Questions tagged [transcendental-number-theory]
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156
questions
2
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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 ...
- 3,240
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0
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150
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 ...
- 13.7k
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0
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56
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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 ...
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votes
2
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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 ...
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153
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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 $\...
- 247
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35
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 ...
- 71
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0
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91
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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 ...
- 807
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1
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237
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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 ...
- 3,240
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votes
1
answer
550
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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 ...
- 856
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1
answer
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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".
...
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181
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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\...
- 141
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1
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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 ...
user44143
5
votes
0
answers
149
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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 ...
- 51
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0
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141
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 $...
- 3,240
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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 ...
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11
votes
4
answers
759
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 ...
- 549
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0
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67
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
$$
0<\left|\ell-\frac{p_n}{q_n}\right|<\dfrac{1}{...
- 467
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0
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165
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Is there any analogue of $\pi$ for non-Archimedean Euclidean fields?
Let us recall that a Euclidean field is an ordered field in which every positive element has a square root. Given a Euclidean field $F$, we can consider the ``Euclidean'' plane $F^2$ endowed with the ...
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5
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answers
134
views
"Interlacement" of transcendental numbers
Let us consider two transcendental numbers whose decimal representation is
$$
0.a_1a_2a_3a_4...
$$
$$
0.b_1b_2b_3b_4...
$$
Is $$0.a_1b_1a_2b_2a_3b_3a_4b_4...$$ also a transcendental number?
- 267
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0
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Are there rational $a,b$ with $a+be=1/\ln 2$?
Are there rational $a$ and $b$ with $$a+be = \frac{1}{\ln 2}\ ?$$
I used the absence of rational solutions repeatedly in this answer.
Here is a proof using Schanuel's conjecture: $e^q$ is ...
user44143
3
votes
1
answer
230
views
Simple estimation of difference of powers of 2 and powers of 3?
1. Question
How to get from the formulas
$$ \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{...
- 133
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votes
0
answers
120
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 ...
- 177
3
votes
1
answer
216
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 ...
- 215
1
vote
0
answers
81
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}^{\...
- 131
6
votes
1
answer
440
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 ...
- 53
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votes
0
answers
333
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 \...
- 161
19
votes
1
answer
816
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: ...
- 9,750
2
votes
1
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115
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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
$$A(x)=\sum_{n\geq 0} \binom{2n}{n}^2 x^n$$
...
- 1,453
8
votes
0
answers
164
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 ...
- 449
3
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0
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154
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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 ...
- 1,633
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votes
2
answers
281
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 $$\alpha+f(\alpha),\...
- 365
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votes
2
answers
1k
views
Why is it easier to prove $e$ is transcendental than $\pi$?
Why is it easier to prove $e$ is transcendental than $\pi$?
I noticed that the proofs of $\pi$'s transcendence are much longer and have more details to check than those of $e.$ My guess is that it's ...
- 713
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1
answer
96
views
Transcendence of values of Fredholm series at algebraic arguments
Let $d$ be an integer greater than $1$ and let $f(z)=\sum_{n\ge0}z^{d^n}$ be the Fredholm series. It is well-known that the Roth-Ridout theorem implies that $f(r)$ is a transcendental number for all ...
- 3,240
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votes
0
answers
88
views
Does logarithms conjecture imply Gelfond conjecture?
Assume that logarithms conjecture is true: Let $a_1,\cdots,a_n$ be algebraic numbers such that $\log(a_1),\cdots,\log(a_n)$ are $\mathbb Q$-linearly independant. Then, $\log(a_1),\cdots,\log(a_n)$ are ...
- 3,240
12
votes
1
answer
603
views
Testing whether $e^x+ax^2+bx+c$ has a zero
What is the simple test with exponential polynomials to determine whether
$$f(x)=e^x+ax^2+bx+c$$ has a positive zero?
This was prompted by the question about discriminants here. We have an ineffective ...
user44143
31
votes
1
answer
1k
views
How to prove that the solution to $x^{x+1}=(x+1)^{x}$ is transcendental?
I was asked by an high school student if there is an algebraic way to find the exact value of the solution to the equation
\begin{equation}\label{eq}
x^{x+1}=(x+1)^x
\end{equation}
Let us define that ...
- 1,303
4
votes
0
answers
197
views
On $\sum_{n=1}^\infty\frac1{p_1+\cdots+p_n}$ and $\sum_{n=1}^\infty\frac1{p_1\cdots p_n}$
For $n=1,2,3,\ldots$ let $p_n$ denote the $n$-th prime number.
Question. Are the two numbers
$$c_1=\sum_{n=1}^\infty\frac1{p_1+\cdots+p_n}\ \ \ \ \text{and}\ \ \ \ c_2=\sum_{n=1}^\infty\frac1{p_1\...
- 13.7k
1
vote
0
answers
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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 ...
- 899
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votes
1
answer
547
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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 ...
user44143
17
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0
answers
629
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 ...
- 72.9k
3
votes
0
answers
325
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 ...
- 31
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votes
0
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93
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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^{...
- 3,240
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votes
1
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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 ...
- 3,240
1
vote
1
answer
138
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=\{...
- 2,986
0
votes
0
answers
114
views
Transcendence of Euler series
Is the Euler series $\sum_{n\ge0}n!X^n\in\mathbb C[[X]]$ transcendental over $\mathbb C(X)$?
- 3,240
1
vote
0
answers
65
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 ...
- 19
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votes
0
answers
90
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 $A:=\sum_{n\ge0}\frac{\alpha^{2^n}}{2-\beta^{2^n}}$ ...
- 3,240
4
votes
1
answer
401
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}\...
- 33.1k
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votes
0
answers
167
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
159
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 ...
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