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ЧЕБЫШЕВСКИЙ СБОРНИК Том 12 Выпуск 2 (2011)
Труды VIII Международной конференции Алгебра и теория чисел: современные проблемы и приложения, посвященной 190-летию Пафнутия Львовича Чебышева и 120-летию Ивана Матвеевича Виноградова
UNIVERSALITY THEOREMS FOR COMPOSITE FUNCTIONS OF ZETA-FUNCTIONS
1 Introduction
After a remarkable Voronin’s work [11], it is known that zeta-functions from a wide class are universal in the sense that their shifts uniformly on compact sets approximate any analytic function, S, M, Voronin in [11] obtained the universality of the Riemann zeta-funetion ((s), s = a + it, We will state the Voronin theorem in a more general modern form. Let D = {s G C : 2 < a < 1}. Denote by K
the class of compact subsets K C D with connected complement, and, for K G K, by H0(K) the class of continuous non-vanishing functions on K which are analytic in the interior of K. Moreover, we will use the notation meas{A} for the Lebesgue measure of a measurable set A C R, Then the Voronin theorem is the following statement.
Suppose that K G K and f (s) G H0(K). Then, for every e > 0,
Proof of the Voronin theorem in the above form can be found, for example, in
The universality property of ((s) has a series of theoretical and practical applications, The Voronin theorem is a significant impact to the approximation theory
* Partially supported by grant No. MP-94 from the Research Council of Lithuania
ANTANAS LAURINCIKAS*
Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania e-mail: [email protected]
of analytic functions, and can be compared to the famous Mergelvan theorem on approximation of analytic functions by polynomials [8], see also [13]. From that theorem, the functional independence of Z(s) follows [12], [4].
For j = 0,..., N, let Fj : Cn ^ C be a continuous function, and identically for
s
N
Y sj Fj (Z (s),Z '(s),...,Z <"-1'(s)) = 0.
j=0
Then Fj(...) = 0 for j = 0,..., N.
Z(s)
example, it is known [1] that the Riemann hypothesis that all non-trivial zeros of Z(s) lie on the critical line a = 2 is equivalent to the statement:
e>0
liminf ^meas { r G [0,T] : sup |Z(s + ir) — Z(s)| < el > 0.
T I seK I
The universality of Z(s) can be applied in the moment problem and other value-distribution problems of the theory of Z(s).
A practical application of the Voronin theorem for estimation of complicated integrals over analytic curves in quantum mechanics is given in [2].
The importance of the universality of the Riemann zeta-function has been observed by a wide circle of mathematicians, and Voronin’s ideas obtained the further fast development. It turned out that some other zeta-functions, among them the Hurwitz and Lerch zeta-functions, zeta-functions attached to certain cusp forms, some rather wide classes of Dirichlet series are universal in the Voronin sense. There exists a Linnik-Ibragimov conjecture [9] that all functions in some half-plane given by Dirichlet series, analytically continuable to the left of the absolute convergence plane and satisfying some natural growth conditions are universal. Many results support that conjecture. On the other hand, there exists non-universal Dirichlet series. For example, let
m
m
if
otherwise.
mo > 1, k > 1,
Then, for a > 1,
m= 1
am
ms
£■
k=1
m
ks
mso
1
Clearly, the function Z(s) is analytic in the whole complex plane, except for simple poles s = 2nir, r G Z, however, is non-universal.
For K G K, denote by H(K) the class of continuous functions on K which are analytic in the interior of K, and, for s G D, define log Z(s) from the value log Z(2) G R by continuous variation along straight lines conneeting the points 2, 2 + it and
a
m
1
1
a + it provided that the path does not pass possible zeros or the pole of Z(s). If it does, then we take
logZ(a + it) = lim logZ(a + i(t + e)).
£^+0
Then it is known [4] that log Z(s) is also universal, the shifts log Z(s + ir), r G R,
K H(K)
Z(s)
The present paper is devoted to the univesality of some classes of composite F(Z(s)) Z(s)
2 Case of the Riemann zeta-funetion
The Riemann zeta-funetion Z(s) is defined, for a > 1, bv the series
1
Z(s) =
s
m
m=1
and can be analytically continued to the whole complex plane, except for a simple s = 1 1
It is not difficult to see that the function ez(s) is also universal.
