouville Numbers
By Mahler classification of Transcendental real numbers (into the sets of $S$-, $T$- and $U$-numbers), we know that
Any Liouville number is a $U$-number.
$\log \alpha$ is either an $S$- or a $T$-...
0
votes
0
answers
79
views
Is $\sum_{x=1}^\infty \frac{1}{x B^x} = -\log{\frac{B-1}{B}} $ simply normal in some base $b$?
A real number is said to be simply normal in an integer base $b$ if its infinite sequence of digits is distributed uniformly.
We have the BBP-type formula (BBP
was discovered by Simon Plouffe in 1995)
...
1
vote
2
answers
150
views
Transcendental functions with two prescribed values
Let $\alpha$ and $\beta$ two algebraic numbers lying in unit ball. Let $T:=(t_k)_k$ be an increasing sequence of positive integers such that $t_{k+1}/t_k$ tends to $1$ as $k\to \infty$.
I would like ...
2
votes
1
answer
323
views
Lang's remark on Lindemann-Weierstrass theorem
On his book "Introduction to transcendental numbers", page 99-100, Lang wrote
"Finally, we note that Lindemann actually proves something slightly stronger than the algebraic ...
0
votes
0
answers
185
views
Are the numbers $\sum_{n=1}^\infty\frac1{p(n)}$ and $\sum_{n=1}^\infty\frac1{q(n)}$ transcendental?
For each positive integer $n$, let $p(n)$ be the number of partitions of $n$ (i.e., the number of ways to write $n$ as a sum of positive integers), and let $q(n)$ be the number of strict partitions of ...
1
vote
0
answers
63
views
Siegel's method for transcendence measure quoted by Mahler
In "Zur Approximation der Exponentialfunktion und des Logarithmus. Teil II" Malher wrote (footnote 5 page 148) that the constant appearing in Satz 4 can be improved by a method communicated ...
65
votes
2
answers
5k
views
To prove irrationality, why integrate?
I have been reading David Angell's lovely book, Irrationality and Transcendence in Number Theory, which has given me some fresh insights even with some of the easier proofs. But the book reminds me of ...
4
votes
0
answers
162
views
Does $p$-adic Baker theorem holds in the given case?
Let $p$ be a prime number, $\mathbb{Q}_p$ the field of $p$-adic numbers, and $\mathbb{C}_p$ the completion of the algebraic closure of $\mathbb{Q}_p$. Let $U_p$ be the units $(1+\mathfrak{m})$ of $\...
2
votes
0
answers
57
views
Apparent singularities and non Fuchsian regular points
I am considering the following function of $z$ on the Riemann sphere:
$$
J(z) = \int_\Delta (L_0+z L_1)^a D^b d^nx
$$
where
$\Delta \in H_n\big(\Bbb{CP}^n\setminus\{L(x)=0\},S\big)$, $S$ being the ...
0
votes
0
answers
108
views
Question about repetition of numbers in the continued fraction of Euler-Mascheroni Constant
I noticed in the Continued Fraction expansion of the Euler-Mascheroni Constant that some numbers recur a lot, like the number 1 or the number 10. Is it known if there are infinitely many of the same ...
4
votes
1
answer
249
views
Transcendence measure: of $\ln(a/b)$
In the book " Number Theory IV Transcendental Numbers" written by Parsin and Shafarevich (book, page 104) it is asserted that to explicit a transcendence measure of a complex number $w$, it ...
10
votes
1
answer
602
views
Baker's theorem for integer combinations of logarithms of integers?
Baker's theorem in transcendental number theory states that
$$
\left|\beta_0 + \sum_{i=1}^n \beta_i \log \alpha_i\right| > H^{-C}
$$
where
$\beta_0, \ldots, \beta_n$ are algebraic numbers, not ...
13
votes
1
answer
4k
views
Is there another controversial statement by Grothendieck apart from 57 being prime?
There is a well-known story about Grothendieck being asked to explain concretely some result involving prime numbers and of his answering "You mean an actual number? All right, take 57".
...
4
votes
0
answers
196
views
Are $\pi$ and $e^{\pi i \sqrt{d}}$ algebraically independent?
Let $-d\in\mathbb{N}$ be square-free. Nesterenko (https://doi.org/10.1070/SM1996v187n09ABEH000158) proved that $e^{\pi\sqrt{-d}}$ and $\pi$ are algebraically independent. Is it known whether $e^{\pi\...
6
votes
1
answer
302
views
What are the simplest sentences which might distinguish Zilber’s field from the complex numbers?
Zilber’s field $\mathbb{B}$ is a field of the same size as the complex numbers $\mathbb{C}$, which satisfies the same first-order sentences about $+$ and $\cdot$. If $\mathbb{B}$ also satisfies the ...
5
votes
0
answers
181
views
Proximity of zeroes of Bessel functions
I have been running into a question for which I found no reference in the litterature. I do not have a strong background in number theory ; for me this is motivated by a question in PDEs (how close ...
