The zeta Function, The Dirichlet/Riemann zeta Function

The Dirichlet/Riemann zeta Function

Let f() map the positive integers into the complex numbers, and let s be a complex number. The Dirichlet zeta function, written ζ(f,s) is the sum of f(n)/n^s, as n runs from 1 to infinity. If f is omited, the constant function 1 is assumed. Thus ζ(s) = ζ(1,s), the sum of 1/n^s. This is the Riemann zeta function. If s is set to the integer k, we obtain ζ(k), as described in the introduction.

There are some functions, such as f(n) = n factorial, that never converge, no matter the value of s. When n exceeds the norm of s, n! increases faster than n^s. The terms do not approach 0, and the series diverges. In this case ζ(f,s) is not defined for any s in the complex plane.

Other functions converge everywhere. The obvious example is f() = 0, whence ζ(f,s) = 0 for every s. Yet there are nonzero functions that converge everywhere as well. Consider f(n) = 1/n!. Just as n! increases faster than any fixed power of n, so 1/n! decreases faster than any power of n. Use the ratio test to show this series converges absolutely. Thus the nonzero function f(n) = 1/n! has ζ(f,s) absolutely convergent over the entire complex plane.

Most functions lie between these two extremes. We often have convergence for high values of s and divergence for low values of s.

Note that ζ(f,0) is simply the sum of f(n). Zeta is defined at the origin iff f is convergent.