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Introduction to analytic number theory. (English) Zbl 0335.10001

Undergraduate Texts in Mathematics. New York-Heidelberg-Berlin: Springer-Verlag. xii, 338 p. DM 36.20; $14.80 (1976).
Das vorliegende Buch gibt eine gründliche und vorbildlich dargestellte Einführung in die elementare Zahlentheorie. Dem Titel „Einführung in die analytische Zahlentheorie“ wird es nicht voll gerecht, da es nur in den Kapiteln 11, 12 und 13 Methoden vermittelt, die man zur analytischen Zahlentheorie zählen darf. Ein zweiter Band mit dem Titel „Modular functions and Dirichlet series in Number Theory“ ist in der Reihe Graduate Texts in Mathematics. Vol. 41, Springer Verlag 1976 erschienen (vgl. die Besprechung im Zbl 0332.10017).
Im einzelnen bietet das Buch: Nach einer historischen Einührung im Chapter 1 elementare Teilbarkeitslehre. Ch. 2. Zahlentheoretische Funktionen. Hierbei wird Wert auf das Faltprodukt gelegt. Ch. 3. Mittelwerte zahlentheoretischer Funktionen. Ch. 4. Elementare Primzahltheorie einschließlich der Tschebytschevschen Formeln. Letztere werden mit Hilfe eines Tauber-Satzes von Shapiro hergeleitet. Ch. 5. Kongruenzen. Ch. 6. Charaktere auf endlichen abelschen Gruppen. Ch. 7. Der Satz von Dirichlet über Primzahlen in Progressionen. Ch. 8. Ramanujan- und Gauß-Summen. Pólya’s Ungleichung für Charaktersummen. Ch. 9. Quadratische Kongruenzen. Ch. 10. Primitivwurzeln. Ch. 11. Dirichlet-Reihen. Ch. 12. \(\zeta\)-Funktion und \(L\)-Reihen mit dem Beweis der Funktionalgleichungen. Ch. 13. Ein Beweis des Primzahlsatzes mit Hilfe des Riemann-Lebesgue-Lemmas. Ch. 14. Partitionen.
Jedes Kapitel enthält eine große Anzahl von Übungsaufgaben.
Das Buch gibt eine ausgezeichnete, auf langer Erfahrung aufbauende Einführung in die elementare Zahlentheorie mit einem Ausblick auf Methoden der analytischen Primzahltheorie. In diesem Sinn ist es jedem an der Zahlentheorie interessierten Studenten zu empfehlen.

MSC:

11-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
11Axx Elementary number theory
11Mxx Zeta and \(L\)-functions: analytic theory
11Lxx Exponential sums and character sums
11A25 Arithmetic functions; related numbers; inversion formulas
11N37 Asymptotic results on arithmetic functions

Citations:

Zbl 0332.10017

Digital Library of Mathematical Functions:

