Zeta function universality

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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. Zeta function universality_sentence_0

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. Zeta function universality_sentence_1

Formal statement Zeta function universality_section_0

A mathematically precise statement of universality for the Riemann zeta-function ζ(s) follows. Zeta function universality_sentence_2

Let U be a compact subset of the strip Zeta function universality_sentence_3

such that the complement of U is connected. Zeta function universality_sentence_4

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. Zeta function universality_sentence_5

Then for any ε > 0 there exists a t ≥ 0 such that Zeta function universality_sentence_6

Even more: the lower density of the set of values t which do the job is positive, as is expressed by the following inequality about a limit inferior. Zeta function universality_sentence_7

where λ denotes the Lebesgue measure on the real numbers. Zeta function universality_sentence_8

Discussion Zeta function universality_section_1

The condition that the complement of U be connected essentially means that U doesn't contain any holes. Zeta function universality_sentence_9

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. Zeta function universality_sentence_10

The function f is not allowed to have any zeros on U. Zeta function universality_sentence_11

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. Zeta function universality_sentence_12

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. Zeta function universality_sentence_13

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. Zeta function universality_sentence_14

The accompanying figure shows the zeta function on a representative part of the relevant strip. Zeta function universality_sentence_15

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. Zeta function universality_sentence_16

Strong colors denote values close to 0 (black = 0), weak colors denote values far away from 0 (white = ∞). Zeta function universality_sentence_17

The picture shows three zeros of the zeta function, at about 1/2 + 103.7i, 1/2 + 105.5i and 1/2 + 107.2i. Zeta function universality_sentence_18

Voronin's theorem essentially states that this strip contains all possible "analytic" color patterns that don't use black or white. Zeta function universality_sentence_19

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. Zeta function universality_sentence_20

The interior of U may be empty, in which case there is no requirement of f being holomorphic. Zeta function universality_sentence_21

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. Zeta function universality_sentence_22

The theorem as stated applies only to regions U that are contained in the strip. Zeta function universality_sentence_23

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. Zeta function universality_sentence_24

In particular, since the zeta function itself is holomorphic, versions of itself are encoded within it at different scales, the hallmark of a fractal. Zeta function universality_sentence_25

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 rather simple, straightforward definition. Zeta function universality_sentence_26

Proof sketch Zeta function universality_section_2

A sketch of the proof presented in (Voronin and Karatsuba, 1992) follows. Zeta function universality_sentence_27

We consider only the case where U is a disk centered at 3/4: Zeta function universality_sentence_28

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. Zeta function universality_sentence_29

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 Zeta function universality_sentence_30

We will first approximate g(s) with the logarithm of certain finite products reminiscent of the Euler product for the ζ-function: Zeta function universality_sentence_31

where P denotes the set of all primes. Zeta function universality_sentence_32

We consider the specific sequence Zeta function universality_sentence_33

where pk denotes the k-th prime number. Zeta function universality_sentence_34

It can then be shown that the series Zeta function universality_sentence_35

is conditionally convergent in H, i.e. for every element v of H there exists a rearrangement of the series which converges in H to v. This argument uses a theorem that generalizes the Riemann series theorem to a Hilbert space setting. Zeta function universality_sentence_36

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. Zeta function universality_sentence_37

The theorem is stated without proof in § 11.11 of (Titchmarsh and Heath-Brown, 1986), the second edition of a 1951 monograph by Titchmarsh; and a weaker result is given in Thm. Zeta function universality_sentence_38

11.9. Zeta function universality_sentence_39

Although Voronin's theorem is not proved there, two corollaries are derived from it: Zeta function universality_sentence_40

Effective universality Zeta function universality_section_3

Some recent work has focused on effective universality. Zeta function universality_sentence_41

Under the conditions stated at the beginning of this article, there exist values of t that satisfy inequality (1). Zeta function universality_sentence_42

An effective universality theorem places an upper bound on the smallest such t. Zeta function universality_sentence_43

Bounds can also be obtained on the measure of these t values, in terms of ε: Zeta function universality_sentence_44

Universality of other zeta functions Zeta function universality_section_4

Work has been done showing that universality extends to Selberg zeta functions Zeta function universality_sentence_45

The Dirichlet L-functions show not only universality, but a certain kind of joint universality that allow any set of functions to be approximated by the same value(s) of t in different L-functions, where each function to be approximated is paired with a different L-function. Zeta function universality_sentence_46

Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/Zeta function universality.