I p-adic numbers. - 1. Basic concepts. - 2. Metrics on the rational numbers. - Exercises. - 3. Review of building up the complex numbers. - 4. The field of p-adic numbers. - 5. Arithmetic in ?p. - Exercises. - II p-adic interpolation of the Riemann zeta-function. - 1. A formula for ?(2k). - 2. p-adic interpolation of the function f(s) = as. - Exercises. - 3. p-adic distributions. - Exercises. - 4. Bernoulli distributions. - 5. Measures and integration. - Exercises. - 6. The p-adic ?-function as a Mellin-Mazur transform. - 7. A brief survey (no proofs). - Exercises. - III Building up ?. - 1. Finite fields. - Exercises. - 2. Extension of norms. - Exercises. - 3. The algebraic closure of ?p. - 4. ?. - Exercises. - IV p-adic power series. - 1. Elementary functions. - Exercises. - 2. The logarithm gamma and Artin-Hasse exponential functions. - Exercises. - 3. Newton polygons for polynomials. - 4. Newton polygons for power series. - Exercises. - V Rationality of the zeta-function of a set of equations over a finite field. - 1. Hypersurfaces and their zeta-functions. - Exercises. - 2. Characters and their lifting. - 3. A linear map on the vector space of power series. - 4. p-adic analytic expression for the zeta-function. - Exercises. - 5. The end of the proof. - Answers and Hints for the Exercises. Language: English