Voronin's Universality Theorem
This extraordinary 1975 result receives surprisingly little coverage (it
was difficult to find a clear and accurate statement anywhere on the WWW in
Let f(z) be any analytic function which is nonzero in the open disc
|z| < r for some 0 < r < 1/4 and continuous up
to the boundary of this disc. Then a disc of radius r centred
on the line Re[s] = 3/4 can always be found in which the zeta function
approximates the behaviour
of f(z) in |z| < r, within any given accuracy.
In other words, given such an f(z), r and
> 0, we can always find some real value t
Note that through a simple translation and rescaling procedure,
we can obtain as a corollary that the (nonzero) behaviour of any analytic function on any open disc in
the complex plane can be reproduced with arbitrary accuracy by the
zeta function acting on one of these discs of radius 1/4 in the right
half of the critical strip.
This theorem first appeared here:
S.M. Voronin, "Theorem on the 'universality' of the Riemann
zeta-function", Izv. Akad. Nauk SSSR, Ser. Matem. 39
(1975) (in Russian). English translation in: Math.
USSR Izvestija 9 (1975) 443-453
This article of the same year appears to be related:
S.M. Voronin, "A theorem on the distribution of values of the
Riemann zeta-function" (Russian), Dokl. Akad. Nauk SSSR 221
(1975) 771. English translation in Soviet Math. Dokl. 16
Universality is covered in this book:
A.A. Karatsuba and S.M. Voronin, The Riemann
Zeta-Function (de Gruyter, 1992)
On p. 377 of M.C. Gutzwiler's Chaos in Classical and Quantum
Mechanics (Springer-Verlag, 1990), he writes:
"Although the Riemann zeta-function is an analytic function with
[a] deceptively simple definition, it keeps bouncing around almost
randomly without settling down to some regular asymptotic pattern.
The Riemann zeta-function displays the essence of chaos in quantum
mechanics, analytically smooth, and yet seemingly unpredictable.
This chaotic feature has been demonstrated more dramatically
by Reich (1980) and Good (1981), in the form of the following amazing
Let D be a disk of radius r > 0 in the complex plane,
centered on one of the points $\mu + im\nu$, where 1/2 < $\mu$ < 1 and
$\nu$ > 0 are fixed, while the integers m > 0; the size r of
these disks is limited by the condition that they have to be inside
the strip 1/2 < z < 1; then choose an arbitrary non-vanishing,
holomorphic function f(z), whose Taylor expansion around
0 converges inside a circle of radius r. Not consider the set
M of integers m > 0, for which the difference
$|f(z - \mu - im\nu) - \zeta(z)| < \epsilon$ in the whole disk around
$\mu + im\nu$ for some fixed $\epsilon < 0$. This set is proved to have a
non-vanishing density, i.e., the number of points in M below
some large N > 0 is a non-vanishing fraction of N with a
lower bound greater that 0; this fraction depends, of course, on the
function f(z) and on $\epsilon$.
In a more intuitive language, the Riemann zeta-function is capable of
fitting any arbitrary smooth function over a finite disk with arbitary
accuracy, and it does so with comparitive ease, since it repeats the
performance like a good actor infinitely many times on a designated set
This is part of the book's section 19.9: "Chaos in Quantum-Mechanical
Scattering" which relates to Selberg zeta
functions a lot of the material in the
scattering and number theory page.
This is from an article by Gutzwiller
in Scientific American (January, 1992):
"The chaos of the Riemann zeta function is particularly apparent in
a theorem that has only recently been proved: the zeta function fits
locally any smooth function. The theorem suggests that the function
may describe all the chaotic behaviour a quantum system can exhibit."
A. Laurincikas, Limit Theorems for the Riemann Zeta-Function,
[AMS review found here]
[review by M.
Jutila:] "Several books on the Riemann zeta-function have appeared in recent
years, but the present one seems to be the first devoted solely to
its probabilistic aspects. Starting from the basis of standard
university courses, this monograph provides an excellent introduction
to topics like the value distribution of the zeta-function or its
"universality" properties. To a considerable extent, the material
covered represents the author's own research in probabilistic number
theory. Illuminating additional information may be found in the notes
after each chapter, and there is an extensive bibliography.
The necessary background from probability theory is given in Chapter 1.
