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Anatomy of Integers

Edited by: Jean-Marie De Koninck Université Laval, Québec, QC, Canada
Andrew Granville Université de Montréal, Montréal, QC, Canada
Florian Luca Universidad Nacional Autonoma de México, Morelia, México
A co-publication of the AMS and Centre de Recherches Mathématiques
Available Formats:
Softcover ISBN: 978-0-8218-4406-9
Product Code: CRMP/46
List Price: $112.00 MAA Member Price:$100.80
AMS Member Price: $89.60 Electronic ISBN: 978-1-4704-3960-6 Product Code: CRMP/46.E List Price:$105.00
MAA Member Price: $94.50 AMS Member Price:$84.00
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This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.
List Price: $168.00 MAA Member Price:$151.20
AMS Member Price: $134.40 Click above image for expanded view Anatomy of Integers Edited by: Jean-Marie De Koninck Université Laval, Québec, QC, Canada Andrew Granville Université de Montréal, Montréal, QC, Canada Florian Luca Universidad Nacional Autonoma de México, Morelia, México A co-publication of the AMS and Centre de Recherches Mathématiques Available Formats:  Softcover ISBN: 978-0-8218-4406-9 Product Code: CRMP/46  List Price:$112.00 MAA Member Price: $100.80 AMS Member Price:$89.60
 Electronic ISBN: 978-1-4704-3960-6 Product Code: CRMP/46.E
 List Price: $105.00 MAA Member Price:$94.50 AMS Member Price: $84.00 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$168.00 MAA Member Price: $151.20 AMS Member Price:$134.40
• Book Details

CRM Proceedings & Lecture Notes
Volume: 462008; 297 pp
MSC: Primary 11;

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.

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.

• Chapters
• Ternary quadratic forms, and sums of three squares with restricted variables
• Entiers ayant exactement $r$ diviseurs dans un intervalle donné
• On the proportion of numbers coprime to a given integer
• Integers with a divisor in ${(}y,2y{]}$
• Power-free values, repulsion between points, differing beliefs and the existence of error
• Anatomy of integers and cyclotomic polynomials
• Parité des valeurs de la fonction de partition $p(n)$ et anatomie des entiers
• The distribution of smooth numbers in arithmetic progressions
• Moyennes de certaines fonctions multiplicatives sur les entiers friables, 4
• Uniform distribution of zeros of Dirichlet series
• On primes represented by quadratic polynomials
• Descartes numbers
• A combinatorial method for developing Lucas sequence identities
• On the difference of arithmetic functions at consecutive arguments
• Pretentious multiplicative functions and an inequality for the zeta-function
• On the distribution of $\omega (n)$
• The Erdős–Kac theorem and its generalizations
• On a conjecture of Montgomery-Vaughan on extreme values of automorphic $L$-functions at 1
• The Möbius function in short intervals
• An explicit approach to hypothesis H for polynomials over a finite field
• On prime factors of integers which are sums or shifted products
• Simultaneous approximation of reals by values of arithmetic functions

• Reviews

• [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
• Requests

Review Copy – for reviewers who would like to review an AMS book
Accessibility – to request an alternate format of an AMS title
Volume: 462008; 297 pp
MSC: Primary 11;

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.

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.

• Chapters
• Ternary quadratic forms, and sums of three squares with restricted variables
• Entiers ayant exactement $r$ diviseurs dans un intervalle donné
• On the proportion of numbers coprime to a given integer
• Integers with a divisor in ${(}y,2y{]}$
• Power-free values, repulsion between points, differing beliefs and the existence of error
• Anatomy of integers and cyclotomic polynomials
• Parité des valeurs de la fonction de partition $p(n)$ et anatomie des entiers
• The distribution of smooth numbers in arithmetic progressions
• Moyennes de certaines fonctions multiplicatives sur les entiers friables, 4
• Uniform distribution of zeros of Dirichlet series
• On primes represented by quadratic polynomials
• Descartes numbers
• A combinatorial method for developing Lucas sequence identities
• On the difference of arithmetic functions at consecutive arguments
• Pretentious multiplicative functions and an inequality for the zeta-function
• On the distribution of $\omega (n)$
• The Erdős–Kac theorem and its generalizations
• On a conjecture of Montgomery-Vaughan on extreme values of automorphic $L$-functions at 1
• The Möbius function in short intervals
• An explicit approach to hypothesis H for polynomials over a finite field
• On prime factors of integers which are sums or shifted products
• Simultaneous approximation of reals by values of arithmetic functions
• [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
Review Copy – for reviewers who would like to review an AMS book
Accessibility – to request an alternate format of an AMS title
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