Rademacher, Hans Topics in analytic number theory. (English) Zbl 0253.10002 Die Grundlehren der mathematischen Wissenschaften. Band 169. Berlin-Heidelberg-New York: Springer-Verlag. ix, 320 p. Cloth DM 88.00; $ 32.60 (1973). An dem vorliegenden Lehrbuch über ausgewählte Gebiete der Analytischen Zahlentheorie hat der Verf. von 1944 bis zu seinem Tode im Jahre 1969 gearbeitet. Seine Schüler E. Grosswald, J. Lehner und M. Newman haben es mit Literaturangaben und einigen Anmerkungen ergänzt. Die Auswahl des Stoffes und dessen vorbildlich klare Darstellung werden jeden zahlentheoretisch-Interessierten begeistern. Der Inhalt des Lehrbuchs ist in folgender Weise gegliedert: I. Analytic tools (Bernoulli polynomials and Bernoulli numbers; The Euler-MacLaurin sum formula; The \(Gamma\)-function and Mellin’s theorem; The Phragmen-Lindelöf theorem; The Poisson sum formula and applications). II. Special functions (The Riemann \(\zeta\)-function; About the prime-number theorem and the zeros of the \(\zeta\)-function; The Eisenstein series; The transformation of \(\log \eta(\tau)\) and the theory of Dedekind sums; The \(\theta\)-functions; Elliptic functions and their applications to number theory). III. Formal power series (Formal power series and the theory of partitions; Ramanujan’s congruences and identities). IV. The circle method (Analytic theory of partitions; Application of the circle method to modular forms of positive dimension). Reviewer: Wolfgang Haneke (Marburg) Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 7 ReviewsCited in 297 Documents MSC: 11-02 Research exposition (monographs, survey articles) pertaining to number theory 11B68 Bernoulli and Euler numbers and polynomials 11M06 \(\zeta (s)\) and \(L(s, \chi)\) 11M35 Hurwitz and Lerch zeta functions 11N05 Distribution of primes 11F11 Holomorphic modular forms of integral weight 11F20 Dedekind eta function, Dedekind sums 11P55 Applications of the Hardy-Littlewood method 11P81 Elementary theory of partitions 11P82 Analytic theory of partitions 11P83 Partitions; congruences and congruential restrictions 11P84 Partition identities; identities of Rogers-Ramanujan type Keywords:analytic number theory; Riemann zeta-function; prime number theorem; Dedekind sums; partitions; Ramanujan’s congruences and identities; circle method × Cite Format Result Cite Review PDF Digital Library of Mathematical Functions: §1.8(iv) Poisson’s Summation Formula ‣ §1.8 Fourier Series ‣ Topics of Discussion ‣ Chapter 1 Algebraic and Analytic Methods §20.7(viii) Transformations of Lattice Parameter ‣ §20.7 Identities ‣ Properties ‣ Chapter 20 Theta Functions §20.7(viii) Transformations of Lattice Parameter ‣ §20.7 Identities ‣ Properties ‣ Chapter 20 Theta Functions Chapter 24 Bernoulli and Euler Polynomials