5
votes
0
answers
151
views
Transcendence measure
Let $a\in\mathbb C$. Suppose that for all $k\in\mathbb N$ ($k>1$), there exist $U_k>0, V_k>0$ such that for every $n\in\mathbb N$, there exists a polynomial $P_n\in\mathbb Z[X]$ with degree $...
7
votes
0
answers
190
views
Can we make a hierarchy of complex numbers by repeatedly iterating the construction that produces periods?
Given a set $S$, which is a subset of the complex numbers, we can form the smallest field which contains $S$, which we will denote by $S_F$ by taking the intersection of all complex fields which ...
11
votes
4
answers
963
views
Compilation of strategies to show that some constant is irrational
I'm looking into expanding my knowledge in ways to show that some constant is irrational. I'm gonna give some examples of irrationality proofs, and I'm interested in learning what strategies you guys ...
1
vote
0
answers
74
views
Liouville numbers with some "special" convergents
Recall that a real number $\ell$ is called a Liouville number, if there exists an infinite sequence of rational numbers $(p_n/q_n)_n$ for which
$$
0<\left|\ell-\frac{p_n}{q_n}\right|<\dfrac{1}{...
3
votes
0
answers
178
views
Is there any analogue of $\pi$ for non-Archimedean Euclidean fields?
Let us recall that a Euclidean field is an ordered field in which every positive element has a square root. Given a Euclidean field $F$, we can consider the ``Euclidean'' plane $F^2$ endowed with the ...
5
votes
0
answers
139
views
"Interlacement" of transcendental numbers
Let us consider two transcendental numbers whose decimal representation is
$$
0.a_1a_2a_3a_4...
$$
$$
0.b_1b_2b_3b_4...
$$
Is $$0.a_1b_1a_2b_2a_3b_3a_4b_4...$$ also a transcendental number?
6
votes
0
answers
214
views
Are there rational $a,b$ with $a+be=1/\ln 2$?
Are there rational $a$ and $b$ with $$a+be = \frac{1}{\ln 2}\ ?$$
I used the absence of rational solutions repeatedly in this answer.
Here is a proof using Schanuel's conjecture: $e^q$ is ...
4
votes
1
answer
387
views
Simple estimation of difference of powers of 2 and powers of 3?
1. Question
How to get from the formulas
$$ \left| \frac{\log 2}{\log 3} - \frac{p}{q} \right| < c_a\frac{1}{q^{B_a}} \ \ \ \ \ \ \ \ \ \ \ (1.1)$$
and / or
$$ \left| \frac{\log 2}{\log 3} - \frac{...
0
votes
0
answers
128
views
Generalization of the zeta values
Consider the constants $c(s_1,s_2,...)=\sum_{n=1}^\infty \frac{1}{n^{s_n}}$ where the $s_i$'s are fixed positive integers and the number of $i$ with $s_i=1$ is finite (so that the previous sum ...
4
votes
1
answer
285
views
Markov constant of $\pi$
Given any $\epsilon > 0$, are there infinitely many $(a,b) \in \mathbb{Z}^2$ with $(a,b) = 1$ such that $$\left|\pi - \frac{a}{b}\right| < \frac{\epsilon}{b^2}?$$
According to this document, if ...
1
vote
0
answers
86
views
Given a partition of a field, construct a partition of its extension
The motivation for my question is the following algebraic consequence of the Continuum Hypothesis ($2^{\aleph_0}=\aleph_1$) by Zoli:
(T1) Assume the Continuum Hypothesis holds. Then $\mathbb{R}^{\...
6
votes
1
answer
446
views
Finite set of numbers whose powers sum up to irrational number
It is well-known that $e/\sqrt{2}$ is irrational.
Indeed, if it was rational, i.e. $p/q$ then $e^2/2 =p^2/q^2.$ Thus, $q^2e^2=2p^2,$ which would imply that $e$ is a root of $q^2x^2=2p^2.$
Now my ...
16
votes
0
answers
345
views
Transcendence of sum of reciprocals of factorials
For $A \subseteq \mathbb{N}$, define $\displaystyle x_A = \sum_{n \in A} \frac{1}{n!}$. It is easy to see that for every infinite $A$, $x_A$ is irrational.
Question: Is there an infinite $A \subseteq \...
19
votes
1
answer
1k
views
Hensel's proof that $e$ is transcendental
When he introduced $p$-adic numbers, Kurt Hensel produced an incorrect local/global proof of the fact that $e$ is transcendental. Apparently, the intended proof goes along the following lines: ...
1
vote
1
answer
124
views
Algebraicity of a generating function and binomial numbers
It is an obvious fact that the sum $\sum_{n\geq 0} \binom{2n}{n} x^n$ defines an algebraic function. I am interested in the variation of this sum, namely
$$A(x)=\sum_{n\geq 0} \binom{2n}{n}^2 x^n$$
...
