Zeta Function Universality - Discussion

Discussion

The condition that the complement of U be connected essentially means that U doesn't contain any holes.

The intuitive meaning of the first statement is as follows: it is possible to move U by some vertical displacement it so that the function f on U is approximated by the zeta function on the displaced copy of U, to an accuracy of ε.

Note that the function f is not allowed to have any zeros on U. This is an important restriction; if you start with a holomorphic function with an isolated zero, then any "nearby" holomorphic function will also have a zero. According to the Riemann hypothesis, the Riemann zeta function does not have any zeros in the considered strip, and so it couldn't possibly approximate such a function. Note however that the function f(s)=0 which is identically zero on U can be approximated by ζ: we can first pick the "nearby" function g(s)=ε/2 (which is holomorphic and doesn't have zeros) and find a vertical displacement such that ζ approximates g to accuracy ε/2, and therefore f to accuracy ε.

The accompanying figure shows the zeta function on a representative part of the relevant strip. The color of the point s encodes the value ζ(s) as follows: the hue represents the argument of ζ(s), with red denoting positive real values, and then counterclockwise through yellow, green cyan, blue and purple. Strong colors denote values close to 0 (black = 0), weak colors denote values far away from 0 (white = ∞). The picture shows three zeros of the zeta function, at about 1/2+103.7i, 1/2+105.5i and 1/2+107.2i. Voronin's theorem essentially states that this strip contains all possible "analytic" color patterns that don't use black or white.

The rough meaning of the statement on the lower density is as follows: if a function f and an ε>0 is given, there is a positive probability that a randomly picked vertical displacement it will yield an approximation of f to accuracy ε.

Note also that the interior of U may be empty, in which case there is no requirement of f being holomorphic. For example, if we take U to be a line segment, then a continuous function f: U → C is nothing but a curve in the complex plane, and we see that the zeta function encodes every possible curve (i.e., any figure that can be drawn without lifting the pencil) to arbitrary precision on the considered strip.

The theorem as stated applies only to regions U that are contained in the strip. However, if we allow translations and scalings, we can also find encoded in the zeta functions approximate versions of all non-vanishing holomorphic functions defined on other regions. In particular, since the zeta function itself is holomorphic, versions of itself are encoded within it at different scales, the hallmark of a fractal.

The surprising nature of the theorem may be summarized in this way: the Riemann zeta functions contains "all possible behaviors" within it, and is thus "chaotic" in a sense, yet it is a perfectly smooth analytic function with a rather simple, straightforward definition.