In mathematics, the universality of zeta functions is the remarkable ability of the Riemann zeta function and other similar functions (such as the Dirichlet L-functions) to approximate arbitrary non-vanishing holomorphic functions arbitrarily well. The universality of the Riemann zeta function was first proven by Sergei Mikhailovitch Voronin in 1975 and is sometimes known as Voronin's universality theorem.
Formal statement A mathematically precise statement of universality for the Riemann zeta function ζ(s) follows. Let U be a compact subset of the strip
{ s ∈ C : 1 2 < R e ( s ) < 1 } {\displaystyle \left\{\ s\in \mathbb {C} :{\frac {\ 1\ }{2}}<\operatorname {\mathrm {Re} } (s)<1\ \right\}}
such that the complement of U is connected. 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. Then for any ε > 0 there exists a t ≥ 0 such that
for all s ∈ U . {\displaystyle s\in U.} Even more: The lower density of the set of values t satisfying the above inequality is positive. More precisely, 0 < lim inf T → ∞ 1 T λ ( { t ∈ [ 0 , T ] : max s ∈ U | ζ ( s + i t ) − f ( s ) | < ε } ) , {\displaystyle 0~<~\liminf _{T\to \infty }~{\frac {1}{\ T\ }}\ \lambda \!\left(\left\{\ t\in [0,T]\;:\;\max _{s\in U}{\Bigl |}\ \zeta (s+it)-f(s)\ {\Bigr |}<\varepsilon \ \right\}\right),} where λ {\displaystyle \lambda } is the Lebesgue measure on the real numbers and lim inf {\displaystyle \liminf } is the limit inferior.
Discussion 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. The function f is not allowed to have any zeros on U. This is an important restriction; if we 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 could not 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. 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 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. 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 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 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.
Proof sketch A sketch of the proof presented in (Voronin and Karatsuba, 1992) follows. We consider only the case where U is a disk centered at 3/4:
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