The prime number theorem / G.J.O. Jameson.

Format:

Book

Author/Creator:

Jameson, G. J. O. (Graham James Oscar), author.

Series:

London Mathematical Society student texts ; 53.

London Mathematical Society student texts ; 53

Language:

English

Subjects (All):

Numbers, Prime.

Physical Description:

1 online resource (x, 252 pages) : digital, PDF file(s).

Place of Publication:

Cambridge : Cambridge University Press, 2003.

Language Note:

English

Summary:

At first glance the prime numbers appear to be distributed in a very irregular way amongst the integers, but it is possible to produce a simple formula that tells us (in an approximate but well defined sense) how many primes we can expect to find that are less than any integer we might choose. The prime number theorem tells us what this formula is and it is indisputably one of the great classical theorems of mathematics. This textbook gives an introduction to the prime number theorem suitable for advanced undergraduates and beginning graduate students. The author's aim is to show the reader how the tools of analysis can be used in number theory to attack a 'real' problem, and it is based on his own experiences of teaching this material.

Contents:

Cover; Series Page; Title; Copyright; Contents; Preface; 1 Foundations; 1.1 Counting prime numbers; 1.2 Arithmetic functions; Exercises; 1.3 Abel summation; Discrete version; Continuous version; 1.4 Estimation of sums by integrals; Euler's summation formula Basic integral estimation; Euler's summation formula; Closer estimation for version 2; Exercises; 1.5 The function li(x); Exercises; 1.6 Chebyshev's theta function; Exercises; 1.7 Dirichlet series and the zeta function The zeta function of a real variable; The function xS for complex s; Basic properties of Dirichlet series

DifferentiabilityExercises; 1.8 Convolutions; Exercises; 2 Some important Dirichlet series and arithmetic functions; 2.1 The Euler product; Exercises; 2.2 The Möbius function; Further series derived from the Euler product; Exercises; 2.3 The series for log ζ(s) and ζ'(s)/ζ(s); The series for log ζ(s); The series ΣpεP(1/ps); The series for ζ'(s)/ζ(s); Exercises; 2.4 Chebyshev's psi function and powers of primes; Estimation of φ(x) and comparison with θ(x); Powers of primes; Chebyshev's lower estimate; Exercises; 2.5 Estimates of some summation functions; The divisor function

Inversion of Dirichlet series by integrals on vertical linesThe Riemann-Lebesgue lemma; The integral version of the fundamental theorem; Further remarks; Exercises; 3.3 An alternative method: Newman's proof; Exercises; 3.4 The limit and series versions of the fundamental theorem; the prime number theorem; The limit and series versions; The prime number theorem; Further special cases of the basic theorems; Exercises; 3.5 Some applications of the prime number theorem Primes in intervals; The nth prime number; Enumemtion of numbers with k prime factors; Exercises

5.2 Connections with the Riemann hypothesis

Notes:

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

Includes bibliographical references (p. 249-250) and index.

ISBN:

1-316-08574-0

1-107-09275-2

0-511-07819-6

1-107-10147-6

0-511-07662-2

1-139-16498-8