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Olivier Bordellès - Arithmetic Tales

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Olivier Bordellès Arithmetic Tales

Arithmetic Tales: summary, description and annotation

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Number theory was once famously labeled the queen of mathematics by Gauss. The multiplicative structure of the integers in particular deals with many fascinating problems some of which are easy to understand but very difficult to solve. In the past, a variety of very different techniques has been applied to further its understanding.

Classical methods in analytic theory such as Mertens theorem and Chebyshevs inequalities and the celebrated Prime Number Theorem give estimates for the distribution of prime numbers. Later on, multiplicative structure of integers leads to multiplicative arithmetical functions for which there are many important examples in number theory. Their theory involves the Dirichlet convolution product which arises with the inclusion of several summation techniques and a survey of classical results such as Hall and Tenenbaums theorem and the Mbius Inversion Formula. Another topic is the counting integer points close to smooth curves and its relation to the distribution of squarefree numbers, which is rarely covered in existing texts. Final chapters focus on exponential sums and algebraic number fields. A number of exercises at varying levels are also included.

Topics in Multiplicative Number Theory introduces offers a comprehensive introduction into these topics with an emphasis on analytic number theory. Since it requires very little technical expertise it will appeal to a wide target group including upper level undergraduates, doctoral and masters level students.

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Olivier Bordells Universitext Arithmetic Tales 2012 10.1007/978-1-4471-4096-2_1 Springer-Verlag London 2012
1. Basic Tools
Olivier Bordells 1
(1)
alle de la Combe 2, Aiguilhe, France
Abstract
This chapter provides the main tools that will be used in the whole text. Particular attention has been paid to the partial summation process which is of constant use in multiplicative number theory. The analytic properties of the divided differences will be used in Chap..
1.1 Euclidean Division
Some results depend on the following axiom.
Axiom 1.1
Any non - empty subset S of 0 contains a smallest element . Furthermore , if S is upper bounded , then it contains also a greatest element .
This result will enable us to get the Euclidean division between two non-negative integers, and thus to study the arithmetic properties of integers.
Theorem 1.2
Given non - negative integers a and b with b 1, there exists a unique couple ( q , r ) of natural numbers such that
q is the quotient and r is the remainder obtained when b is divided into a - photo 1
q is the quotient and r is the remainder obtained when b is divided into a .
Proof
Let S be the set defined by
The set S is clearly a subset of 0 and since a S Using Axiom 11 we infer - photo 2
The set S is clearly a subset of 0 and Picture 3 since a S . Using Axiom 1.1 we infer that S contains a smallest element denoted by r . Thus, r is a non-negative integer and we call q the integer satisfying r = a bq .
Let us show that r < b . If r = a bq b , then a b ( q +1)0 so that a b ( q +1) S , and therefore r = a bq a b ( q +1) since r is the smallest element in S . The latest inequality easily gives 10 which is obviously impossible. We thus proved that r < b .
To show the uniqueness, suppose there exists another pair ( q , r ) of integers such that Picture 4 , Picture 5 and a = bq + r with 0 r < b . Since a = bq + r , we deduce that b ( q q )= r r , and then b | q q |=| r r |, and thus b | r r | since Picture 6 implies that | q q |1. But the inequalities 0 r < b and 0 r < b imply that | r r |< b giving a contradiction. The proof is complete.
Remark 1.3
One can compute q and r . In , we have
Arithmetic Tales - image 7
and the inequalities Arithmetic Tales - image 8 imply Arithmetic Tales - image 9 so that
Remark 14 There exists a version of the Euclidean division in The result is - photo 10
Remark 1.4
There exists a version of the Euclidean division in . The result is similar to that of Theorem 1.2, except that the condition 0 r < b must be replaced by 0 r <| b |. We leave the details to the reader.
Remark 1.5
The particular case r =0 is interesting in itself. We will say that b divides a denoted by b a . Thus, b a is equivalent to the existence of an integer q such that a = bq . Recall that one of the most important properties is the following
Lemma 16 Let a b and n be a positive integer Then In particular if a - photo 11
Lemma 1.6
Let a , b and n be a positive integer . Then
In particular if a b then a b a n b n Proof Indeed the - photo 12
In particular , if a , b , then ( a b )( a n b n ).
Proof
Indeed, the right-hand side is equal to
Arithmetic Tales - image 13
as required.
Proposition 1.7
For all | x |<1 we have
Arithmetic Tales - image 14
(1.1)
Proof
This is well known and left to the reader.
1.2 Binomial Coefficients
This subject is well known and proofs and examples can easily be found in any book of combinatorial theory. We only recall here the main properties required.
Definition 1.8
Let n and k {0,, n }. The binomial coefficient Arithmetic Tales - image 15 is defined by the formula
Arithmetic Tales - image 16
Together with well-known results such as Newtons formula
valid for all a b it may be useful to have at our disposal some basic - photo 17
valid for all a , b , it may be useful to have at our disposal some basic estimates for binomial coefficients.
Proposition 1.9
Let n 2 be an integer .
(i)
For all integers Arithmetic Tales - image 18 , we have
Arithmetic Tales - image 19
(ii)
For all integers Arithmetic Tales - image 20 , we have
iii We have Proof The first bounds follow easily from - photo 21
(iii)
We have
Proof The first bounds follow easily from The remaining inequalities are - photo 22
Proof
The first bounds follow easily from
The remaining inequalities are immediate consequences of the following Stirling - photo 23
The remaining inequalities are immediate consequences of the following Stirling type estimates []
12 valid for all n Remark 110 With a little more work it can be - photo 24
(1.2)
valid for all n .
Remark 1.10
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