Journal ArticleDOI
Modular functions and transcendence questions
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In this paper, it was shown that and are algebraically independent and Bertrand's conjecture on algebraic independence over of the values at algebraic points of a modular function and its derivatives.Abstract:
We prove results on the transcendence degree of a field generated by numbers connected with the modular function . In particular, we show that and are algebraically independent and we prove Bertrand's conjecture on algebraic independence over of the values at algebraic points of a modular function and its derivatives.read more
Citations
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Journal ArticleDOI
Transcendental values of the digamma function
M. Ram Murty,N. Saradha +1 more
TL;DR: In this paper, it was shown that all the numbers ψ ( a / q ) + γ, ( a, q ) = 1, 1 ⩽ a ⩾ q, are transcendental.
Journal ArticleDOI
Périodes de 1-motifs et transcendance
TL;DR: In this paper, the elliptico-toric conjecture (CET) was proposed, which is equivalent to the generalized Grothendieck's conjecture of periods (CPG)K applied to 1-motives defined over K of the kind M=[Zr∏j=1nEj×Gms] of the elliptic analogue.
Journal ArticleDOI
On the binary expansions of algebraic numbers
TL;DR: In this article, the density of 1's in the binary expansion of real algebraic numbers has been established, and bounds on the number of 1-bits in the expansion have been established.
Journal ArticleDOI
Transcendental values of certain Eichler integrals
TL;DR: In this article, the transcendence of certain Eichler integrals associated to Eisenstein series and more generally to modular forms using functional identities due to Ramanujan, Grosswald, Weil et al.
Book ChapterDOI
Elliptic Functions and Transcendence
TL;DR: In this paper, the main properties of elliptic functions that are involved in transcendence theory are surveyed, including linear independence, algebraic independence, and abelian independence, even when many results are only special cases of statements valid for abelians.
References
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Book
A Course in Arithmetic
TL;DR: In this article, the theorem on arithmetic progressions modular forms is proved for finite fields p-adic fields Hilbert symbol quadratic forms over Qp, and over Q integral quadratics forms with discriminant +-1.
Book
Collected Papers of Srinivasa Ramanujan
TL;DR: In this article, Ramanujan's letters to G. H. Hardy are annotated with a set of simultaneous equations and solutions for the questions and answers of Question 330 of Prof. Sanjana.
Journal ArticleDOI
Une preuve de la conjecture de Mahler-Manin
Book
Elliptic Functions and Transcendence
TL;DR: A lower bound for non-vanishing linear forms is given in this paper for elliptic functions with complex multiplication, where linear forms in algebraic points are assumed to vanish without complex multiplication.