Lectures on Elementary Number Theory by Hans Rademacher (PDF)

Description

Using a background of analysis and algebra, the reader is led to the fundamental theorems of number theory; the uniqueness of prime number factorization and the reciprocity law of quadratic residues. Cyclotomy is treated in some detail because of its own significance and as a framework for the elegant theorems on Gaussian sums. Asymptotic laws are discussed as a foretaste of analytic number theory; also, Dirichlet’s theorem about primes in an arithmetic progression and V. Brun’s theorem on twin primes.

User’s Reviews

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

⭐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-contained treatment 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.

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Download Lectures on Elementary Number Theory PDF
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