For those who want to learn more about the topics covered on this site, or want to improve their mathematical knowledge in order to understand them, on this page we recommend some books that we liked and that we believe are also valid for self-taught study. Prices were updated on .
What Is Mathematics? An Elementary Approach to Ideas and Methods (R. Courant, H. Robbins)
We recommend this text to all people who are passionate about mathematics but who have only studied it in school, due to the difficulty (and perhaps even the fear) of approaching university-level mathematics. It is perfect for approaching “advanced” mathematics in a non-traumatic way. It covers various topics, in order to have an idea of what the different areas of mathematics are, analyzing the salient aspects of each. A good foundation in high school level mathematics is required. Chapter I can be used as an introduction to number theory (including topics covered on this site), while Chapters VI and VII can be used as an introduction to mathematical analysis (a prerequisite for understanding much of this site). The price is quite fair compared to the quality of the exposition and the vastness of the topics covered.
An Introduction to the Theory of Numbers (G. H. Hardy, E. M. Wright)
We strongly recommend this text to start a serious study of number theory. It is difficult to study it all, due to the vastness of the topics covered, but it is possible to concentrate on the chapters of greatest interest and use the others as a reference if necessary. It should be read very carefully because, as often happens in academic level texts, the language is very concise and often the most basic steps are skipped, therefore an excellent preparation in school mathematics topics is recommended, up to real analysis in one variable. This text contains practically all number theory topics that can be tackled without university-level math knowledge (although it may be useful to have attended the first year of a science degree course). We used this text as a source for our path on the “elementary” proof of the prime number theorem
and its subpaths.
This text is not only an excellent source for studying mathematical analysis (even as self-taught students), but it is also an excellent starting point for becoming good mathematicians. The first chapters devote ample space to fundamental issues such as set theory and the properties of the various numerical sets, aspects often overlooked by other texts but very important for a serious study of mathematics. The final appendix is also valuable; it explains how the language of mathematics is made: what are and how can they be statements, proofs, logical operators, etc., all very important notions but often taken for granted. The exposition is very clear but no less rigorous for this. The author is one of the greatest contemporary experts in number theory, winner of numerous prizes including the Fields medal (equivalent to the Nobel Prize); you can consider him a modern “genius” of mathematics, if you like this title (however, the author himself invites to be wary of that title
). The author’s passion for mathematics is perceptible in some explanations and in many passages in which he stimulates the reader to ask certain questions. Highly recommended text.
Popular books and other genres
The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics (M. Du Sautoy)
Excellent popular text to get a general idea of the problems related to prime numbers (not just the Riemann hypothesis). The style is very discursive, sometimes fictionalized, full of historical details and curious anecdotes. In short, it is a very pleasant read from which you can learn many things. We highly recommend it to all number theory enthusiasts.
Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (J. Derbyshire)
This popular text allows you to fully understand the Riemann Hypothesis and the central role it has in number theory, for example by linking it to the prime number theorem. Some chapters focus on historical aspects, so they take on a discursive style, while others are technical in nature. Those people with a good knowledge of school mathematics will have no difficulty in following the technical parts, also because the text briefly and clearly explains the basic notions required (complex numbers, matrices, big Oh notation, etc.). Towards the end it gets a little complicated to follow, but it’s a good text.
Correspondence of Leonhard Euler with Christian Goldbach (L. Euler, C. Goldbach)
We could not fail to mention this text of high historical value. It contains the entire correspondence (translated into English) between Euler and Goldbach: in fact, the two mathematicians not only discussed the famous Conjecture, but also many other mathematical topics, mostly related to number theory. This text offers the rare opportunity to take a trip back in time, learning the mathematical notions from the words of the authors themselves, rather than from the countless subsequent reworkings. The price of the paper version is fairly substantial, but the electonig version is downloadable for free