Arithmetical Functions
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 T...
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| Format: | eBook |
| Language: | English |
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Berlin, Heidelberg
Springer Berlin Heidelberg
1970, 1970
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| Edition: | 1st ed. 1970 |
| Series: | Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics
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| Subjects: | |
| Online Access: | |
| Collection: | Springer Book Archives -2004 - Collection details see MPG.ReNa |
Table of Contents:
- 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
- § 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
- § 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 generatingfunction of p(n
- § 5. The Dedekind ?-function
- § 6. The Hardy-Ramanujan formula