Pillars of Transcendental Number Theory

1. Preliminaries.- 2. e and pi are Transcendental.- 3. Theorem of Hermite-Lindemann-Weierstrass.- 4. Theorem of Gelfond and Schneider.- 5. Extensions due to Ramachandra.- 6. Diophantine Approximation and Transcendence.- 7. Roth's Theorem.- 8. Baker's Theorems and some Applications.- 9. Ground Work for the Proof of Baker's theorem.- 10. Proof of Baker's Theorem.- 11. Subspace Theorem and Some Applications.

Saradha Natarajan is an INSA Senior Scientist at the DAE Center for Excellence in Basic Sciences (CEBS) at the University of Mumbai. Earlier, she was a Professor of Mathematics at the Tata Institute of Fundamental Research, Mumbai, until 2016. She was a postdoctoral fellow at Concordia University, Canada; Macquarie University, Australia; National Board of Higher Mathematics (NBHM), India. She is an elected fellow of the Indian National Science Academy (INSA). She obtained her Ph.D. in 1983 under the guidance of Professor T. S. Bhanumurthy from Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai.

Her area of specialization is number theory, in general, and transcendental number theory and Diophantine equations, in particular. She has published several papers in international journals of repute. She has made substantial contributions to the conjectures of Erdos on perfect powers in arithmetic progressions, where combinatorial and computational methods, linear forms in logarithms and modular method are combined. For instance, it is shown that product of k (>1) successive terms from arithmetic progression with common difference d is cube or higher power only for d large. She has also made significant contributions to Thue equations and Diophantine approximations, especially towards conjectures of Bombieri, Mueller and Schmidt on number of solutions of Thue inequalities for forms in terms of number of non-zero coefficients of the form. Using new induction technique, an old result of Siegel on the number of primitive solutions of Thue inequalities was improved significantly. In the area of transcendence, she has obtained best possible simultaneous approximation measures for values of exponential function and Weierstarss elliptic function. Further, significant lower bounds were shown for the Ramanujan tau-function for almost all primes p. Some problems in elementary number theory have also attracted her attention, for example

"In this small volume, the authors assemble the classical proofs of some of the important
theorems from transcendental number theory mostly through the lens of Diophantine analysis. Written for the advanced reader ... . The book concludes with the seminal results of Baker's linear forms in logarithms and Schmidt's Subspace Theorem." (Edward B. Burger, Mathematical Reviews, April, 2022)