Theorem 1. Suppose that K G K and f (s) G H0(K). Then, for every e > 0,
lim inf — meas { r G [0, T] : sup |ez(s+iT) — f (s)| < el > 0.
T^~ T [ s€K J
PROOF. Since f (s) G H0(K), we can define a branch log f (s) of the logarithm of f (s) such that logf (s) G H0(K), Then, bv the Voronin theorem stated in the
e>0
liminf ^meas {r G [0,T]: sup |Z(s + ir) — log f (s)| < > 0, (1)
T^~ T [ s€K 3Mk J
where MK = max ( max |f (s)|, 1 I, Using the inequalitv |ez — 1| < |z|e|z|, z G C, we y s€K I
find that, for r G R satisfying the inequality
e
suP |Z(s + ir) — log f (s) 1 < iTjr-j-, s€K 3Mk
sup |ez(s+iT) — f (s)| = sup |ez(s+iT) — elog f(s)|
s€K s€K
< sup |f (s)||Z(s + ir) — logf (s)|e|z(s+iT)-logf(s)| < e.
s€K
This together with (1) proves the theorem, □
In [6], some classes of universal composite functions F(Z(s)) were described. We comment and state results of [6] in a more convenient form.
Denote bv H(D) the space of analytic functions on D equipped with the topology of uniform convergence on compacta.
Theorem 2. Suppose that a function F : H(D) ^ H(D) is such, that, for every polynomial p = p(s) and every K G K, there exists an element g G F-1{p} C H(D), g(s) = 0 on K. Moreover, for every K G K, let there exist constants c > 0 ^ > 0, K1 G K
sup |F(gi(s)) — F(g2(s))| < c sup |gi(s) — g2(s)|^
s€K sGKi
g1, g2 G H(D) K G K f(s) G H(K) e > 0
liminf ^meas { r G [0, T] : sup |F(Z(s + ir)) — f (s)| < e 1 > 0.
T^~ T [ s€K J
Theorem 2 is a direct corollary of the Voronin theorem, and of the Mergelvan
theorem which asserts that, under the hypotheses of the theorem, there exists a
p(s)
e
sup |f (s) — P(s)| <2. (2)
s€K 2
The theorem was inspired by the universality of the derivative Z;(s) which, in view of the integral Cauchy formula, satisfies the hypotheses of Theorem 2 with ft = 1, Clearly, the theorem remains true for Z(n)(s), n G N.
The next theorems are based on using the probabilistic approach for the proof of universality for zeta-functions.
Theorem 3. Suppose that F : H(D) ^ H(D) is a continuous function such that, for every open set G G H(D), the set (F-1G) n {g G H(D) : g-1(s) G
H(D) or g(s) = 0} is non-empty. Let K G K and f (s) G H(K). Then the assertion of Theorem 2 is true.
Theorem 3 is theoretical, it is difficult to check its hypotheses. We will give a sketch of the proof.
Denote by B(H(D)) the class of Borel sets of the space H(D), It is known, see, for example, [5], that the probability measure
T^meas {r G [0,T]: Z(s + ir) G A} , A G B(H(D)),
converges weakly to some probability measure P^ as T ^ to. Moreover, the support of the measure P^ '1S the set {g G H(D) : g-1(s) G H(D) or g(s) = 0}, The
F
;1meas {r G [0, T] : F(Z(s + ir)) G A} , A G B(H(D)),
to the measure PzF 1 as T ^ to, where PzF 1 is understood as
PzF-1 (A) = Pc(F-1A), A gB(H(D)).
From the hypothesis of the theorem on the pre-images of open sets, it is deduced that the support of the measure PzF-1 is the whole of H(D), These two facts together with properties of the weak convergence and support lead to the inequality
liminf ^meas < r G [0, T] : sup |F(Z(s + ir)) — p(s)| < - 1 > PzF-1(G) > 0,
T[ seK 2 J
where
-
G = {g G H(D) : sup |g(s) — p(s)| <-},
seK 2
p(s)
The hypothesis of Theorem 3 that, for every open set G G H(D), the set (F-1G)n {g G H(D) : g-1(s) G H(D) or g(s) = 0} is non-empty can be replaced by a
stronger but simpler one.
Theorem 4. Suppose that F : H(D) ^ H(D) is a continuous function
such that, for every polynomial p(s), the set (F-1{p}) n {g G H(D) : g-1(s) G
H(D) or g(s) = 0} K G K f(s) G H(K)
of Theorem 2 is true.
