scholarly journals On Ramachandra's Contributions to Transcendental Number Theory

2013 ◽  
Vol Volume 34-35 ◽  
Author(s):  
Michel Waldschmidt
Keyword(s):  
Number Theory ◽  
Transcendental Number ◽  
Additive Functions ◽  
Baker’S Method ◽  
Open Questions ◽  
Transcendence Measure ◽  
International Audience

International audience The first part of this paper is a survey on Ramachandra's contribution to transcendental number theory included in his 1968 paper in Acta Arithmetica. It includes a discussion of pseudo-algebraic points of algebraically additive functions. The second part deals with applications to density statements related with a conjecture due to B.~Mazur. The next part is a survey of other contributions of Ramachandra to transcendence questions (on the numbers $2^{\pi^k}$, a note on Baker's method, an easy transcendence measure for $e$). Finally, related open questions are raised.

scholarly journals Diophantine conditions in small divisors and transcendental number theory

2003 ◽  
Vol 9 (6) ◽  
pp. 1401-1409
Author(s):  
E. Muñoz Garcia ◽  
◽  
R. Pérez-Marco ◽  
Keyword(s):  
Number Theory ◽  
Transcendental Number ◽  
Small Divisors

NOTE ON FOURIER–STIELTJES COEFFICIENTS OF COIN-TOSSING MEASURES

2020 ◽  
Vol 102 (3) ◽  
pp. 479-489
Author(s):  
XIANG GAO ◽  
SHENGYOU WEN
Keyword(s):  
Number Theory ◽  
Lower Bound ◽  
Transcendental Number ◽  
Linear Forms In Logarithms ◽  
Multiplicative Semigroup ◽  
Linear Forms ◽  
Coin Tossing

It is known that the Fourier–Stieltjes coefficients of a nonatomic coin-tossing measure may not vanish at infinity. However, we show that they could vanish at infinity along some integer subsequences, including the sequence ${\{b^{n}\}}_{n\geq 1}$ where $b$ is multiplicatively independent of 2 and the sequence given by the multiplicative semigroup generated by 3 and 5. The proof is based on elementary combinatorics and lower-bound estimates for linear forms in logarithms from transcendental number theory.

NON-VANISHING OF DERIVATIVES OF L-FUNCTIONS ATTACHED TO HILBERT MODULAR FORMS

2012 ◽  
Vol 08 (04) ◽  
pp. 1099-1105 ◽  
Author(s):  
NAOMI TANABE
Keyword(s):  
Number Theory ◽  
Modular Forms ◽  
Critical Value ◽  
Transcendental Number ◽  
Hilbert Modular Forms ◽  
Hilbert Modular ◽  
Derivatives Of

This paper is to show a non-vanishing property of the derivative of certain L-functions. For certain primitive holomorphic Hilbert modular forms, if the central critical value of the standard L-function does not vanish, then neither does its derivative. This is a generalization of a result by Gun, Murty and Rath in the case of elliptic modular forms. Some applications in transcendental number theory deduced from this result are discussed as well.

scholarly journals Set Theory INC# ∞# Based on Innitary Intuitionistic Logic with Restricted Modus Ponens Rule. Hyper Inductive Denitions. Application in Transcendental Number Theory. Generalized Lindemann-Weierstrass Theorem

2021 ◽  
pp. 70-119
Author(s):  
Jaykov Foukzon
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Intuitionistic Logic ◽  
Theoretical Language ◽  
Modus Ponens ◽  
Transcendental Number ◽  
Weierstrass Theorem ◽  
Induction Principle ◽  
Intuitionistic Set Theory ◽  
Intuitionistic Arithmetic

In this paper intuitionistic set theory INC#∞# in infinitary set theoretical language is considered. External induction principle in nonstandard intuitionistic arithmetic were derived. Non trivial application in number theory is considered.The Goldbach-Euler theorem is obtained without any references to Catalan conjecture. Main results are: (i) number ee is transcendental; (ii) the both numbers e + π and e − π are irrational.

scholarly journals The work of K. Ramachandra in algebraic number theory

2013 ◽  
Vol Volume 34-35 ◽  
Author(s):  
M. Ram Murty
Keyword(s):  
Number Theory ◽  
Algebraic Number ◽  
Cyclotomic Field ◽  
Algebraic Number Theory ◽  
Class Numbers ◽  
Cyclotomic Fields ◽  
Relative Class ◽  
International Audience ◽  
Abelian Fields ◽  
Fundamental Units

International audience We give a brief survey of three papers of K. Ramachandra in algebraic number theory. The first paper is based on his thesis and appeared in the Annals of Mathematics and titled, ``Some Applications of Kronecker's Limit Formula.'' The second paper determines a system of fundamental units for the cyclotomic field and is titled, ``On the units of cyclotomic fields.'' This appeared in Acta Arithmetica. The third deals with relative class numbers and is titled, ``The class number of relative abelian fields.'' This appeared in Crelle's Journal.

Transcendental Number Theory

1975 ◽  
Vol 59 (410) ◽  
pp. 280
Author(s):  
H. Halberstam ◽  
Alan Baker
Keyword(s):  
Number Theory ◽  
Transcendental Number
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