Theory of Numbers: A Textbook is aimed at students of Mathematics who are graduates or even undergraduates. Very little prerequisites are needed. The reader is expected to know the theory of functions of a real variable and in some chapters complex integration and some simple principles of complex function theory are assumed. The entire book is self contained except theorems 7 and 9 of chapter 11 which are assumed. The most ambitious chapter is chapter 11 where the most attractive result on difference between consecutive primes is proved. References to the latest developments like Heath-Brown’s work and the work of R.C. Baker, G. Harman and J. Pintz alongwith readable accounts of Brun’s sieve and also of Apery’s Theorem on irrationality of zeta (3) are given. Finally the reader is acquainted with Montgomery-Vaughan Theorem in the last chapter. It is hoped that the reader will enjoy the leisurely style of presentation of many important results.

Preface / Introduction and Reference Books and Articles / Elementary Estimates for p(x) and Allied Functions / Simple W Results Based on Simple Properties of z(s) / Landau’s Theorem on the Singularity of Dirichlet Series with Positive Coefficients / Roth’s Theorem on Square – Free Integers / Landau-Ramanujan-Ingham-Wiener-Ikehara Approach to the Prime Number Theorem / Elementary Results of Chebyshev and the Advanced Prime Number Theorems of Hadamard and de la Vallee Poussin / A Theorem of Tijdeman (also Erdo² s , Jutila, Ramachandra and Shorey) / Infinitely many zeros of z(s) in s ³ ½ – d / Infinity of zeros of z(s) in s ³ ½ / Some Properties of G(s) and the order of z(s) in s £ 0 / Difference Between Consecutive Primes / Miscellaneous Results / Brun’s Sieve / Roth’s Theorem and Szemeredi’s Theorem / Erdo² s Szemeredi Sieve / Irrationality of z(3) / A Remark on S. Srinivasan’s Theorem / Montgomery-Vaughan Theorem / Index.

Graduate and Undergraduate Students in Mathematics