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Questions tagged [transcendental-number-theory]

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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 ...
joaopa's user avatar
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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 ...
Zhi-Wei Sun's user avatar
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1 vote
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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 ...
joaopa's user avatar
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64 votes
2 answers
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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 ...
Timothy Chow's user avatar
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4 votes
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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 $\...
ANG's user avatar
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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 ...
0x11111's user avatar
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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 ...
Benjamin Warren's user avatar
4 votes
1 answer
237 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 ...
joaopa's user avatar
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10 votes
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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 ...
Dave R's user avatar
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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". ...
Georges Elencwajg's user avatar
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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\...
Zephos's user avatar
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6 votes
1 answer
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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 ...
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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 ...
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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 $...
joaopa's user avatar
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7 votes
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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 ...
JoshuaZ's user avatar
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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 ...
Pinteco's user avatar
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1 vote
0 answers
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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}{...
Jean's user avatar
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3 votes
0 answers
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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 ...
Taras Banakh's user avatar
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5 votes
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"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?
Kei's user avatar
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6 votes
0 answers
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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 ...
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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{...
Daniel S.'s user avatar
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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 ...
Nick Belane's user avatar
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 ...
acreativename's user avatar
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}^{\...
T.Ch.'s user avatar
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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 ...
Guido Li's user avatar
16 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 \...
Sam's user avatar
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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: ...
Olivier's user avatar
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2 votes
1 answer
115 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 $$A(x)=\sum_{n\geq 0} \binom{2n}{n}^2 x^n$$ ...
Johnny Cage's user avatar
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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 ...
SorcererofDM's user avatar
3 votes
0 answers
154 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 ...
Seewoo Lee's user avatar
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0 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),\...
Archer's user avatar
  • 365
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 ...
Display name's user avatar
6 votes
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 ...
joaopa's user avatar
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3 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 ...
joaopa's user avatar
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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 ...
user avatar
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 ...
gigi's user avatar
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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\...
Zhi-Wei Sun's user avatar
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1 vote
0 answers
49 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 ...
lyrically wicked's user avatar
6 votes
1 answer
547 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 ...
user avatar
17 votes
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 ...
Timothy Chow's user avatar
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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 ...
Sazed's user avatar
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0 votes
0 answers
93 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^{...
joaopa's user avatar
  • 3,240
8 votes
1 answer
385 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 ...
joaopa's user avatar
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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=\{...
Vincent Granville's user avatar
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 $\mathbb C(X)$?
joaopa's user avatar
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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 ...
user161812's user avatar
2 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 $A:=\sum_{n\ge0}\frac{\alpha^{2^n}}{2-\beta^{2^n}}$ ...
joaopa's user avatar
  • 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}\...
Liviu Nicolaescu's user avatar
3 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 $W(1)=\Omega$ ...
user142929's user avatar
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 ...
Mark Lewko's user avatar
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