Publ. Mat. 54 (2010), 209–219

EFFECTIVE UNIFORM APPROXIMATION BY THE
RIEMANN ZETA-FUNCTION
Ramūnas Garunkštis, Antanas Laurinčikas,
Kohji Matsumoto, Jörn Steuding, and Rasa Steuding
Dedicated to Prof. Dr. Wolfgang Schwarz on the occasion of his 75th birthday

Abstract
We apply an effective multidimensional Ω-result of Voronin inorder to obtain effective universality-type theorems for the Riemann zeta-function. We further use this approach to study approximation properties of linear combinations of derivatives of thezeta-function.

1. History of Voronin’s work and related results
In 1975, Voronin [17] proved a remarkable universality theoremfor ζ(s) which states, roughly speaking, that any non-vanishing analytic
function can be approximated uniformly by certain purely imaginary
shifts of the zeta-function in the critical strip:
Theorem 1. Let 0 < r < 14 and suppose that g(s) is a non-vanishingcontinuous function on the disk |s| ≤ r which is analytic in the interior.
Then, for any ǫ > 0,




1
3
lim inf meas τ ∈ [0, T ] : max ζ s + + iτ − g(s) < ǫ > 0.
T →∞ T|s|≤r
4
It is natural to ask for an effective version of this theorem. Given an
admissible function g and some positive ǫ, is it possible to give an upperbound for the first τ > 0 such that
3max ζ s + + iτ − g(s) < ǫ,4|s|≤r
or is it possible to give a lower bound for the positive lower density
in Theorem 1? This is indeed a very difficult problem and neither the2000 Mathematics Subject Classification. 11M06, 30E10.
Key words. Riemann zeta-function, universality, uniform approximation.

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known proofs of Voronin’s universality Theorem 1 nor its generalizations
or extensions are effective. However, there are some remarkable attempts
towards a solution of this problem.
First of all, Good [4] combined Voronin’s universality Theorem 1 withthe work of Montgomery [10] on extreme values of the zeta-function,
which enabled him to complement Voronin’s qualitative picture with
Montgomery’s quantitative estimates. His result is rather complicatedand a more satisfying statement was recently given by Garunkštis [1],building on Good’s ideas. Garunkštis proved that if f (s) is analytic in|s| ≤ 0.05 with max|s|≤0.05 |f (s)| ≤ 1, then for any 0 < ǫ < 12 there exists
a real number τ with(1)0 ≤ τ ≤ exp exp 10ǫ−13such that3
max log ζ s + + iτ − f (s) < ǫ,4|s|≤0.0001

and further





3
1
lim inf meas τ ∈ [0, T ] : max log ζ s + + iτ − f (s) < ǫ
T →∞ T4|s|≤0.0001
≥ exp −ǫ−13 .

The original theorem is too complicated to be given here. Laurinčikas [6],[7] found another approach which gives conditional effective results subject to certain assumptions on the speed of convergence of the related
limit distribution. However, the rate of convergence of weakly convergent probability measures related to the space of analytic functions is
not understood very well. Steuding [12], [13] obtained non-trivial upperbounds for the upper density of universality.
The purpose of the present paper is to give another approach toobtain effective statements on the approximation by ζ(s). The origin of Voronin’s universality Theorem 1 seems to be a density theorem
from [15]:Theorem 2. For any fixed numbers s1 , . . . , sn with 21 < Re sk < 1 for
1 ≤ k ≤ n and sk 6= sℓ for k 6= ℓ, the set{(ζ(s1 + it), . . . , ζ(sn + it)) : t ∈ R}
is dense in C . Moreover, for any fixed number s in 21 < σ < 1,n{(ζ(s + iτ ), ζ ′ (s + iτ ), . . . , ζ (n−1) (s + iτ )) : τ ∈ R}
is dense in Cn .

Effective Uniform Approximation

211

This theorem is a significant improvement of classic results due toHarald Bohr and his collaborators from the first half of the twentiethcentury. Of course, Theorem 2 is a simple consequence of Theorem 1,
but Theorem 2 was proved earlier than Theorem 1. In [16], Voronin extended his approach in order to prove the joint functional independenceof Dirichlet L-functions to pairwise inequivalent characters. Actually,
this paper may be regarded as forerunner of his universality Theorem 1.
As indicated by Laurinčikas [8], the missing link for obtaining (joint) universality was the absence of a rearrangement theorem for Hilbert spaces
which was proved by Pečerskiı̆ [11] on Voronin’s request. Garunkštis [2]found a slight modification of Voronin’s proof which does not make useof the rearrangement theorem.
Voronin’s universality Theorem 1 can be seen as an infinite dimensional analogue of the second part of Theorem 2: the truncated Taylor
series of the target function g(s) can be approximated by the truncated
Taylor series of a certain shift of zeta. This idea was cultivated by Matsumoto [9] and in the next section we shall combine the idea from [9]with the following Theorem 3 in order to obtain our effective results inthe present paper.
In 1988, Voronin [18] proved the following, remarkable effective version of the second part of Theorem 2:
Theorem 3. Let σ ∈ ( 12 , 1), ǫ ∈ (0, 1) and b = (b0 , b1 , . . . , bn−1 ) ∈ Cn
with |b0 | > ǫ be fixed. Then the system of inequalities|ζ (k) (σ + it) − bk | < ǫ

