Despite appearances, this is not an introductory text; the “elementary” in the title is used in the technical sense of “not depending on complex analysis”. This book is a 1958 translation of the first one-sixth of Landau’s 1927 three-volume work Vorlesungen über Zahlentheorie, with nine pages of exercises added. The rest of the 1927 work deals with applying “new theories” to classical problems in number theory; specifically it treats the Goldbach Conjecture, the Waring problem, the Prime Number Theorem and equidistribution of primes in residue classes, the Gauss circle problem, and Fermat’s Last Theorem.
Since 1927 there have been many even newer theories applied to these problems, and the three-volume work is largely obsolete today, but this elementary part is still valuable. The only part of it that is really out of date is the chapter on Brun’s theorem on twin primes. In 1927 the modern theory of sieves was less than 10 years old, and it has evolved tremendously since then, especially in the last 20 or 30 years. A modern treatment of Brun’s theorem can be found in Cojocaru & Murty’s An Introduction to Sieve Methods and their Applications.
This is not a textbook by modern standards, because it doesn’t have enough exercises, but it is still useful as a reference because its proofs are complete and it goes into depth in the results it handles. It is especially strong on quadratic forms and the representation of numbers as a sum of two, three, or four squares, where it proves the characterizations of the numbers that are representable and (except for three squares) gives the formulas for the number of representations. This is then carried on to a derivation of the Dirichlet class number formula for classes of binary quadratic forms.
The proof of Dirichlet’s theorem on primes in arithmetic progressions is also very good and almost up-to-date (there are now simpler proofs of the non-vanishing of the L-series). The treatments of quadratic residues and reciprocity and of Pell’s equation, although conventional, are very clear and complete and I often refer to this book when I have questions about these subjects. My favorite part of the book, although it’s not something you would use very often, is the four different proofs of the evaluation of Gauss sums.
A recent book that has some overlap in topics and has somewhat the same approach of showing what can be done without a lot of effort and preparation, is Pollack’s Not Always Buried Deep: A Second Course in Elementary Number Theory.
Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.