The plan of this book had its inception in a course of lectures on arithmetical functions given by me in the summer of 1964 at the Forschungsinstitut fUr Mathematik of the Swiss Federal Institute of Technology, Zurich, at the invitation of Professor Beno Eckmann. My Introduction to Analytic Number Theory has appeared in the meanwhile, and this book may be looked upon as a sequel. It presupposes only a modicum of acquaintance with analysis and number theory. The arithmetical functions considered here are those associated with the distribution of prime numbers, as well as the partition function and the divisor function. Some of the problems posed by their asymptotic behaviour form the theme. They afford a glimpse of the variety of analytical methods used in the theory, and of the variety of problems that await solution. I owe a debt of gratitude to Professor Carl Ludwig Siegel, who has read the book in manuscript and given me the benefit of his criticism. I have improved the text in several places in response to his comments. I must thank Professor Raghavan Narasimhan for many stimulating discussions, and Mr. Henri Joris for the valuable assistance he has given me in checking the manuscript and correcting the proofs. K. Chandrasekharan July 1970 Contents Chapter I The prime number theorem and Selberg's method § 1. Selberg's fonnula . . . . . . 1 § 2. A variant of Selberg's formula 6 12 § 3. Wirsing's inequality . . . . . 17 § 4. The prime number theorem. .

Inhalt

I The prime number theorem and Selberg's method.- § 1. Selberg's formula.- § 2. A variant of Selberg's formula.- § 3. Wirsing's inequality.- § 4. The prime number theorem.- § 5. The order of magnitude of the divisor function.- Notes on Chapter I.- II The zeta-function of Riemann.- § 1. The functional equation.- § 2. The Riemann-von Mangoldt formula.- § 3. The entire function ?.- § 4. Hardy's theorem.- § 5. Hamburger's theorem.- Notes on Chapter II.- III Littlewood's theorem and Weyl's method.- § 1. Zero-free region of ?.- § 2. Weyl's inequality.- § 3. Some results of Hardy and Littlewood and of Weyl.- § 4. Littlewood's theorem.- § 5. Applications of Littlewood's theorem.- Notes on Chapter III.- IV Vinogradov's method.- § 1. A refinement of Littlewood's theorem.- § 2. An outline of the method.- § 3. Vinogradov's mean-value theorem.- § 4. Vinogradov's inequality.- § 5. Estimation of sections of ?(s) in the critical strip.- § 6. Chudakov's theorem.- § 7. Approximation of ?(x).- Notes on Chapter IV.- V Theorems of Hoheisel and of Ingham.- § 1. The difference between consecutive primes.- § 2. Landau's formula for the Chebyshev function ?.- § 3. Hoheisel's theorem.- § 4. Two auxiliary lemmas.- § 5. Ingham's theorem.- § 6. An application of Chudakov's theorem.- Notes on Chapter V.- VI Dirichlet's L-functions and Siegel's theorem.- § 1. Characters and L-functions.- § 2. Zeros of L-functions.- § 3. Proper characters.- § 4. The functional equation of L(s,?).- § 5. Siegel's theorem.- Notes on Chapter VI.- VII Theorems of Hardy-Ramanujan and of Rademacher on the partition function.- § 1. The partition function.- § 2. A simple case.- § 3. A bound for p(n).- § 4. A property of the generating function of p(n.- § 5. The Dedekind ?-function.- § 6. The Hardy-Ramanujan formula.- § 7. Rademacher's identity.- Notes on Chapter VII.- VIII Dirichlet's divisor problem.- § 1. The average order of the divisor function.- § 2. An application of Perron's formula.- § 3. An auxiliary function.- § 4. An identity involving the divisor function.- § 5. Voronoi's theorem.- § 6. A theorem of A. S. Besicovitch.- § 7. Theorems of Hardy and of Ingham.- § 8. Equiconvergence theorems of A. Zygmund.- § 9. The Voronoi identity.- Notes on Chapter VIII.- A list of books.