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E-grāmata: Arithmetical Functions

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Arithmetical FunctionsPalielināt
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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. .

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Springer Book Archives
I The prime number theorem and Selbergs method.- §
1. Selbergs
formula.- §
2. A variant of Selbergs formula.- §
3. Wirsings 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. Hardys theorem.- §
5. Hamburgers theorem.- Notes on
Chapter II.- III Littlewoods theorem and Weyls method.- §
1. Zero-free
region of ?.- §
2. Weyls inequality.- §
3. Some results of Hardy and
Littlewood and of Weyl.- §
4. Littlewoods theorem.- §
5. Applications of
Littlewoods theorem.- Notes on
Chapter III.- IV Vinogradovs method.- §
1. A
refinement of Littlewoods theorem.- §
2. An outline of the method.- §
3.
Vinogradovs mean-value theorem.- §
4. Vinogradovs inequality.- §
5.
Estimation of sections of ?(s) in the critical strip.- §
6. Chudakovs
theorem.- §
7. Approximation of ?(x).- Notes on
Chapter IV.- V Theorems of
Hoheisel and of Ingham.- §
1. The difference between consecutive primes.- §
2. Landaus formula for the Chebyshev function ?.- §
3. Hoheisels theorem.-
§
4. Two auxiliary lemmas.- §
5. Inghams theorem.- §
6. An application of
Chudakovs theorem.- Notes on
Chapter V.- VI Dirichlets L-functions and
Siegels theorem.- §
1. Characters and L-functions.- §
2. Zeros of
L-functions.- §
3. Proper characters.- §
4. The functional equation of
L(s,?).- §
5. Siegels 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 generatingfunction of p(n.- §
5. The Dedekind ?-function.- §
6. The Hardy-Ramanujan formula.- §
7. Rademachers identity.- Notes on
Chapter VII.- VIII Dirichlets divisor problem.- §
1. The average order of
the divisor function.- §
2. An application of Perrons formula.- §
3. An
auxiliary function.- §
4. An identity involving the divisor function.- §
5.
Voronois 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.
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