Number Theory

The Indian National Science Academy on the occasion ofthe Golden Jubilee Celebration (Fifty years of India's Independence) decided to publish a number of monographs on the selected fields. The editorial board of INS A invited us to prepare a special monograph in Number Theory. In reponse to this assignment, we invited several eminent Number Theorists to contribute expository/research articles for this monograph on Number Theory. Al­ though some ofthose invited, due to other preoccupations-could not respond positively to our invitation, we did receive fairly encouraging response from many eminent and creative number theorists throughout the world. These articles are presented herewith in a logical order. We are grateful to all those mathematicians who have sent us their articles. We hope that this monograph will have a significant impact on further development in this subject. R. P. Bambah v. C. Dumir R. J. Hans-Gill A Centennial History of the Prime Number Theorem Tom M. Apostol The Prime Number Theorem Among the thousands of discoveries made by mathematicians over the centuries, some stand out as significant landmarks. One of these is the prime number theorem, which describes the asymptotic distribution of prime numbers. It can be stated in various equivalent forms, two of which are: x (I) K(X) '" -I - as x --+ 00, ogx and Pn '" n log n as n --+ 00. (2) In (1), K(X) denotes the number of primes P ::s x for any x > O.

A Centennial History of the Prime Number Theorem.- Non-homogeneous Problems: Conjectures of Minkowski and Watson.- On the Oscillation Theorems of Pringsheim and Landau.- Modular Equations in Ramanujan s Lost Notebook.- The abc-conjecture.- On Values of Linear and Quadratic Forms at Integral Points.- Variants of the Second Borel-Cantelli Lemma and their Applications in Metric Number Theory.- Pythagorean Triples.- Integer Points in Plane Regions and Exponential Sums.- Artin s Conjecture for Polynomials Over Finite Fields.- Continuous Homomorphisms as Arithmetical Functions, and Sets of Uniqueness.- Hamburger s Theorem on ? (s) and the Abundance Principle for Dirichlet Series with Functional Equations.- A Survey of Number Theory and Cryptography.- Recent Developments in the Mean Square Theory of the Riemann Zeta and Other Zeta-Functions.- Algebraic Curves Over Finite Fields with many Rational Points and their Applications.- A Report on Artin s Holomorphy Conjecture.- Siegel s Main Theorem for Quadratic Forms.- Pfister s Work on Sums of Squares.- Notes on the Prime Number Theorem-I.- Sums of Squares: An Elementary Method.- Solution of the Basic Problems of Discrete Geometry on the Plane.- Exponential Diophantine Equations Involving Products of Consecutive Integers and Related Equations.- Algebraic Independence of Transcendental Numbers: A Survey.