This is a leisure introduction into basic analytic properties of Dirichlet series, which culminates in proofs of the prime number theorem and Dirichlet's theorem on primes in an arithmetic progression. The book is accessible even to undergraduate students - the prerequisites include standard courses in real and complex analysis but hardly any number theory. The first two chapters collect background material on Abel's summation and elementary properties of Dirichlet series. Chapter 3 treats (two version of) a Tauberian theorem relating the behaviour of Dirichlet's series f(s)= ∑n≥1 a(n)n-s and the function A(x)= ∑n≥1a(n). The prime number theorem is obtained as a special case for f(s)=-ζ'(s)/ ζ(s). These results are sharpened in Chapter 5 to include estimates for error terms and for zero-free regions of ζ(s). Chapter 4 is devoted to Dirichlet's theorem and Chapter 6 to an 'elementary' proof of the prime number theorem. Several appendices list background results from real analysis, others discuss numerical calculations of π(x) and historical background

Reviewer:

jnek