INTRODUCTION

There has developed in recent years a p-adic analysis comparable in scope to the classical analysis of real and complex functions, but with many salient differences. A particular object of study has been the so-called p-adic L-functions, which appear to hold the key to several deep unsolved problems. In this chapter we shall develop the theory sufficiently far to explain the slightly sophisticated relationship between p-adic L-functions and the classical L-functions of analytic number theory and to prove one or two nontrivial theorems. Incidentally, this will put the proof of the von Staudt-Clausen relations of Chapter 1, §3 into a natural setting. The final section, which is somewhat divorced from the rest, explains a theorem of Mahler about a development of continuous functions on ℤp as infinite series. Necessarily, this chapter is rather a pot pourri.

The natural domain to work is the field Ωp ⊃ Qp introduced in Chapter 8, §3. All we shall need here, however, is that Ωp is both complete and algebraically closed.

ELEMENTARY FUNCTIONS

If a function of a complex variable is defined by a power series with coefficients in Q then it is natural to use the same power series to define a function of a variable in Ωp (or, rather, in the domain of convergence). One might expect the new function to have properties analogous to the old, and in some cases this is an immediate consequence of the “permanence of algebraic form” (an old Victorian mathematical tool which fell into disrepute because it was mishandled, but which remains a most potent weapon in the mathematician's arsenal).