Anatomy of Integers

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J. M. de Koninck, Andrew Granville, Florian Luca
American Mathematical Soc., Jan 1, 2008 - Mathematics - 297 pages
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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 functionssuch as the partition function. More specific topics include the Erdos-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 andunexpected consequences. The volume includes contributions from leading participants in this active area of research, such as Kevin Ford, Carl Pomerance, Kannan Soundararajan and Gerald Tenenbaum.
 

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Contents

What is anatomy?
vii
Ternary quadratic forms and sums of three squares with restricted variables
1
Entiers ayant exactement r diviseurs dans un intervalle donné
19
On the proportion of numbers coprime to a given integer
47
Integers with a divisor in y 2y
65
Powerfree values repulsion between points differing beliefs and the existence of error
81
Anatomy of integers and cyclotomic polynomials
89
Parité des valeurs de la fonction de partition pn et anatomie des entiers
97
Descartes numbers
167
A combinatorial method for developing Lucas sequence identities
175
On the difference of arithmetic functions at consecutive arguments
179
Pretentious multiplicative functions and an inequality for the zetafunction
191
On the distribution of 𝛚n
199
The ErdosKac theorem and its generalizations
209
On a conjecture of MontgomeryVaughan on extreme values of automorphic Lfunctions at 1
217
The Môbius function in short intervals
247

The distribution of smooth numbers in arithmetic progressions
115
Moyennes de certaines fonctions multiplicatives sur les entiers friables 4
129
Uniform distribution of zeros of Dirichlet series
143
On primes represented by quadratic polynomials
159
An explicit approach to hypothesis H for polynomials over a finite field
259
On prime factors of integers which are sums or shifted products
275
Simultaneous approximation of reals by values of arithmetic functions
289
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