Contents  1 Estimating Primes  1 1 Asymptotic Notation  1 2 Partial Summation  1 3 Arithmetic Functions and Dirichlet convolution  1 4 Dirichlet Series  2 Elementary Estimates for Primes  2 1 Merten s Theorems  2 2 Sieve Methods  2 3 Selberg Sieve  3 The Riemann Zeta Function  3 1 Partial fraction approximation of    3 2 Zero free region  Index  What is analytic number theory   Study of number theoretic problems using analysis  real  complex  Fourier      Also tools from combinatorics  probability     What kind of problems are studied   A variety of problems about integers  especially primes   Are there infinitely many primes   Euclid   300BC   Are there infinitely many primes starting with 7 in base 10   follows from prime number theorem   Are there infinitely many primes ending with 7 in base 10   follows from Dirichlet s theorem   Are there infinitely many primes with 49  of the digits being 7 in base 10  would follow from the Riemann hypothesis  Are there infinitely many pairs of primes differing by 2   twin prime conjecture   Key feature  To show that a set  of primes  is infinite  want to estimate the number of elements   x   Definition  Define          x         primes   x         p   x 1       Euclid showed  lim x         x         Theorem   Prime number theorem      lim x         x   log x x   1           x     x log x   Conjectured  Legendre  Gauss  Proved  Hadamard  de la Vall e Poussin