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Questions tagged [transcendental-number-theory]

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Characteristics of numbers that satisfy inequality related to diophantine approximation with perfect squares

This question has come up in my algorithms and physics research. I apologize if this is very basic, but I am new to number theory and it seems this is a number-theoretic question. What can we say ...
groupoid's user avatar
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2 votes
0 answers
55 views

Are there any known Khinchin reals for which the asymptotics of "average" of their coefficients seems experimentally known?

We can define a Khinchin Real and recall the definition of Khinchin's Constant A real number $r$ is a Khinchin real if given the simple continued fraction expansion of $r$ as $$ r = a_0 + \cfrac{1}{...
Sidharth Ghoshal's user avatar
4 votes
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774 views

One of the numbers $\zeta(5), \zeta(7), \zeta(9), \zeta(11)$ is irrational

I am reading an interesting paper One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational by Zudilin. We fix odd numbers $q$ and $r$, $q\geq r+4$ and a tuple $\eta_0,\eta_1,...,\eta_q$ of positive ...
Max's user avatar
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6 votes
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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 ...
Alex Pawelko's user avatar
2 votes
0 answers
239 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 ...
Zerox's user avatar
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22 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 ...
Matthew Albano's user avatar
3 votes
0 answers
82 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 $T$-...
Jean's user avatar
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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) ...
joro's user avatar
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2 answers
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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 ...
Jean's user avatar
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2 votes
1 answer
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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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65 votes
2 answers
6k 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 ...
Timothy Chow's user avatar
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4 votes
0 answers
164 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 $\...
MAS'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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0 answers
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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 L. Warren's user avatar
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 ...
joaopa's user avatar
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10 votes
1 answer
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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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13 votes
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". ...
Georges Elencwajg's user avatar
4 votes
0 answers
202 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\...
Zephos's user avatar
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7 votes
1 answer
309 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 ...
user avatar
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 ...
Micka's user avatar
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5 votes
0 answers
156 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 $...
joaopa's user avatar
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7 votes
0 answers
196 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 ...
JoshuaZ's user avatar
  • 6,725
11 votes
4 answers
1k 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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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
179 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 ...
Taras Banakh's user avatar
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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?
Kei's user avatar
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6 votes
0 answers
216 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 ...
user avatar
4 votes
1 answer
422 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
  • 143
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 ...
Nick Belane's user avatar
4 votes
1 answer
297 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 ...
Siddharth Iyer's user avatar
1 vote
0 answers
87 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
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 ...
Guido Li's user avatar
16 votes
0 answers
349 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
  • 161
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: ...
Olivier's user avatar
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1 vote
1 answer
125 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
  • 1,543
8 votes
0 answers
197 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
179 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
307 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),\...
Beta'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
112 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
  • 3,801
3 votes
0 answers
101 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
  • 3,801
12 votes
1 answer
899 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
  • 1,333
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\...
Zhi-Wei Sun's user avatar
  • 14.6k
1 vote
0 answers
52 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
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 ...
user avatar
17 votes
0 answers
721 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
  • 79.5k
3 votes
0 answers
352 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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