for

0≤k<n

has a solution t ∈ [T, 2T ] provided that


8
8
(2)
T ≥ C(n, σ) exp exp 5A(n, b, ǫ) 1−σ + σ−1/2 ,where C(n, σ) is a positive, effectively computable constant, depending
only on n and σ, and
n2

kbk
A(n, b, ǫ) = |log|b0 || +ǫPwith kbk := 0≤k<n |bk |.

This theorem is also given in the monograph of Karatsuba and Voronin [5]; unfortunately, Voronin called his results Ω-theorems which does
not really match the statement. The proof of Theorem 3 is pretty muchthe same as the one for Theorem 2 with two exceptions which allow theeffectiveness of the statement: firstly, the use of systems of linear equations in as many unknowns as equations (solvable by Cramer’s rule with

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a solution in an explicit range), which replaces the ineffective rearrangement theorem; secondly, Fourier analysis as substitute for Kronecker’s
diophantine approximation theorem.
Because of Voronin’s untimely death in 1997 we will never knowwhether this reasoning was Voronin’s inspiration for [18]; we believethat Voronin was aware of these applications of his late results from [18]to the effectivity problem in universality and, if he was, we may speculatethat he was not completely satisfied.

2. Effective uniform approximation
Let K = {s ∈ C : |s − s0 | ≤ r}, where r is a positive real number ands0 = σ0 + it0 with σ0 ∈ ( 12 , 1). Notice that K is not necessarily includedin the strip 12 < Re s < 1. Assume that we are given an analytic targetfunction g : K → C. Our main tool is the Taylor series expansion of g:g(s) =∞X
g (k) (s0 )

k=0

k!

(s − s0 )k ,valid for all s ∈ K. By Cauchy’s formula,
Z
k!
g(s)
(k)
g (s0 ) =ds,
2πi |s−s0 |=r (s − s0 )k+1
where the integral is taken over the circle |s−s0 | = r in counterclockwise
direction. Hence,|g (k) (s0 )| ≤ k!M r−kwhere M := max|s−s0 |=r |g(s)|. We fix a number δ0 ∈ (0, 1). Theng (k) (s0 )(s − s0 )k ≤ M δ0kk!

for|s − s0 | ≤ δ0 r.

In order to apply Theorem 3 we suppose that g(s0 ) 6= 0. Then we canchoose a sufficiently small ǫ satisfying 0 < ǫ < |g(s0 )|. There exists a
positive integer n = n(δ0 , ǫ, M ) such that(3)

Σ1 := g(s) −X g (k) (s0 )ǫ(s − s0 )k <
k!
3

0≤k<n

for

|s − s0 | ≤ δ0 r.

Effective Uniform Approximation

213

Let 0 < δ ≤ δ0 . Then, needless to say, (3) is valid for |s − s0 | ≤ δr. Now
we apply Theorem 3 with bk = g (k) (s0 ) and 3ǫ in place of ǫ. Then, forany sufficiently large T , there exists t1 ∈ [T, 2T ] such that
ǫfor 0 ≤ k < n;(4)
|ζ (k) (σ0 + it1 ) − g (k) (s0 )| <3
the condition with respect to T is


8
8
+
(5)
T ≥ C(n, σ0 ) exp exp 5A(n, g, ǫ/3) 1−σ0 σ0 −1/2 ,where g is the vector composed from the coefficients in the Taylor series
expansion for g. In particular, the first component of g is g(s0 ). Thus,
Σ2 :=X ζ (k) (σ0 + it1 )X g (k) (s0 )(s − s0 )k −(s − s0 )k
k!
k!

0≤k<n

(6)
<

0≤k<n

ǫ
ǫ X (δr)k< exp(δr)
3k!
3

for |s − s0 | ≤ δr.

0≤k<n

Now let τ = t1 − t0 ; then σ0 + it1 = s0 + iτ . We use the Taylor expansionof ζ(s) on the disc K + iτ . For this purpose it is necessary that the poleof ζ(s) is not included in this disc. This is achieved if(7)

T > r,

which is satisfied if T is sufficiently large. Under this assumption, wehave∞Xζ (k) (s0 + iτ )(s − s0 )kζ(s + iτ ) =k!
k=0

for s ∈ K. Put
M (τ ) = max |ζ(s + iτ )|.
|s−s0 |=rThen, again by Cauchy’s formula,ζ (k) (s0 + iτ )(s − s0 )k ≤ M (τ )δ kk!

for|s − s0 | ≤ δr.

