Olivier Bordellès - Arithmetic Tales: Advanced Edition (Universitext)

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Significant advances have been made in number theory since the book Arithmetic Tales was published back in 2012. The most important breakthrough is probably the landmark result that Zhang proved on the bounded gaps between prime numbers. One of the main achievements of the proof is to establish a suitable extension of the BombieriVinogradov theorem.

This naturally led me to write a thoroughly revised and updated version of the book. To begin with, the chapters devoted to the knowledge required from undergraduate mathematics students have all been removed. In particular, this includes practically all of the content of my earlier book Thmes dArithmtiques, with only the most necessary overlapping content completely rewritten in summarized form. The prerequisites for this second edition are therefore essentially all known results in classic elementary number theory. Besides, the sections previously called Further Developments have become full sections that have substantially been expanded. In addition, the number of exercises, all corrected as in the first edition, has increased by almost 50%.

Each chapter has been revised, updated and enhanced, and Chap. in particular has undergone some drastic alterations. Dirichlet series are now to be found at the beginning of the chapter immediately after the basic theory. A quite simple asymptotic formula has been derived for multiplicative functions close to the divisor functions. Each mean value of the usual arithmetic function is given in detail, including the important Hooleys divisor function. A section, that is rarely mentioned in the literature and dealing with arithmetic functions of several variables, is provided. Ten well-known problems in analytic number theory have also been selected and given with almost self-contained proofs. The results that are beyond the scope of this book are given without proof but always with one or several references, so that the reader who is eager to learn more about the subject may go deeper into the analysis and pursue further so as to absorb the most profound results of the subject area.

This revised edition would have been impossible without the patience, availability and enthusiasm of Remi Lodh, whom I wish to thank for his interest in this project. I would also like to thank most warmly my wife, Vronique, for her rigorous and faithful translation of this work and for her unrelenting support. I am also most grateful to my colleagues Anne-Lise Camus, Florian Daval and Igor E. Shparlinski for their careful proof-reading and to Wadysaw Narkiewicz and Titus W. Hilberdink for their suggestions. Last but not least, I would particularly like to thank my two children, Nicolas, a graduate student in Civil and Mining Engineering at the Ecole des Mines, and Marie, who follows a 2-year undergraduate intensive course in Mathematics, Physics and Biology.