How to use this text

Chapters 1, 2, 3, 5 and 6 are suitable for a course emphasizing arithmetic

functions and the two classical highlights of analytic number theory: Dirich-

let’s theorem on primes in arithmetic progressions, and the Prime Number

Theorem.

Chapters 1, 2, 3 and 4 are suitable for a syllabus with an emphasis on

elementary methods, for students with little knowledge of analysis. Unfor-

tunately the Prime Number Theorem is not covered.

Chapters 1, 3, 4, 5 and 6 are suitable for a syllabus with a Diophantine

emphasis, but also including a proof of the Prime Number Theorem.

Chapters 1, 3, 5, 6, 8 and 10 are suitable for a syllabus with an emphasis

on analytic methods, concentrating on the Riemann zeta function and the

distribution of primes.

Chapters 1, 3, 5, 8 and 9 are suitable for a syllabus with an emphasis

on the analytic theory of number fields. Students following such a syllabus

should either have some knowledge of algebraic number theory, or else have

a good knowledge of abstract algebra and do some reading. Here the Prime

Number Theorem is established by means of the Ikehara theorem.

Chapters 1, 3, 5 and 7 are suitable for a syllabus aiming at the Prime

Number Theorem for arithmetic progressions. The ordinary Prime Number

Theorem is established as a corollary at the end of the course.

To help with the planning of syllabi a diagram of dependencies between

chapters has been provided; see Figure 1 on page xvi. To supplement this,

more specific comments may prove useful: Unless the material on Artin

L-functions in Chapter 9 is covered, Section 3.8 may be substituted for Sec-

tions 3.4 and 3.5. This saves a little time, and the construction of Dirichlet

xv