**CRM Proceedings & Lecture Notes**

Volume: 46;
2008;
297 pp;
Softcover

MSC: Primary 11;

**Print ISBN: 978-0-8218-4406-9
Product Code: CRMP/46**

List Price: $112.00

AMS Member Price: $89.60

MAA Member Price: $100.80

**Electronic ISBN: 978-1-4704-3960-6
Product Code: CRMP/46.E**

List Price: $105.00

AMS Member Price: $84.00

MAA Member Price: $94.50

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# Anatomy of Integers

Share this page *Edited by *
*Jean-Marie De Koninck; Andrew Granville; Florian Luca*

A co-publication of the AMS and Centre de Recherches Mathématiques

The book is mostly devoted to the study of the prime factors of integers, their size and their quantity, to good bounds on the number of integers with different properties (for example, those with only large prime factors) and to the distribution of divisors of integers in a given interval. In particular, various estimates concerning smooth numbers are developed. A large emphasis is put on the study of additive and multiplicative functions as well as various arithmetic functions such as the partition function. More specific topics include the Erdős–Kac Theorem, cyclotomic polynomials, combinatorial methods, quadratic forms, zeta functions, Dirichlet series and \(L\)-functions. All these create an intimate understanding of the properties of integers and lead to fascinating and unexpected consequences. The volume includes contributions from leading participants in this active area of research, such as Kevin Ford, Carl Pomerance, Kannan Soundararajan and Gérald Tenenbaum.

Titles in this series are co-published with the Centre de Recherches Mathématiques.

#### Readership

Undergraduates and graduate students and research mathematicians interested in Erdős–type elementary number theory, smooth numbers, and distribution of prime factors of integers, partitions, etc.

#### Reviews & Endorsements

[I]t is quite satisfying to read in these papers some pretty descriptions of the insights behind the methods of proof, so that the application of technical anatomical results seems natural rather than confounding.

-- Greg Martin, University of British Columbia

# Table of Contents

## Anatomy of Integers

- Cover Cover11
- Title page i2
- Contents iii4
- What is anatomy? v6
- Ternary quadratic forms, and sums of three squares with restricted variables 18
- Entiers ayant exactement 𝑟 diviseurs dans un intervalle donné 1926
- On the proportion of numbers coprime to a given integer 4754
- Integers with a divisor in (𝑦,2𝑦] 6572
- Power-free values, repulsion between points, differing beliefs and the existence of error 8188
- Anatomy of integers and cyclotomic polynomials 8996
- Parité des valeurs de la fonction de partition 𝑝(𝑛) et anatomie des entiers 97104
- The distribution of smooth numbers in arithmetic progressions 115122
- Moyennes de certaines fonctions multiplicatives sur les entiers friables, 4 129136
- Uniform distribution of zeros of Dirichlet series 143150
- On primes represented by quadratic polynomials 159166
- Descartes numbers 167174
- A combinatorial method for developing Lucas sequence identities 175182
- On the difference of arithmetic functions at consecutive arguments 179186
- Pretentious multiplicative functions and an inequality for the zeta-function 191198
- On the distribution of 𝜔(𝑛) 199206
- The Erdős–Kac theorem and its generalizations 209216
- On a conjecture of Montgomery-Vaughan on extreme values of automorphic 𝐿-functions at 1 217224
- The Möbius function in short intervals 247254
- An explicit approach to hypothesis H for polynomials over a finite field 259266
- On prime factors of integers which are sums or shifted products 275282
- Simultaneous approximation of reals by values of arithmetic functions 289296
- Back Cover Back Cover1308