The Prime Number TheoremCambridge University Press, 17 d’abr. 2003 - 252 pàgines "Counter 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 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."--Publisher's description. |
Continguts
Foundations | 1 |
11 Counting prime numbers | 2 |
12 Arithmetic functions | 6 |
13 Abel summation | 9 |
14 Estimation of sums by integrals Eulers summation formula | 18 |
15 The function li𝒙 | 28 |
16 Chebyshevs theta function | 33 |
17 Dirichlet series and the zeta function | 38 |
Prime numbers in residue classes Dirichlets theorem | 147 |
41 Characters of finite abelian groups | 148 |
42 Dirichlet characters | 153 |
43 Dirichlet Lfunctions | 160 |
44 Prime numbers in residue classes | 173 |
Error estimates and the Riemann hypothesis | 178 |
51 Error estimates | 179 |
52 Connections with the Riemann hypothesis | 192 |
18 Convolutions | 50 |
Some important Dirichlet series and arithmetic functions | 56 |
21 The Euler product | 57 |
22 The Mōbius function | 61 |
23 The series for log 𝛇8 and 𝛇ˡ8𝛇8 | 70 |
24 Chebyshevs psi function and powers of primes | 75 |
25 Estimates of some summation functions | 82 |
26 Mertenss estimates | 89 |
The basic theorems | 98 |
31 Extension of the definition of the zeta function | 99 |
32 Inversion of Dirichlet series the integral version of the fundamental theorem | 114 |
Newmans proof | 124 |
34 The limit and series versions of the fundamental theorem the prime number theorem | 130 |
35 Some applications of the prime number theorem | 138 |
53 The zerofree region of the zeta function | 195 |
An elementary proof of the prime number theorem | 206 |
61 Framework of the proof | 207 |
62 Selbergs formulae and completion of the proof | 212 |
Complex functions of a real variable | 223 |
Double series and multiplication of series | 225 |
Infinite products | 228 |
Differentiation under the integral sign | 230 |
The O o notation | 232 |
Computing values of 𝜋𝒙 | 234 |
Table of primes | 238 |
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