H(D)
K G K g
an open set G and K G K, then, by the Mergelvan theorem, for every - > 0, there p = p ( s ) -
p(s) G
Theorem 3,
Theorem 4 can be applied to obtain the universality of the function
C1Z;(s) +---+ CrZ(r)(s), C1, ..., cr G C \ {0}.
Let, for brevity,
Ho(D) = {g G H(D) : g-1(s) G H(D) or g(s) = 0}.
Then it is not difficult to see that, for arbitrary continuous function F : H(D) ^ H(D), the support of the measure PzF-1 is the closure of the set F(H0(D)), Really, let g be an arbitrary element of F(H0(D)^^d G be any open neighbourhood of g. Then there exists an element g1 G H0(D) such that F(g1) = g. This means that F-1G is an open neighbourhood of g^. Sinee H0(D) is the support of of the measure Pz, we have that Pz(F-1G) > 0, Hence,
PzF-1(G) = Pz(F-1G) > 0.
Moreover,
Pz F-1(F(H0(D))) = Pz (H0(D)) = 1.
H(D) H(D)
space), two latter relations show that the support of PzF-1 is a closure of the set F (H0(D)).
The above remarks give the following statement.
Theorem 5. Suppose that F : H(D) ^ H(D) is a continuous function. Let K C D be an arbitrary compact subset, and f (s) G F(H0(D)). Then the assertion of Theorem 2 is frwe.
Now we present a special class of functions F related to Theorem 5, For F : H(D) ^ H(D) and a1,..., ar G C, define
Hai...ar(D) = {g G H(D) : (g(s) — a,)-1 G H(D), j = 1,..., r} U {F(0)}.
Theorem 6. Suppose that F : H(D) ^ H(D) is a continuous function such that F(H0(D)) D Hai,...,ar(D). For r = 1, lei K C K, and f(s) 6e a continuous = a1 K K r > 2 K C D
arbitrary compact set, and f (s) G Hai,...,ar(D). Then the assertion of Theorem 2is true.
For the proof of Theorem 6, we apply the following arguments. In the case r = 1,
f(s) = a1 K
p(s)
sup|f(s) — eP(s) — a1| <-. (3)
seK
2
The function f1(s) = ep(s) + a1 is analytic on D and is not equal to a1; i, e,, f1(s) G
Hai (D), Since F(H0(D)) D Hai (D), by a remark before Theorem 5, the function
f1(s) is a an element of the support of the measure PzF-1, Therefore, setting
G1 = {g G H(D) : sup |g(s) — f1(s)| < -},
seK 2
we have that PzF-1(G1) > 0, Hence, as in the case of Theorem 3,
1 f £ liminf —meas < r G [0, T] : sup |F(Z(s + ir)) — f1(s)| < -
T^~ ^1 seK 2
> PzF-1 (G1) > 0.
Combining this with (3) gives the theorem in the case r =1, r>2
A simple form of the set Hai,...,ar (D) allows to obtain the universality of F(Z(s)) for some elementary functions F. For example, if F(g) = 2g—3, we find by Theorem 6 with r1 = 1 and a1 = —3, that every continuous = —3 on K function which is analytic in the interior of K can be approximated uniformly on K by shifts 2Z (s + ir) — 3. If r = 2 and a1 = —1, a0 = 1, we have the universality of the function sin(Z(s))
eig(s) _ e-ig(s)
L-1T— = f w
in functions g G H0(D),
f(s) D s0 G D
f (z)dz = 0
for s G D. Then, by Theorem 5, the function f (s) can be approximated by shifts Z (s + ir )Z/(s + ir)-
s
3 Case of the Hurwitz zeta-funetion
Let a, 0 < a < 1, be a fixed parameter. The Hurwitz zeta-function Z(s,a) is defined, for a > 1, by the series
Z (s,a) = ^
m=0
(m + a)s ’
and has analytic continuation to the whole complex plane, except for a simple pole at the point s = 1 with residue 1. Clearly, we have that Z(s, 1) = Z(s) and
Z (s, 1) = (2' — 1)Z (s).
Analytical properties of the function Z(s, a) are closely related to the arithmetical nature of the parameter a. This is also reflected in the universality of Z(s,a). We have the following universality theorem.
