Results 1  10
of
22
Double integrals and infinite products for some classical constants via analytic continuations of Lerch’s transcendent, Ramanujan J. (to appear); Eprint
, 2005
"... Abstract. The twofold aim of the paper is to unify and generalize on the one hand the double integrals of Beukers for ζ(2) and ζ(3), and those of the second author for Euler’s constant γ and its alternating analog ln(4/π), and on the other hand the infinite products of the first author for e, and o ..."
Abstract

Cited by 11 (1 self)
 Add to MetaCart
Abstract. The twofold aim of the paper is to unify and generalize on the one hand the double integrals of Beukers for ζ(2) and ζ(3), and those of the second author for Euler’s constant γ and its alternating analog ln(4/π), and on the other hand the infinite products of the first author for e, and of the second author for π and e γ. We obtain new double integral and infinite product representations of many classical constants, as well as a generalization to Lerch’s transcendent of Hadjicostas’s double integral formula for the Riemann zeta function, and logarithmic series for the digamma and Euler beta functions. The main tools are analytic continuations of Lerch’s function, including Hasse’s series. We also use Ramanujan’s polylogarithm formula for the sum of a certain series involving
Open Diophantine Problems
 MOSCOW MATHEMATICAL JOURNAL
, 2004
"... Diophantine Analysis is a very active domain of mathematical research where one finds more conjectures than results. We collect here a number of open questions concerning Diophantine equations (including Pillai’s Conjectures), Diophantine approximation (featuring the abc Conjecture) and transcendent ..."
Abstract

Cited by 10 (3 self)
 Add to MetaCart
Diophantine Analysis is a very active domain of mathematical research where one finds more conjectures than results. We collect here a number of open questions concerning Diophantine equations (including Pillai’s Conjectures), Diophantine approximation (featuring the abc Conjecture) and transcendental number theory (with, for instance, Schanuel’s Conjecture). Some questions related to Mahler’s measure and Weil absolute logarithmic height are then considered (e. g., Lehmer’s Problem). We also discuss Mazur’s question regarding the density of rational points on a variety, especially in the particular case of algebraic groups, in connexion with transcendence problems in several variables. We say only a few words on metric problems, equidistribution questions, Diophantine approximation on manifolds and Diophantine analysis on function fields.
Transcendence of Periods: The State of the Art
, 2005
"... function, multiple zeta values (MZV). AMS subject classifications 11J81 11J86 11J89 Abstract: The set of real numbers and the set of complex numbers have the power of continuum. Among these numbers, those which are “interesting”, which appear “naturally”, which deserve our attention, form a countabl ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
function, multiple zeta values (MZV). AMS subject classifications 11J81 11J86 11J89 Abstract: The set of real numbers and the set of complex numbers have the power of continuum. Among these numbers, those which are “interesting”, which appear “naturally”, which deserve our attention, form a countable set. Starting from this point of view we are interested in the periods as defined by M. Kontsevich and D. Zagier. We give the state of the art on the question of the arithmetic nature of these numbers: to decide whether a period is a rational number, an irrational algebraic number or else a transcendental number is the object of a few theorems and of many conjectures. We also consider the approximation of such numbers by rational or algebraic numbers. Acknowledgment This is an english updated version of the paper in french: Transcendance de périodes: état des connaissances,
A hypergeometric approach, via linear forms involving logarithms, to irrationality criteria for Euler’s constant
 CRM Conference Proceedings of CNTA 7
, 2002
"... Abstract. Using an integral of a hypergeometric function, we give necessary and sufficient conditions for irrationality of Euler’s constant γ. The proof is by reduction to known irrationality criteria for γ involving a Beukerstype double integral. We show that the hypergeometric and double integral ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
Abstract. Using an integral of a hypergeometric function, we give necessary and sufficient conditions for irrationality of Euler’s constant γ. The proof is by reduction to known irrationality criteria for γ involving a Beukerstype double integral. We show that the hypergeometric and double integrals are equal by evaluating them. To do this, we introduce a construction of linear forms in 1, γ, and logarithms from Nesterenkotype series of rational functions. In the Appendix, S. Zlobin gives a changeofvariables proof that the series and the double integral are equal. 1.
