The Zeta Function


The zeta function, ζ(s) = Σ(1/ks) = 1 + 1/2s + 1/3s + ..., is usually referred to as Riemann's, because he elucidated so many of its properties, but was already known to Euler. It is valuable in the theory of prime numbers and analysis, and also as the value of certain integrals in modern physics.

Euler expressed the zeta function as a remarkable infinite product, 1/ζ(s) = Π(1 - p-s), where p are the prime numbers 2, 3, 5, 7, ...! How do the prime numbers get in here? Well, from the original series Euler first subtracted all the terms with denominators that were multiples of 2, which yielded ζ(s)(1 - 2-s). Then multiples of 3 were eliminated, then multiples of 5, and so on, which is just the Sieve of Eratosthenes. When this is completed in the limit, we have Euler's Product. How simple this is once you have seen it! It is typical of Euler's brilliance to have seen this.

The variable s can be complex, and ζ(s) is an analytic function of s (it is a uniformly convergent series of analytic functions), but we shall confine ourselves here only to real values of s. If you want to calculate ζ(s), the infinite series looks promising, but it converges discouragingly slowly. Euler's Product does much better. The primes 2, 3, 5, 7 and 11 already give ζ(4) = 1.8023, while we have to go to k > 13 for a similar precision using the series. The exact value for ζ(4) is π4/90. ζ(2n) can be expressed in terms of the Bernoulli number B2n, and there are other curious relations.

The integral can be expressed as a zeta function. This is an integral that is useful in physics.

References

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed., (Cambridge: C.U.P., 1958), Chapter XIII.


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Composed by J. B. Calvert
Created 12 November 2000
Last revised