Number Theory i

Preface Among the various branches of mathematics, number theory is characterized to a lesser degree by its primary subject ("integers") than by a psychologi­ cal attitude. Actually, number theory also deals with rational, algebraic, and transcendental numbers, with some very specific analytic functions (such as Dirichlet series and modular forms), and with some geometric objects (such as lattices and schemes over Z). The question whether a given article belongs to number theory is answered by its author's system of values. If arithmetic is not there, the paper will hardly be considered as number-theoretical, even if it deals exclusively with integers and congruences. On the other hand, any mathematical tool, say, homotopy theory or dynamical systems may become an important source of number-theoretical inspiration. For this reason, com­ binatorics and the theory of recursive functions are not usually associated with number theory, whereas modular functions are. In this report we interpret number theory broadly. There are compelling reasons to adopt this viewpoint. First of all, the integers constitute (together with geometric images) one of the primary subjects of mathematics in general. Because of this, the history of elementary number theory is as long as the history of all mathematics, and the history of modern mathematic began when "numbers" and "figures" were united by the concept of coordinates (which in the opinion of LR. Shafarevich also forms the basic idea of algebra).

This is an indispensable reference work for specialists in number theory and other mathematicians who need access to some of these results in their own fields of research.

In 1791 Gauss made the following assertions (collected works, Vol. 10, p.ll, Teubner, Leipzig 1917): Primzahlen unter a ( = 00 ) a la Zahlen aus zwei Factoren lla· a la (warsch.) aus 3 Factoren 1 (lla)2a –- 2 la et sic in info In more modern notation, let 1tk(X) denote the number of integers not exceeding x which are made up of k distinct prime factors, k = 1, 2, …. Then his assertions amount to the asymptotic estimate x (log log X)k-l ( ) 1tk X '" –"';"'-"–-"::–:-'-,- (x-..oo). log x (k-1)! The case k = 1, known as the Prime Number Theorem, was independently established by Hadamard and de la Vallee Poussin in 1896, just over a hundred years later.

This handbook covers a wealth of topics from number theory, special attention being given to estimates and inequalities. As a rule, the most important results are presented, together with their refinements, extensions or generalisations.

This handbook covers a wealth of topics from number theory, special attention being given to estimates and inequalities. As a rule, the most important results are presented, together with their refinements, extensions or generalisations.