8
votes
0
answers
189
views
Key ideas behind p-adic Baker's theorem
I'm trying to understand Kunrui Yu's series of papers [1 2 3] on lower bounds of linear forms of p-adic logarithms (i.e., p-adic Baker's theorem). I know the proof of the usual Baker's theorem through ...
3
votes
0
answers
176
views
Transcendentality of Coleman integral
I wonder if there's any work that considers the algebracity/transcendentality of Coleman integral (over $\mathbb{Q}_{p}$). The reason I think about this is because, for hyperelliptic curves, there are ...
0
votes
2
answers
299
views
Transcendence on $ \alpha+f(\alpha), \alpha f(\alpha) $ and $ \alpha/f(\alpha) $ where $ \alpha$ is transcendental
Let $f(x)$ be a real transcendental function with algebraic coefficients. So $f(x)$ and $x$ are algebraically independent. Let $\alpha$ be a transcendental number, are the numbers $$\alpha+f(\alpha),\...
15
votes
2
answers
1k
views
Why is it easier to prove $e$ is transcendental than $\pi$?
Why is it easier to prove $e$ is transcendental than $\pi$?
I noticed that the proofs of $\pi$'s transcendence are much longer and have more details to check than those of $e.$ My guess is that it's ...
6
votes
1
answer
102
views
Transcendence of values of Fredholm series at algebraic arguments
Let $d$ be an integer greater than $1$ and let $f(z)=\sum_{n\ge0}z^{d^n}$ be the Fredholm series. It is well-known that the Roth-Ridout theorem implies that $f(r)$ is a transcendental number for all ...
3
votes
0
answers
100
views
Does logarithms conjecture imply Gelfond conjecture?
Assume that logarithms conjecture is true: Let $a_1,\cdots,a_n$ be algebraic numbers such that $\log(a_1),\cdots,\log(a_n)$ are $\mathbb Q$-linearly independant. Then, $\log(a_1),\cdots,\log(a_n)$ are ...
12
votes
1
answer
819
views
Testing whether $e^x+ax^2+bx+c$ has a zero
What is the simple test with exponential polynomials to determine whether
$$f(x)=e^x+ax^2+bx+c$$ has a positive zero?
This was prompted by the question about discriminants here. We have an ineffective ...
31
votes
1
answer
1k
views
How to prove that the solution to $x^{x+1}=(x+1)^{x}$ is transcendental?
I was asked by an high school student if there is an algebraic way to find the exact value of the solution to the equation
\begin{equation}\label{eq}
x^{x+1}=(x+1)^x
\end{equation}
Let us define that ...
4
votes
0
answers
203
views
On $\sum_{n=1}^\infty\frac1{p_1+\cdots+p_n}$ and $\sum_{n=1}^\infty\frac1{p_1\cdots p_n}$
For $n=1,2,3,\ldots$ let $p_n$ denote the $n$-th prime number.
Question. Are the two numbers
$$c_1=\sum_{n=1}^\infty\frac1{p_1+\cdots+p_n}\ \ \ \ \text{and}\ \ \ \ c_2=\sum_{n=1}^\infty\frac1{p_1\...
1
vote
0
answers
51
views
Is the “amalgamation” of an enumerated infinite collection of absolutely normal real numbers always absolutely normal?
Let $S$ denote an enumerated infinite collection of absolutely normal (i.e. normal in all integer bases greater than or equal to $2$) real numbers (with no repetitions) such that any natural number ...
6
votes
1
answer
555
views
Do we have an algorithm for comparing $e^e$ with rationals?
Do we have an algorithm for comparing $e^e$ with rationals, with a known time to convergence?
In a non-constructive sense, there obviously is an algorithm.
If $e^e$ is some rational $q_0$, then we ...
17
votes
0
answers
705
views
Picture of Lambert's proof that $\pi$ is irrational?
With a suitably generous notion of "picture proof" or "proof without words" or "geometric proof," there do exist such proofs of the irrationality of square roots and even ...
3
votes
0
answers
348
views
Galois theory of periods of algebraic varieties PhD project
I finished my MMath at Exeter in July of this year and I would like to undertake a PhD at the interface of algebraic geometry, number theory and Galois theory. More specifically, I recently came ...
0
votes
0
answers
96
views
Transcendence à la Liouville
Let $\alpha\in\overline{\mathbb Q}^*$. How to prove that for a non ultimately constant sequence $(\varepsilon_n)_n$ with $\varepsilon_n\in\{0,1\}$, the number $\sum_{n\ge0}\frac{\varepsilon_n}{\alpha^{...
8
votes
1
answer
439
views
Irrationality measure of powers
Let $\alpha$ be an irrational number. Denote by $\mu(\alpha)$ its irrationality measure. Can one say anything about $\mu(\alpha^n)$ for every $n\in\mathbb N$?
Even more, one knows that $\mu(e)=2$. Can ...