Zeta function universality
The universality of the Riemann zeta function was first proven by Sergei Mikhailovitch Voronin [ru] in 1975[1] and is sometimes known as Voronin's universality theorem.
A mathematically precise statement of universality for the Riemann zeta function ζ(s) follows.
Let f : U → ℂ be a continuous function on U which is holomorphic on the interior of U and does not have any zeros in U.
Even more: The lower density of the set of values t satisfying the above inequality is positive.
The condition that the complement of U be connected essentially means that U does not 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.
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.
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 does not 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.
Voronin's theorem essentially states that this strip contains all possible "analytic" color patterns that do not 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 are given, then there is a positive probability that a randomly picked vertical displacement it will yield an approximation of f to accuracy ε.
For example, if we take U to be a line segment, then a continuous function f : U → C is 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.
[2] The surprising nature of the theorem may be summarized in this way: the Riemann zeta function contains "all possible behaviors" within it, and is thus "chaotic" in a sense, yet it is a perfectly smooth analytic function with a straightforward definition.
We consider only the case where U is a disk centered at 3/4: and we will argue that every non-zero holomorphic function defined on U can be approximated by the ζ-function on a vertical translation of this set.
Passing to the logarithm, it is enough to show that for every holomorphic function g : U → C and every ε > 0 there exists a real number t such that We will first approximate g(s) with the logarithm of certain finite products reminiscent of the Euler product for the ζ-function: where P denotes the set of all primes.
is a sequence of real numbers, one for each prime p, and M is a finite set of primes, we set We consider the specific sequence and claim that g(s) can be approximated by a function of the form
We set where pk denotes the k-th prime number.
Because of a relationship between the norm in H and the maximum absolute value of a function, we can then approximate our given function g(s) with an initial segment of this rearranged series, as required.
By a version of the Kronecker theorem, applied to the real numbers
(which are linearly independent over the rationals) we can find real values of t so that
Although Voronin's theorem is not proved there, two corollaries are derived from it: Some recent work has focused on effective universality.
Under the conditions stated at the beginning of this article, there exist values of t that satisfy inequality (1).
An effective universality theorem places an upper bound on the smallest such t. For example, in 2003, Garunkštis proved that if
Bounds can also be obtained on the measure of these t values, in terms of ε:
[5]: 210 Work has been done showing that universality extends to Selberg zeta functions.
[7] [8]: Section 4 A similar universality property has been shown for the Lerch zeta function
