“Zeta function universality”的意思、由来-开放百科全书

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^[1] 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.

such that the complement of U is connected. Let {{nowrap|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 {{nowrap|ε > 0}} there exists a {{nowrap|t ≥ 0}} such that

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.

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 {{nowrap|1=f(s) = 0}} which is identically zero on U can be approximated by ζ: we can first pick the "nearby" function {{nowrap|1=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 {{nowrap|1/2 + 103.7i}}, {{nowrap|1/2 + 105.5i}} and {{nowrap|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 {{nowrap|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 {{nowrap|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.^[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 rather simple, straightforward definition.

Proof sketch

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 {{nowrap|g : U → C}} and every {{nowrap|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:

is a sequence of real numbers, one for each prime p, and M is a finite set of primes, we set

and claim that g(s) can be approximated by a function of the form

for a suitable set M of primes. The proof of this claim utilizes the Bergman space, falsely named Hardy space in (Voronin and Karatsuba, 1992),^[3] in H of holomorphic functions defined on U, a Hilbert space. We set

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. 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

is approximated by

. Further, for some of these values t,

approximates

, finishing the proof.

The theorem is stated without proof in § 11.11 of (Titchmarsh and Heath-Brown, 1986),^[4]

the second edition of a 1951 monograph by Titchmarsh; and a weaker result is given in Thm. 11.9. Although Voronin's theorem is not proved there, two corollaries are derived from it:

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.

Universality of other zeta functions

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.^[7]

A similar universality property has been shown for the Lerch zeta function

, at least when the parameter α is a transcendental number.

Sections of the Lerch zeta-function have also been shown to have a form of joint universality.

References

1. ^Voronin, S.M. (1975) "Theorem on the Universality of the Riemann Zeta Function." Izv. Akad. Nauk SSSR, Ser. Matem. 39 pp.475-486. Reprinted in Math. USSR Izv. 9, 443-445, 1975
2. ^{{Cite arxiv| last = Woon| first = S.C.| title = Riemann zeta function is a fractal| date = 1994-06-11| eprint = chao-dyn/9406003}}
3. ^¹{{Cite book| publisher = Walter de Gruyter| isbn = 3-11-013170-6| last = Karatsuba| first = A. A.|author2=Voronin, S. M.| title = The Riemann Zeta-Function| date = July 1992| page = 396}}
4. ^{{Cite book | last1 = Titchmarsh | first1 = Edward Charles | last2 = Heath-Brown | first2 = David Rodney ("Roger") | title = The Theory of the Riemann Zeta-function | publisher = Oxford U. P. | edition = 2nd | location = Oxford | year = 1986 | pages = 308–309 | isbn = 0-19-853369-1}}
5. ^{{ cite journal |author1=Ramūnas Garunkštis |author2=Antanas Laurinčikas |author3=Kohji Matsumoto |author4=Jörn Steuding |author5=Rasa Steuding |title=Effective uniform approximation by the Riemann zeta-function |journal=Publicacions Matemàtiques |volume=54 |date=2010 |pages=209–219 |jstor=43736941 |doi=10.5565/publmat_54110_12}}
6. ^{{ cite journal |author1=Paulius Drungilas |author2=Ramūnas Garunkštis |author3=Audrius Kačėnas |title=Universality of the Selberg zeta-function for the modular group |journal=Forum Mathematicum |volume=25 |issue=3 |pages=– |date=2013 |doi=10.1515/form.2011.127 |issn=1435-5337}}
7. ^{{cite journal|author=B. Bagchi|title=A Universality theorem for Dirichlet L-functions|journal=Mathematische Zeitschrift|year=1982|volume=181|issue=3|pages=319–334|doi=10.1007/BF01161980}}
8. ^¹²{{cite conference |title=A survey on the theory of universality for zeta and L-functions |author=Kohji Matsumoto |date= 2013 |publisher=World Scientific |conference=The 7th China–Japan Seminar on Number Theory |book-title=Plowing and Starring Through High Wave Forms. Proceedings of the 7th China–Japan Seminar |volume=11 |pages=95–144 |location=Fukuoka, Japan |arxiv=1407.4216 |isbn=978-981-4644-92-1 |bibcode=2014arXiv1407.4216M}}