Topics in Analytic Number Theory (Grundlehren der mathematischen Wissenschaften, 169) by Hans Rademacher (PDF)

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At the time of Professor Rademacher’s death early in 1969, there was available a complete manuscript of the present work. The editors had only to supply a few bibliographical references and to correct a few misprints and errors. No substantive changes were made in the manu­ script except in one or two places where references to additional material appeared; since this material was not found in Rademacher’s papers, these references were deleted. The editors are grateful to Springer-Verlag for their helpfulness and courtesy. Rademacher started work on the present volume no later than 1944; he was still working on it at the inception of his final illness. It represents the parts of analytic number theory that were of greatest interest to him. The editors, his students, offer this work as homage to the memory of a great man to whom they, in common with all number theorists, owe a deep and lasting debt. E. Grosswald Temple University, Philadelphia, PA 19122, U.S.A. J. Lehner University of Pittsburgh, Pittsburgh, PA 15213 and National Bureau of Standards, Washington, DC 20234, U.S.A. M. Newman National Bureau of Standards, Washington, DC 20234, U.S.A. Contents I. Analytic tools Chapter 1. Bernoulli polynomials and Bernoulli numbers ……. . 1 1. The binomial coefficients ………………………………. . 1 2. The Bernoulli polynomials ……………………………… . 4 3. Zeros of the Bernoulli polynomials ……………………….. . 7 4. The Bernoulli numbers ………………………………… . 9 5. The von Staudt-Clausen theorem ………………………… . 10 6. A multiplication formula for the Bernoulli polynomials ……….. .

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐perfection.

⭐This book is a real gem by a great mathematician who, although not a Jew, had to leave Hitler’s Germany and had to settle for a low position in a backwater US university; this is a posthumous book edited by Emil Grosswald and J. Lehner in the years following professor Rademacher’s death.Written by a master of the subject, this book contains:1) Bernoulli numbers and Von staudt-Clausen theorem.2) Euler-Mac laurin formula, Gamma function and the Mellin transform.3) Phraëgmen-Lindelöf method.4) Riemann’ Zeta function and the Prime Number Theorem, and discussion of the Riemann hypothesis.5) Iseki formula and the dedekind Eta function.6) Theta functions.7) the circle method and first and foremost, Rademacher’s own proof of the celebrated convergent series for the partition number (this formula includes a former asymptotic formula proved by Hardy and Ramanujan): his greatest achievement; it can also be found in Apostol’s: “Modular functions and Dirichlet series in number theory” (yet another gem…).Rademacher does not delve deeply into the prime number theorem (for example, he does not discuss the subject of the error term in the PNT) as his main interest was in the partitions number theory but he gives a lot of important tools to work with in analytic number theory; this work is to be compared with another of his books: “Lectures on elementary number theory” (this one can be found easily enough) since in”topics…” , Rademacher makes extensive use of complex analysis from the very start, culminating with the lengthy and difficult proof of his formula using sophisticated contour integrals, whereas in “Lectures on elementary …” , he avoids calling on complex analysis although he gives a proof of Dirichlet theorem (prime numbers in arithmetic sequences) and also of Brun’s theorem (on twin primes).A real collector, as it is so hard to find…(this book has been reissued lately but in a cheap softcover format).

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