p-adic number, p-adic analysis, and zeta functions.

Includes index.

Includes bibliographical references (p. 129-131)

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.