§24.13(i) Bernoulli Polynomials ‣ §24.13 Integrals ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials
§24.17(iii) Number Theory ‣ §24.17 Mathematical Applications ‣ Applications ‣ Chapter 24 Bernoulli and Euler Polynomials
§24.5(ii) Other Identities ‣ §24.5 Recurrence Relations ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials
§24.8(i) Fourier Series ‣ §24.8 Series Expansions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials
(25.11.13) ‣ §25.11(v) Special Values ‣ §25.11 Hurwitz Zeta Function ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
(25.11.14) ‣ §25.11(v) Special Values ‣ §25.11 Hurwitz Zeta Function ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
(25.11.16) ‣ §25.11(v) Special Values ‣ §25.11 Hurwitz Zeta Function ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
(25.11.1) ‣ §25.11(i) Definition ‣ §25.11 Hurwitz Zeta Function ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
(25.11.25) ‣ §25.11(vii) Integral Representations ‣ §25.11 Hurwitz Zeta Function ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
(25.11.2) ‣ §25.11(i) Definition ‣ §25.11 Hurwitz Zeta Function ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
(25.11.5) ‣ §25.11(iii) Representations by the Euler–Maclaurin Formula ‣ §25.11 Hurwitz Zeta Function ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
(25.11.9) ‣ §25.11(iv) Series Representations ‣ §25.11 Hurwitz Zeta Function ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
§25.11 Hurwitz Zeta Function ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
§25.11(iv) Series Representations ‣ §25.11 Hurwitz Zeta Function ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
§25.11(vii) Integral Representations ‣ §25.11 Hurwitz Zeta Function ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
§25.11(x) Further Series Representations ‣ §25.11 Hurwitz Zeta Function ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
(25.13.1) ‣ §25.13 Periodic Zeta Function ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
(25.13.2) ‣ §25.13 Periodic Zeta Function ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
(25.13.3) ‣ §25.13 Periodic Zeta Function ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
§25.13 Periodic Zeta Function ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
(25.15.10) ‣ §25.15(ii) Zeros ‣ §25.15 Dirichlet L -functions ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
(25.15.1) ‣ §25.15(i) Definitions and Basic Properties ‣ §25.15 Dirichlet L -functions ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
(25.15.2) ‣ §25.15(i) Definitions and Basic Properties ‣ §25.15 Dirichlet L -functions ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
(25.15.3) ‣ §25.15(i) Definitions and Basic Properties ‣ §25.15 Dirichlet L -functions ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
(25.15.4) ‣ §25.15(i) Definitions and Basic Properties ‣ §25.15 Dirichlet L -functions ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
(25.15.5) ‣ §25.15(i) Definitions and Basic Properties ‣ §25.15 Dirichlet L -functions ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
(25.15.6) ‣ §25.15(i) Definitions and Basic Properties ‣ §25.15 Dirichlet L -functions ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
(25.15.9) ‣ §25.15(ii) Zeros ‣ §25.15 Dirichlet L -functions ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
§25.15(i) Definitions and Basic Properties ‣ §25.15 Dirichlet L -functions ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
§25.15 Dirichlet L -functions ‣ Related Functions ‣ Chapter 25 Zeta and Related Functions
(25.16.1) ‣ §25.16(i) Distribution of Primes ‣ §25.16 Mathematical Applications ‣ Applications ‣ Chapter 25 Zeta and Related Functions
(25.16.3) ‣ §25.16(i) Distribution of Primes ‣ §25.16 Mathematical Applications ‣ Applications ‣ Chapter 25 Zeta and Related Functions
§25.16(i) Distribution of Primes ‣ §25.16 Mathematical Applications ‣ Applications ‣ Chapter 25 Zeta and Related Functions
§25.16 Mathematical Applications ‣ Applications ‣ Chapter 25 Zeta and Related Functions
(25.2.11) ‣ §25.2(iv) Infinite Products ‣ §25.2 Definition and Expansions ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions
(25.2.1) ‣ §25.2(i) Definition ‣ §25.2 Definition and Expansions ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions
(25.2.3) ‣ §25.2(ii) Other Infinite Series ‣ §25.2 Definition and Expansions ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions
(25.2.6) ‣ §25.2(ii) Other Infinite Series ‣ §25.2 Definition and Expansions ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions
(25.2.7) ‣ §25.2(ii) Other Infinite Series ‣ §25.2 Definition and Expansions ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions
(25.2.8) ‣ §25.2(iii) Representations by the Euler–Maclaurin Formula ‣ §25.2 Definition and Expansions ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions
§25.2(i) Definition ‣ §25.2 Definition and Expansions ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions
§25.2 Definition and Expansions ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions
§25.2(iv) Infinite Products ‣ §25.2 Definition and Expansions ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions
(25.4.1) ‣ §25.4 Reflection Formulas ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions
(25.4.2) ‣ §25.4 Reflection Formulas ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions
(25.4.3) ‣ §25.4 Reflection Formulas ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions
(25.4.4) ‣ §25.4 Reflection Formulas ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions
§25.4 Reflection Formulas ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions
(25.5.20) ‣ §25.5(iii) Contour Integrals ‣ §25.5 Integral Representations ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions
§25.5(iii) Contour Integrals ‣ §25.5 Integral Representations ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions
§25.5(i) In Terms of Elementary Functions ‣ §25.5 Integral Representations ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions
(25.6.2) ‣ §25.6(i) Function Values ‣ §25.6 Integer Arguments ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions
(25.6.3) ‣ §25.6(i) Function Values ‣ §25.6 Integer Arguments ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions
(25.6.4) ‣ §25.6(i) Function Values ‣ §25.6 Integer Arguments ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions
§25.6(i) Function Values ‣ §25.6 Integer Arguments ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions
Chapter 25 Zeta and Related Functions
§27.10 Periodic Number-Theoretic Functions ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory
§27.11 Asymptotic Formulas: Partial Sums ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory
§27.13(iii) Waring’s Problem ‣ §27.13 Functions ‣ Additive Number Theory ‣ Chapter 27 Functions of Number Theory
§27.13(i) Introduction ‣ §27.13 Functions ‣ Additive Number Theory ‣ Chapter 27 Functions of Number Theory
§27.13(iv) Representation by Squares ‣ §27.13 Functions ‣ Additive Number Theory ‣ Chapter 27 Functions of Number Theory
§27.14(ii) Generating Functions and Recursions ‣ §27.14 Unrestricted Partitions ‣ Additive Number Theory ‣ Chapter 27 Functions of Number Theory
§27.14(i) Partition Functions ‣ §27.14 Unrestricted Partitions ‣ Additive Number Theory ‣ Chapter 27 Functions of Number Theory
§27.14(v) Divisibility Properties ‣ §27.14 Unrestricted Partitions ‣ Additive Number Theory ‣ Chapter 27 Functions of Number Theory
§27.2(i) Definitions ‣ §27.2 Functions ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory
§27.3 Multiplicative Properties ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory
§27.4 Euler Products and Dirichlet Series ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory
§27.5 Inversion Formulas ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory
§27.5 Inversion Formulas ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory
§27.6 Divisor Sums ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory
§27.8 Dirichlet Characters ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory
§27.9 Quadratic Characters ‣ Multiplicative Number Theory ‣ Chapter 27 Functions of Number Theory
Chapter 27 Functions of Number Theory