Then the main objects of study, the Dirichlet series and polynomials,
are introduced in Chapter 2. Basic properties of Riemann's
zeta-function and Dirichlet's L-functions are surveyed; it may
be a bit misleading for a non-specialist that the zero-free region
for the zeta-function (in Theorem 8.6) is not given in the sharpest
known form. The actual topic of the book begins in Chapter 3 with
a discussion of limit theorems for the modulus of the zeta-function
in the half-plane Re[s] > 1/2 (in the first place near the
critical line). The theory of moments of the zeta-function (going back
to Ramachandra and Heath-Brown) plays an important role here. In the
next chapter, the value distribution of the zeta-function is studied
more generally. Chapter 5 deals with limit theorems of the
zeta-function in the space of analytic functions, as a preparation
for a proof of Voronin's universality theorem in Chapter 6. Then, in
Chapter 7, a limit theorem for the zeta-function in the space of
continuous functions is established under the assumption of the
Riemann hypothesis. In the last two chapters, preceding limit
theorems pertaining to the zeta-function are generalized to wider
classes of functions, first to Dirichlet L-functions, and finally
to Dirichlet series with multiplicative coefficients."
K. Bitar, "Path
integrals and Voronin's theorem on the universality of the Riemann zeta
function", Nucl. Phys. Proc. Suppl. 26 (1992) 656
[abstract:] "We explore a new approach to the path integral for a latticized
quantum theory. This talk is based on work with N. Khuri and H. Ren."
The work in question is this:
K.M. Bitar, N.N. Khuri and H.C. Ren, "Path integrals and Voronin's
theorem on the universality and the Riemann zeta function", Annals
of Physics 67 (1991) 172-196
K. Bitar, N.N. Khuri and H.C. Ren, "Path integrals and discrete
sums", Physical Review Letters 67 (1991) 781-784
K. Bitar, "A study of the Riemann zeta function" (online notes)
"The well known Riemann zeta function has many interesting
properties and has proven to be useful in many applications in
physics. In our attempt to use one such property, one described by
Voronin's theorems, a new characteristic distribution was discovered
numerically and then calculated analytically. This distribution allows
the use of the Riemann zeta function as a generator of pseudo random
. . .
Knowing these distributions allows the use of the zeta function in
evaluating the path integrals for quantum mechanical systems. We have
tested this on simple systems such as the anharmonic oscillator with
good results. Further more since the zeta function is an analytic
function with known properties its use in these applications may lead
to a definition of the path integral in the continuum."
"Riemann zeta function is a fractal" (preprint 06/94)
"[We] infer three corollaries from Voronin's theorem [on the
'universality' of the Riemann zeta function]. The first is interesting,
the second is a strange and amusing consequence, and the third is ludicrous
and shocking (but a consequence nevertheless)."
"Corollary 1 ("interesting") Riemann zeta function is a
Woon's innovation here is to devise analytic function f
based on the zeta function itself (involving translations and rescalings),
in order to show that zeta replicates its own behaviour infinitely often at all scales.
"Since ao can be arbitrarily chosen, there are
self-similarities at all scales. Therefore, Riemann zeta function
is a fractal."
He goes on to show that you can choose f to be based on
the zeta function not only via translation and dilation, but also
invovling rotation and reflection. The result is that we have
self-similarities between discs at different scales and orientations."
"Riemann zeta function is fractal in the sense that the Mandelbrot
set is fractal (self-similarities between a region bounded by a
closed loop C and other regions bounded by closed Cm'
of the same shape at smaller scales and/or orientations). The fractal
property of zeta is not "infinitely recursive" as in Koch snowflake.
Such infinite recursions in a function will render the function
non-differentiable, whereas zeta is infinitely differentiable. So,
the manifold of zeta function is not of fractal dimension."
"All Dirichlet L-functions are also fractal. This follows
from the remark following Voronin's theorem in Voronin's paper."
"Corollary 2 ("strange and amusing") Riemann zeta function is a
'library' of all possible smooth continuous line drawings in a plane."
Imagine all the ways an analytic function can map a line segment,
say the vertical diameter of |z| < 1/4, into the complex
plane. Woon points out that every imaginable kind of
shape (an outline Mickey Mouse is used as an example) can be drawn in this way.
It then follows from the Universality Theorem that the zeta function's behaviour
on segments of Re[s] = 3/4 (or any other line between 1/2
and 1) can replicate any such shape.
"Corollary 3 ("ludicrous and shocking") Riemann zeta function is a
concrete "representation" of the giant book of theorems referred to
by Paul Halmos."
Woon explains that you can represent arbitrarily long Morse code
messages as oscillating curves representing 'signals'. Every
possible one of these messages is reproducible to within a workable accuracy
by the zeta function. So the entire Encylopedia Britannica could be
deduced as a Morse Code transmission encoded as a wave which was the
image of a vertical segment of length 1/2 on Re[s] = 3/4 under
the Riemann zeta function.
"So... the entire human knowledge are already encoded in zeta function."
"Hence, Riemann zeta function is probably one of the most
remarkable functions because it is a concrete "representation"
(in group theory sense) of "the God's giant book of theorems" that
Paul Halmost spoke of - all possible theorems and texts are already
encoded in some form in Riemann zeta function, and repeated infinitely
many times. Although a white noise function and an infinite sequence
of random digits are also concrete "representations", Riemann zeta
function is not white noise or random but well-defined.
