| The Prime Numbers and Their Distribution |
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| Written by Giulia Biagini | ||||
| Sunday, 14 January 2007 | ||||
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Basic Information
This book gives a general and pleasing overview on many topics about the distribution of prime numbers. Its goal is to provide insights of different nature on that theme and this is performed through the illustration of conjectures, methods, results and even (very concise) proofs.
The volume is divided into five chapters, they are: Genesis: from Euclid to Chebyshev; The Riemann Zeta Function; Stochastic Distribution of Prime Numbers; An Elementary Proof of the Prime Number Theorem; The Major Conjectures. All of them are almost independent one to another, so you may skip the ones you are not interested in without any problem.
Then the book discusses both the theme of decomposition (existence and uniqueness) and that of congruences (Gauss' notation, Euler's totient function, Fermat's little theorem, Euler formula, Wilson's theorem). Follows a quick cryptographic intermezzo about the well known RSA cryptosystem. The subsequent paragraph shows what a quadratic residue is, introduces the Legendre Symbol and also proves the Girard-Fermat theorem. Always about the infinitude of the set of primes, some consequences of the Euler's proof are also described. Other famous results reported here are: the sieve of Eratosthenes, Chebyshev's Theorems and Mertens' ones, about which are also presented the Mobius function, the Legendre formula, the Mangoldt function and the Euler's constant. Finally, the book proposes a survey on Brun's sieve and the twin prime conjecture.
Another important aspect is analytic continuation: for example, using it together with the functional equation for the zeta function (already discovered by Riemann and proved in this paragraph with the Poisson formula) allows us to define \zeta(s) in the half-plane \sigma \leq 1/2. There are also two consequences of the functional equation: the first one is that the point s = 1 is the only singularity of zeta in the whole complex plane, while the second one is about the value of \zeta(2n) for n \geq 1 (for this point, the Bernoulli number is introduced). The book proposes also an explanation on the relation between the fact that the zeta function has no zero on the line \sigma = 1 (due to Hadamard and La Vallée-Poussin) and the prime number theorem (in order to accomplish this it recalls the Perron formulae and the Riemann-Lebesgue lemma). Follows the exposition of the Riemann Hypothesis (all the non trivial zeros of \zeta(s) lie on the axis of symmetry \sigma = 1/2) and a survey on the role of the Riemann Hypothesis on the distribution of primes (including the Selberg, Levinson, Conrey and Vinogradov-Korobov results). The paragraph ends with an inspection of some famous outcome divided into two categories: the oscillation theorems by Phragmén-Landau, Littlewood, Hardy and Dirichlet; the theorems about the localization of integers in short intervals by Vinogradov-Korobov, Ingham, Huxley and Iwaniec-Jutila.
Other cited results are: the Siegel-Walfisz theorem, the Pólya-Vinogradov inequality, something concerning the so called Siegel Zero and the Bombieri-Vinogradov theorem. Follows an exposition of the Cramér's model, which is a method able to produce conjectures regarding primes. Its relevance is also confirmed by results such as Littlewood's one. Then, the Cramer's conjecture is introduced as a weaker consequence of the Borel-Cantelli lemma and also the Poisson distribution and the Gallagher result are presented to show the Cramér's usage in the study of the primes density. In particular, this model is consistent with the Riemann Hypothesis but not even with the Hardy-Littlewood conjecture. The book reports briefly also the results by Selberg, Maier (first refutation of the model) and Rankin (Buchstab's function and Iwaniec theorem are cited through the lines). The uniform distribution modulo one is also exposed, in particular the Weyl's uniform distribution criterion is described and so also other contributions suitable for this context (by van der Corput, Dirichlet, Vinogradov, Vaughan and Daboussi). Finally, some words are spent about the geometric vision of primes distribution.
This choice allows them to introduce some of the basic mathematics commonly studied in analytic number theory: integration by parts; convolution of arithmetic functions; a more detailed explanation about the Mobius function and its meaning in the prime number theorem; a discussion about integers free of large, or small, prime factors; the Dickman function. Then the elementary proof of the Prime Number Theorem (the Daboussi version revisited) is illustrated.
In my opinion this is really a good book for those interested in this
area, it provides interesting ideas and motivates to read more about the subject. Even if this is a very concise text the number of notions discussed here is amazing!
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