Customer Reviews (1)

sadly out of print
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...
... Read more

Product Description
What is so special about the number 30? How many colors are needed to color a map? Do the prime numbers go on forever? Are there more whole numbers than even numbers? These and other mathematical puzzles are explored in this delightful book by two eminent mathematicians. Requiring no more background than plane geometry and elementary algebra, this book leads the reader into some of the most fundamental ideas of mathematics, the ideas that make the subject exciting and interesting. Explaining clearly how each problem has arisen and, in some cases, resolved, Hans Rademacher and Otto Toeplitz's deep curiosity for the subject and their outstanding pedagogical talents shine through. ... Read more

Customer Reviews (3)

A wonderful read
I really enjoyed this book and would recommend it to high school students with an interest in studying math in college. Typically, high school students are not exposed to well written proofs. This book would remedy the situation.

The Real Beauty and Exhiliration of Math
This is a superb book, and the best way to first sample the delights of math for its own sake. Don't let the subtitle "Selections from Mathematics for the Amateur" turn you off. True, the math involved is elementary, not going beyond high school algebra and geometry. But the material won't be found in most textbooks, having been chosen with elegance and beauty, rather than utility, in mind (which isn't to say that it doesn't have its important uses). Even the professional mathematician who has already seen most of the contents will profit from studying the book's exemplary treatment of its topics. The authors are some of the best math expositors who have ever lived. Each idea in the book is developed admirably, and strikes the perfect balance between conciseness and lucidity. You probably can't do better in choosing an introduction to prime and perfect numbers, the four-color problem, regular polyhedra, Pythagorean triples, pedal triangles, periodic decimal fractions, Waring's problem, the number 30 (sic!), and many other fascinating intellectual tibdits.

At Dover's low price, this book is a great buy. Get it! If you have more mathematical knowledge, then I also recommend "Proofs from the Book" by Aigner, et al., which is written for mathematicians in a similar style.

An excellent collection of problems at the elementary level
Written by two top-notch mathematicians, requiring the mathematics that a high school student can do, this book takes you on a journey that is easy to follow and yet shows the power of mathematics. It is an excellent bookfor high school enrichement or early undergraduates.

Even if you are amathematician you will be suprized at many of the results and enjoy tryingto solve some of the problems. Many of the problems are quite hard. Eventhough some of the problems are the standard ones, many are off the beatentrack. No matter what your level of expertise, I am sure you will enjoythis book. ... Read more

Product Description
This book contains papers presented at the Hans Rademacher Centenary Conference, held at Pennsylvania State University in July 1992. The astonishing breadth of Rademacher's mathematical interests is well represented in this volume. The papers collected here range over such topics as modular forms, partitions and $q$-series, Dedekind sums, and Ramanujan type identities. Rounding out the volume is the opening paper, which presents a biography of Rademacher. This volume is a fitting tribute to a remarkable mathematician whose work continues to influence mathematics today. ... Read more

Product Description
These two volumes contain all the papers published by Hans Rademacher, either alone or as joint author, essentially in chronological order. Included also are a collection of published abstracts, a number of papers that appeared in institutes and seminars but are only now being formally published, and several problems posed and/or solved by Rademacher. The editor has provided notes for each paper, offering comments and making corrections. He has also contributed a biographical sketch.

The earlier papers are on real variables, measurability, convergence factors, and Euler summability of series. This phase of Rademacher's work culminates in a paper of 1922, in which he introduced the systems of orthogonal functions now known as the Rademacher functions.

After this, a new period in Rademacher's career began, and his major effort was devoted to the theory of functions of a complex variable and number theory. Some of his most important contributions were made in these fields. He perfected the sieve method and used it skillfully in the study of algebraic number fields; he studied the additive prime number theory of these fields; he generalized Goldbach's Problem; and he began his work on the theory of the Riemann zeta function, modular functions, and Dedekind sums (now often—and justly—called Dedekind-Rademacher sums). To this period also becomes what has become known as the Rademacher-Brauer formula.

Rademacher came to the United States as a refugee in 1934. In the years that followed, he obtained some of his most important results in connection with the Fourier coefficients of modular forms of positive dimensions. His general method may be considered a modification and improvement of the Hardy-Ramanujan-Littlewood circle method. He also published additional papers on Dedekind-Rademacher sums (with A. Whiteman), general number theory (with H. S. Zuckerman), and modular functions (also with Zuckerman).

During the last decade of his life—the 1960s—he continued his work on these problems and devoted considerable attention to general analysis—especially harmonic analysis—and to analytic number theory.

All of the papers in Volume I and ten of those in Volume II are in German. One paper is in Hungarian.

The volumes are part of the MIT Press series Mathematicians of Our Time (Gian-Carlo Rota, general editor). ... Read more

Product Description
Using a background of analysis and algebra, the reader is led to thefundamental theorems of number theory; the uniqueness of prime numberfactorization and the reciprocity law of quadratic residues. Cyclotomyis treated in some detail because of its own significance and as aframework for the elegant theorems on Gaussian sums. Asymptotic lawsare discussed as a foretaste of analytic number theory; also,Dirichlet's theorem about primes in an arithmetic progression andV. Brun's theorem on twin primes. ... Read more

Customer Reviews (1)

A gem of a number theory book
This is my favorite of all time number theory books. It does a lot of pretty high-powered number theory starting with things that a well-prepared and committed math undergrad should know: strong algebra and decent background in real analysis: no previous number theory required.

Prof. Rademacher starts naively with Farey fractions (fractions with bounded denominator arranged by size). He then proceeds in a highly readable way to do the basics: prime factorization, the GCD and its properties, congruences, and rational approximations. The proofs are elementary and very well presented.

He then moves on to some more exciting material: primitive roots and a beautiful exposition of the contruction of the 17-sided polygon. Gauss' insights are displayed clearly, and explanations and motivation are supplied everywhere. Rademacher is not afraid to display computations at length, but these are organized in such a way as to make the general principles crystal clear. This section of the book ends with a nice proof of Quadratic Reciprocity, using "finite Fourier series".

This is followed by a pleasant diversion on lattice points, mobius inversion, and a nice application of the zeta function as a measure of the density of lattice points with relatively prime coordinates.

Now come the real treats.

First, there is a totally self-containedtreatment of Dirichlet's theorem on primes in arithmetic progressions, including a quick intro to group characters and L-series. This is a great introduction to the methods of analytic number theory.

Finally, the book concludes with a hard-to-find-elsewhere account of V. Brun's theorem on prime pairs: the sum, of the reciprocals of primes that differ by 2, converges. This is one of the hardest chapters in the book, but the result and the insight gained in proving it is well worth the effort. ... Read more