文档介绍：Lectures on Analytic Number Theory By H. Rademacher Tata Institute of Fundamental Research, Bombay 1954-55 Lectures on Analytic Number Theory By H. Rademacher Notes by K. Balagangadharan and V. Venugopal Rao Tata Institute of Fundamental Research, Bombay 1954-1955 Contents I Formal Power Series 1 1 Lecture 2 2 Lecture 11 3 Lecture 17 4 Lecture 23 5 Lecture 30 6 Lecture 39 7 Lecture 46 8 Lecture 55 II Analysis 59 9 Lecture 60 10 Lecture 67 11 Lecture 74 12 Lecture 82 13 Lecture 89 iii CONTENTS iv 14 Lecture 95 15 Lecture 100 III Analytic theory of partitions 108 16 Lecture 109 17 Lecture 118 18 Lecture 124 19 Lecture 129 20 Lecture 136 21 Lecture 143 22 Lecture 150 23 Lecture 155 24 Lecture 160 25 Lecture 165 26 Lecture 169 27 Lecture 174 28 Lecture 179 29 Lecture 183 30 Lecture 188 31 Lecture 194 32 Lecture 200 CONTENTS v IV Representation by squares 207 33 Lecture 208 34 Lecture 214 35 Lecture 219 36 Lecture 225 37 Lecture 232 38 Lecture 237 39 Lecture 242 40 Lecture 246 41 Lecture 251 42 Lecture 256 43 Lecture 261 44 Lecture 264 45 Lecture 266 46 Lecture 272 Part I Formal Power Series 1 Lecture 1 Introduction In additive number theory we make reference to facts about addition in 1 contradistinction to multiplicative number theory, the foundations of which were laid by Euclid at about 300 . Whereas one of the principal concerns of the latter theory is the deconposition of numbers into prime factors, addi- tive number theory deals with the position of numbers into summands. It asks such questions as: in how many ways can a given naturalnumber be ecpressed as the sum of other natural numbers? Of course the postion into primary summands is trivial; it is therefore of interest to restrict in some way the nature of the summands (such as odd numbers or even numbers or per- fect squares) or the number of summands allowed. These are questions typical of those which will arise in this course. We shall have occasion to study the properties ofV-funct