Zeta function universality
Encyclopedia
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 function
s 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.
Let U be a compact subset
of the strip
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
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.
where λ denotes the Lebesgue measure
on the real numbers.
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.
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.
If
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
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), in H of holomorphic functions defined on U, a Hilbert space
. We set
where pk denotes the k-th prime number. It can then be shown that the series
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, 1986).
. 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. Sections of the Lerch zeta-function have also been shown to have a form of joint universality.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, 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 function
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
s 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
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
of the strip
such that the complement
Complement (set theory)
In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...
of U is connected
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
. Let f : U → C be a continuous function
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
on U which is holomorphic on the interior
Interior (topology)
In mathematics, specifically in topology, the interior of a set S of points of a topological space consists of all points of S that do not belong to the boundary of S. A point that is in the interior of S is an interior point of S....
of U and does not have any zeros in U. Then for any ε > 0 there exists a 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.
where λ denotes the Lebesgue measure
Lebesgue measure
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...
on the real numbers.
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
Riemann hypothesis
In mathematics, the Riemann hypothesis, proposed by , is a conjecture about the location of the zeros of the Riemann zeta function which states that all non-trivial zeros have real part 1/2...
, 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
Fractal
A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
.
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.
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:
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
Logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...
, 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.
If
is a sequence of real numbers, one for each prime p, and M is a finite set of primes, we setWe consider the specific sequence
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 spaceBergman space
In complex analysis, a branch of mathematics, a Bergman space, named after Stefan Bergman, is a function space of holomorphic functions in a domain D of the complex plane that are sufficiently well-behaved at the boundary that they are absolutely integrable...
, falsely named Hardy space
Hardy space
In complex analysis, the Hardy spaces Hp are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper...
in (Voronin and Karatsuba, 1992), in H of holomorphic functions defined on U, a Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
. We set
where pk denotes the k-th prime number. It can then be shown that the series
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, 1986).
Universality of other zeta functions
A similar universality property has been shown for the Lerch zeta-functionLerch zeta function
In mathematics, the Lerch zeta-function, sometimes called the Hurwitz–Lerch zeta-function, is a special function that generalizes the Hurwitz zeta-function and the polylogarithm...
. 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. Sections of the Lerch zeta-function have also been shown to have a form of joint universality.
Further reading
- A. A. Karatsuba and S. M. Voronin, The Riemann-Zeta Function, Walter de Gruyter, July 1992
External links
- Voronin's Universality Theorem, by Matthew R. Watkins
- X-Ray of the Zeta Function Visually oriented investigation of where zeta is real or purely imaginary. Gives some indication of how complicated it is in the critical strip.