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6 pages, 253 KB

Open AccessArticle

Schanuel’s Conjecture and the Transcendence of Power Towers

by Eva Trojovská and Pavel Trojovský

Mathematics 2021, 9(7), 717; https://doi.org/10.3390/math9070717 - 26 Mar 2021

Cited by 1 | Viewed by 3011

We give three consequences of Schanuel’s Conjecture. The first is that

P {(e)}^{Q (e)}

and

P {(π)}^{Q (π)}

are transcendental, for any non-constant polynomials [...] Read more.

We give three consequences of Schanuel’s Conjecture. The first is that

P {(e)}^{Q (e)}

and

P {(π)}^{Q (π)}

are transcendental, for any non-constant polynomials

P (x), Q (x) \in \bar{Q} [x]

. The second is that

π \neq α^{β}

, for any algebraic numbers

α

and

β

. The third is the case of the Gelfond’s conjecture (about the transcendence of a finite algebraic power tower) in which all elements are equal. Full article

(This article belongs to the Special Issue New Developments in Number Theory)

7 pages, 242 KB

Open AccessArticle

On the Nature of γ-th Arithmetic Zeta Functions

by Pavel Trojovský

Symmetry 2020, 12(5), 790; https://doi.org/10.3390/sym12050790 - 8 May 2020

Cited by 1 | Viewed by 1872

Abstract

Symmetry and elementary symmetric functions are main components of the proof of the celebrated Hermite–Lindemann theorem (about the transcendence of

e^{α}

, for algebraic values of

α

) which settled the ancient Greek problem of squaring the circle. In this paper, we [...] Read more.

Symmetry and elementary symmetric functions are main components of the proof of the celebrated Hermite–Lindemann theorem (about the transcendence of

e^{α}

, for algebraic values of

α

) which settled the ancient Greek problem of squaring the circle. In this paper, we are interested in similar results, but for powers such as

e^{γ log n}

. This kind of problem can be posed in the context of arithmetic functions. More precisely, we study the arithmetic nature of the so-called γ-th arithmetic zeta function

ζ_{γ} (n) : = n^{γ}

(

= e^{γ log n}

), for a positive integer n and a complex number

γ

. Moreover, we raise a conjecture about the exceptional set of

ζ_{γ}

, in the case in which

γ

is transcendental, and we connect it to the famous Schanuel’s conjecture. Full article

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