The generalizedEulerconstant function γ(z) and a generalization of Somos’s quadratic recurrence constant
 J. Math. Anal. Appl
"... We define the generalizedEulerconstant function γ(z) = ∑∞ n=1 zn−1 () 1 n+1 n − log ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
We define the generalizedEulerconstant function γ(z) = ∑∞ n=1 zn−1 () 1 n+1 n − log
Transcendance de périodes: État des connaissances
 Advances in Mathematics, Vol
"... AMS subject classifications. 11J81 11J86 11J89 Abstract. Les nombres réels ou complexes forment un ensemble ayant la puissance du continu. Parmi eux, ceux qui sont 〈〈intéressants 〉〉, qui apparaissent 〈〈naturellement 〉〉, qui méritent notre attention, forment un ensemble dénombrable. Dans cet état d’e ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
AMS subject classifications. 11J81 11J86 11J89 Abstract. Les nombres réels ou complexes forment un ensemble ayant la puissance du continu. Parmi eux, ceux qui sont 〈〈intéressants 〉〉, qui apparaissent 〈〈naturellement 〉〉, qui méritent notre attention, forment un ensemble dénombrable. Dans cet état d’esprit nous nous intéressons aux périodes au sens de Kontsevich et Zagier. Nous faisons le point sur l’état de nos connaissances concernant la nature arithmétique de ces nombres: décider si une période est un nombre rationnel, algébrique irrationnel ou au contraire transcendant est l’objet de quelques théorèmes et de beaucoup de conjectures. Nous précisons aussi ce qui est connu sur l’approximation diophantienne de tels nombres, par des nombres rationnels ou algébriques. 1. Introduction. Dans leur article [37] intitulé 〈〈Periods 〉〉, M. Kontsevich et D. Zagier introduisent la notion de périodes en en donnant deux définitions dont ils disent qu’elles sont équivalentes; il proposent une conjecture, deux principes et cinq problèmes. Le premier principe est le suivant: 〈〈chaque fois que vous rencontrez un nouveau nombre et que vous voulez savoir s’il est transcendant,
A FAMILY OF CRITERIA FOR IRRATIONALITY OF EULER’S
, 2005
"... Abstract. Following earlier results of Sondow, we propose another criterion of irrationality for Euler’s constant γ. It involves similar linear combinations of logarithm numbers Ln,m. To prove that γ is irrational, it suffices to prove that, for some fixed m, the distance of dnLn,m (dn is the least ..."
Abstract
 Add to MetaCart
Abstract. Following earlier results of Sondow, we propose another criterion of irrationality for Euler’s constant γ. It involves similar linear combinations of logarithm numbers Ln,m. To prove that γ is irrational, it suffices to prove that, for some fixed m, the distance of dnLn,m (dn is the least common multiple of the n first integers) to the set of integers Z does not converge to 0. A similar result is obtained by replacing logarithms numbers by rational numbers: it gives a sufficient condition involving only rational numbers. Unfortunately, the chaotic behavior of dn is an obstacle to verify this sufficient condition. All the proofs use in a large manner the theory of Padé approximation. 1.
In =
, 2003
"... Abstract. The aim of the paper is to relate computational and arithmetic questions about Euler’s constant γ with properties of the values of the qlogarithm function, with natural choice of q. By these means, we generalize a classical formula for γ due to Ramanujan, together with Vacca’s and Gosper’ ..."
Abstract
 Add to MetaCart
Abstract. The aim of the paper is to relate computational and arithmetic questions about Euler’s constant γ with properties of the values of the qlogarithm function, with natural choice of q. By these means, we generalize a classical formula for γ due to Ramanujan, together with Vacca’s and Gosper’s series for γ, as well as deduce irrationality criteria and tests and new asymptotic formulas for computing Euler’s constant. The main tools are Eulertype integrals and hypergeometric series.