Author	Chandrasekharan, K. author
Title	Arithmetical Functions [electronic resource] / by K. Chandrasekharan
Imprint	Berlin, Heidelberg : Springer Berlin Heidelberg, 1970
Connect to	http://dx.doi.org/10.1007/978-3-642-50026-8
Descript	XI, 236 p. 1 illus. online resource

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 ยง 1. Selberg's fonnula . . . . . . 1 ยง 2. A variant of Selberg's formula 6 12 ยง 3. Wirsing's inequality . . . . . 17 ยง 4. The prime number theorem.

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

1. Mathematics
2. Number theory
3. Mathematics
4. Number Theory