## Theory of Numbers: A TextbookTheory 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 along with 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. |

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### Contents

Preface vii | 1 |

Simple 1 results based on simple properties s | 29 |

Landaus theorem on the singularity of Dirichlet series | 35 |

Copyright | |

17 other sections not shown

### Common terms and phrases

absolute constant absolute convergence absolute O-constant analytic continuation arithmetic progressions bounded power Cauchy's theorem chapter choose coefficients completes the proof complex numbers consider constant depending Corollary D. R. HEATH-BROWN deduce defined Dirichlet series ERDOS estimate finite order fixed following Theorem G. H. HARDY gives H. L. MONTGOMERY half-plane Hence holds hypothesis I. M. VINOGRADOV infinity integer interval it,X K. F. Roth L-series last expression Lemma Lemma 9 log(l logn logp logT logx modulus natural numbers notations Note number of primes number satisfying O-constant depending O-expression obtain positive constant positive integer prime factor prime number theorem proof of Theorem prove the following prove Theorem RAMACHANDRA Ramanujan real number rectangle Remark result Riemann zeta function Riemann zeta-function shows sieve simple pole square-free sufficiently large suppose Theorem 13 theory TIJDEMAN uniformly upper bound zero of C(s