The zeta Function, An Introduction

Introduction

If k is an integer, zeta(k), written ζ(k), is the sum of 1/n^k as n runs from 1 to infinity. We've already seen that ζ(1) produces the harmonic series, which does not converge, while ζ(2), the sum of inverse squares, does converge. By dominance, ζ(k) converges for any k > 2.

As we shall see, this function generalizes in several different ways. For instance, the exponent k could be a real or complex number, and that's just a start.

The zeta function, in all its generality, is essential for analytic number theory - so perhaps I should describe it there. But it is also used in algebraic number theory. On the other hand, complex calculus is used to evaluate the zeta function, so perhaps it belongs there. Finally, the function is defined as a convergent series, so maybe it belongs here, under sequences and series. Apparently I found this last point compelling, because here it is.