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

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