On Ramachandra’s Contributions to Transcendental Number Theory

The title of this lecture refers to Ramachandra’s paper in Acta Arithmetica [36], which will be our central subject: in section 1 we state his Main Theorem, in section 2 we apply it to algebraically additive functions. Next we give new consequences of Ramachandra’s results to density problems; for instance we discuss the following question: let E be an elliptic curve which is defined over the field of algebraic numbers, and let Γ be a finitely generated subgroup of algebraic points on E; is Γ dense in E(C) for the complex topology? The other contributions of Ramachandra to transcendental number theory are dealt with more concisely in section 4. Finally we propose a few open problems. The author wishes to convey his best thanks to the organizer of the Madras Conference of July 1993 in honor of Professor Ramachandra’s 60th birthday, R. Balasubramanian, for his invitation to participate, which provided him the opportunity to write this paper. Next he is grateful to the organizer of the Bangalore Conference of December 2003 in honor of Professor Ramachandra’s 70th birthday, K. Srinivas, for his invitation to participate, which provided him the opportunity to publish this paper. He is also glad to express his deep gratitude to Professor K. Ramachandra for the inspiring role of his work and for his invitation to the Tata Institute as early as 1976.