Alternatively, from the point of view of information theory, even
though Riemann zeta function is well-defined, its mappings in the
right half of the critical strip are random enough to encode
arbitrary large amount of information - the "entropy" of its mapping
This article is also encoded somewhere in Riemann zeta function as it
is being written!"
S.C. Woon, "Fractals
of the Julia and Mandelbrot sets of the Riemann zeta function" (preprint 12/98)
"Computations of the Julia and Mandelbrot sets of the Riemann zeta
function and observations of their properties are made. In the
appendix section, a corollary of Voronin's theorem is derived and a
scale-invariant equation for the bounds in Goldbach conjecture is
R. Garunkstis, "The effective universality
theorem for the Riemann zeta-function" (preprint 02/03)
[abstract:] "It is known that Voronin's universality theorem for the
Riemann zeta-function is ineffective. For some partial cases we obtain
the effective version of this theorem."
Ineffective in this case means that upper bounds cannot be derived
for t as a function of .
R. Garunkstis, "On the Voronin's universality
theorem for the Riemann zeta-function"
[abstract:] "We present a slight modification of the Voronin's
proof for the universality of the Riemann zeta-function. The difference
(and simplification) is that we do not use the rearrangement of terms
in functional series."
[The following is based on AMS notes found here:]
Corresponding results for Dedekind zeta-functions, for all
Dirichlet L-functions and other Dirichlet series, for Hurwitz
and Lerch zeta-functions, for certain Euler products and other
related functions were obtained by Voronin, Reich, Gonek, Laurincikas
and co-workers and Bagchi.
Some functions were also found where the requirement that f be
zero-free can be dropped; examples are the derivative of the Riemann
zeta-function, log, Hurwitz and Lerch zeta-functions and others; see
Gonek '79, Bagchi '81-'82, Gavrilov and Kanatnikov '82,
Voronin '77, and Laurincikas and Garunkstis '97-'99. For
treatments of these topics in textbooks refer to the 1992 and 1996 textbooks
A. Reich, "Universelle werteverteilung von Eulerprodukten", Nachr.
Akad. Wiss. Göttingen Math.-Phys. Kl. II (1977) 1-17
A. Reich, "Werteverteilung von zetafunktionen", Arch. Math.
34 (1980) 440-451
A. Reich, "Zur Universalität und Hypertranszendenz der
Dedekindschen zetafunktion", Abh. Braunschweig. Wiss. Ges. 33
S.M. Gonek, "Analytic properties of zeta and L-functions",
Thesis, Univ. of Michigan, Ann Arbor, 1979.
A. Laurincikas, "The universality theorem" (Russian), Litovsk.
Mat. Sb. 23 no. 3 (1983) 53-62. English transl. in:
Lithuanian Math. J. 23 (1983) 283-289
A. Laurincikas, "The universality theorem. II" (Russian), Litovsk.
Mat. Sb. 24 no. 2 (1984) 113-121. English transl. in:
Lithuanian Math. J. 24 (1984) 143-149
A. Laurincikas, "On the universality of the Riemann zeta-function"
(Russian), Liet. Mat. Rink. 35 (1995) 502-507. English
transl. in: Lithuanian Math. J. 35 (1995) 399-402
A. Laurincikas, "The universality of the Lerch zeta-function"
(Russian), Liet. Mat. Rink. 37 (1997) 367-375. English
transl. in: Lithuanian Math. J. 37 (1997), 275-280
A. Laurincikas, "On the Lerch zeta function with rational
parameters" (Russian), Liet. Mat. Rink. 38 (1998),
113-124. English transl. in: Lithuanian Math. J. 38 (1998)
A. Laurincikas, "On the Matsumoto zeta-function", Acta Arith.
84 (1998) 1-16
A. Laurincikas and K. Matsumoto, "The universality of zeta-functions attached to
certain cusp forms" (preprint)
A. Laurincikas, K. Matsumoto and J. Steuding,
universality of L-functions associated with new forms",
Izvestiya: Mathematics 67 no. 1 (2003) 77
B. Bagchi, "The statistical behaviour and universality properties
of the Riemann zeta function and other allied Dirichlet series",
Thesis, Indian Statistical Institute, Calcutta, 1981.
B. Bagchi, "A joint universality theorem for Dirichlet
L-functions, Math. Z. 181 (1982), 319-334.
V.I. Gavrilov and A.N. Kanatnikov, "An example of a universal
holomorphic function" (Russian), Dokl. Akad. Nauk SSSR
65 (1982), 274-276. English transl. in: Soviet Math.
Dokl. 26 (1982), 52-54.