Theorem 7. Suppose that a is a transcendental or rational =1, \ number. Let
K C K and f (s) G H(K). Then, for every - > 0
liminf — meas | r G [0, T] : sup |Z(s + ir, a) — f (s)| < £ 1 > 0.
T^~ T I seK )
Theorem 7 was obtained independently by S, M, Gonek [3] and B, Bagehi [1], see also [9] and [7]. In the eases a = 1 and a = 1, the funetion Z(s, a) is also universal, however, in these eases, f (s) G H0(K). The ease of algebraic irrational a remains an open problem.
If a is transcendental, then the set {log(m + a) : m G N U {0}} is linearly independent over the field of rational numbers, and for the proof of universality of Z(s, a) H(D)
In the ease of rational a = q, (a, q) = 1, the universality of Z(s,a) is deduced from the joint universality of Dirichlet L-funetions L(s,x) and the equality
Z (s -) = q-s ^
x
where x runs over all Diriehlet characters modulo q,
F(Z(s, a)) Z(s)
a
our method uses the probabilistic approach.
Obviously, the function ez(s,a) is universal with all transcendental or rational a, and the approximated function f (s) G H0(D), The proof of this fact completely coincides with that of Theorem 1,
Z(s, a)
Theorem 8. Suppose that a is a transcendental or rational = 1, 2 number, and that a function F : H(D) ^ H(D) is such that, for every polynomial p = p(s), there exists an element g G F-1{p} C H(D). Moreover, for every K gK, let there exist constants c > 0 ^ > 0, and a set K1 G K such that
sup |F(g1(s)) — F(g2(s))| < c sup |g1(s) — g2(s)|^
seK seKi
for all g1 ,g2 G H(D). Let K G K and f (s) G H(K). Then, for every - > 0,
liminf —meas < r G [0, T] : sup |F(Z(s + ir, a)) — f (s)| < £ 1 > 0.
T^~ T [ seK J
Theorem 8 is a direct consequence of Theorem 7 and the Mergelvan theorem. Theorem 8 together with the integral Cauchy formula gives the universality for Z(n)(s, a), n G N.
Theorem 9. Suppose that a is a transcendental number, and that F : H(D) ^ H(D) is a continuous function such, that, for every open set G G H(D), the set F-1G is non-empty. Let K G K and f (s) G H(K). Then the assertion of Theorem 8is true.
a
probability measure
—meas {r G [0,T]: Z(s + ir, a) G A} , A G B(H(D)),
converges weakly to the explicitly given probability measure Pz as T ^ to, and the support of PZ is the whole of H(D), From this and the hypotheses of Theorem 9, we deduce that the probability measure
—meas {r G [0,T]: F(Z(s + ir, a)) G A} , A G B(H(D)),
converges weakly to PzF-1, and the support of PzF-1 is the whole of H(D), These two facts, as in the case of Theorem 3, imply the assertion of Theorem 9,
F-1G
pre-image of a polynomial,
a
F : H(D) ^ H(D) is a continuous function such that, for every polynomial
P = p(s), the s et F-1{p} is non-empty. Let K G Km d f (s) G H (K). Then the assertion of Theorem 8 is true.
Theorem 10 implies the universality of the linear combination c1Z/(s, a) + ■ ■ ■ + crZ(r)(s, a), c1,..., cr G C \ {0}.
Z(s, a)
Theorem 11. Suppose that a is a transcendental number, and that F : H(D) ^ H(D) K C D
f(s) G F(H(D))
For F : H(D) ^ H(D^d a1,..., ar G C, define
Hi,.,ar (D) = {g G H(D) : (g(s) — )-1 G H(D), j = 1,..., r}.
Theorem 12. Suppose that a is a transcendental number, and that F : H(D) ^ H(D) is a continuous function such that F(H(D)) D Hai ...ar(D). For r = 1, fei K C K, and let f (s) 6e a continuous = a1 function on K which is analytic in the interior of K. For r > 2, fei K C D be an arbitrary compact subset, and f (s) G Hai)...>ar(D). Then the assertion of Theorem 8 is true.
A proof of Theorem 12 runs in a similar way as that of Theorem 6,
r = 2 -1 = — 1 -2 = 1
of the function sinh(Z(s,a)).
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Received 12.10.2011
