Tenenbaum G. Introduction to Analytic and Probabilistic Number Theory

3rd Edition. — Cambridge University Press 1995 (reprint 2015). — 466 p. — (Cambridge Studies in Advanced Mathematics 46). – ISBN: 082189854X.These twin motivations have led us to employ a slight variation of the traditional subdivision into text, notes and exercises. Thus, the main text, although generally restricted to statements proved in full detail, may also contain comments on additional references when they provide a useful background for a first reading. Conversely, the notes often give way to statements, and even proofs, of related results which may be omitted on first contact. In a similar way, the exercises serve a double purpose. Some are traditionally designed to facilitate the mastery of concepts introduced in the text itself. Others, mainly in Part III, lead to genuine research results which are sometimes new. In this context we have tried to break away from an unfortunate modern tendency by only proposing exercises which are soluble without excessive ingenuity or exceptional technical skill. We usually avoid questions to which the answers are not provided; the results aimed at are systematically stated at the outset and the main steps are indicated. This part of the book may therefore be used, even without making the effort of solving the problems, as an informal source of references. Complete solutions will appear shortly as a joint volume wElementary methods
Some tools from real analysis
Prime numbers
Arithmetic functions
Average orders
Sieve methods
Extremal orders
Summation formulae
The Riemann zeta function
The prime number theorem and the Riemann hypothesis
The Selberg—Delange method
Two arithmetic applications
Tauberian theorems
Prime numbers in arithmetic progressions
Probabilistic methods
Densities
Limiting distribution of arithmetic functions
Distribution of additive functions and mean values of multiplicative functions
Integers free of large prime factors. The saddle-point method
Integers free of small prime factors