Hence,
Σ3 := ζ(s + iτ ) −X ζ (k) (s0 + iτ )(s − s0 )k
k!

0≤k<n

(8)
=

∞
X
ζ (k) (s0 + iτ )

k=n

k!

(s − s0 )k ≤ M (τ )δn1−δfor |s − s0 | ≤ δr.

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Putting (3)–(8) together, we find|ζ(s + iτ ) − g(s)| ≤ Σ1 + Σ2 + Σ3 <ǫδn
ǫ
+ exp(δr) + M (τ )
.
3 3
1−δ

Now choose δ > 0 such thatδnǫ= (2 − exp(δr));1−δ
3this is possible since the left hand-side tends to zero as δ → 0, while
the right hand-side tends to 3ǫ > 0, resp. when δ → 1 the left-hand side
tends to infinity, but the right-hand side remains bounded. We thus have
proved(9)

M (τ )

Theorem 4. Let σ0 ∈ ( 21 , 1), g : K → C continuous, g(s0 ) 6= 0 andanalytic for |s − s0 | < r. Then, for any ǫ ∈ (0, |g(s0 )|) there exist real
numbers τ ∈ [T, 2T ] and δ = δ(ǫ, g, τ ) > 0 defined by (9) such thatmax

|s−s0 |≤δr

|ζ(s + iτ ) − g(s)| < ǫ,

where T = T (g, ǫ, σ0 ) has to satisfy (5) and (7), and δ is effectively
computable.
Corollary 5. The assertion of Theorem 4 holds even if g(s0 ) = 0, if we
replace (5) by (10) below.
Proof: Suppose g(s0 ) = 0. Set gǫ (s) = g(s) + ǫ/2, so that gǫ (s0 ) 6= 0.
Since ǫ/3 ∈ (0, |gǫ (s0 )|), we can apply Theorem 4 to find τ ∈ [T, 2T ]and δ > 0 for whichmax|s−s0 |≤δr

|ζ(s + iτ ) − gǫ (s)| < ǫ/3holds, and T has to satisfy
(10)



8
8
+
T ≥ C(n, σ) exp exp 5A(n, gǫ , ǫ/3) 1−σ0 σ0 −1/2 ,where gǫ = (ǫ/2, g (1)(s0 ), . . . , g (n−1) (s0 )). This clearly implies the assertion of the corollary.
Remark. By the same idea it is possible to show that a slightly modified
e = (ǫ/2, b1, . . . , bn−1 ).
version of Theorem 3 holds even if b0 = 0. Let b
Applying Theorem 3 we find that the system of inequalities |ζ (k) (σ +it) − ebk | < ǫ/3 for 0 ≤ k < n (where eb0 = ǫ/2) has a solution t ∈ [T, 2T ],provided that88e ǫ/3) 1−σ0 + σ0 −1/2 .
(11)
T ≥ C(n, σ) exp exp 5A(n, b,

Effective Uniform Approximation

215

This t is a solution of the system |ζ (k) (σ + it) − bk | < ǫ for 0 ≤ k < n,as desired.
It is remarkable that there is no objection on g to be non-vanishingon K as in the universality Theorem 1. Indeed, the statement of Theorem 1 contradicts the Riemann hypothesis if g has a zero in K (since any
such zero would generate via Rouché’s theorem many zeros of ζ(s + iτ );see [14, §8.1]). However, it seems that there is an inner mechanism which
prevents to obtain an extraordinarily good approximation of the targetfunction. We observe that a small ǫ leads to a big T in (5) and further
that the bigger T , the smaller we have to choose δ via (9). This follows
from the fact that the zeta-function is unbounded inside the critical strip
and so the quantity M (τ ) is increasing with T to infinity. We illustrate
this by an example. Let s0 be a complex number with 12 < σ0 < 1 andb > 0. We put g(s) = b+s−s0; then g is an entire function with a simple
zero in s0 − b. So we may choose n = 2 in the above proof of Theorem 4(and there is no error term coming from (3)). We observe that (9) holds
for some sufficiently small δ > 0. Now assume that 0 < b < δr. Notethat δr < log 2 holds because the right-hand side of (9) is positive under
our choice of δ. If we could choose ǫ > 0 such that
(12)

max

|s−s0 |=δr|ζ(s + iτ ) − g(s)| < ǫ <min|s−s0 |=δr|g(s)| = δr − b,then we would obtain a disproof of the Riemann hypothesis (if we additionally suppose that 21 < σ0 − δr and σ0 + δr < 1 in order to have azero off the critical line). Indeed, the zero of g(s) in s = s0 − b lies insidethe disk of radius δr, and so by Rouché’s theorem ζ(s + iτ ) vanishes forsome value of s with |s − s0 | < δr too. The second inequality of (12)can be fulfilled by choosing ǫ sufficiently small, comparing with the valueof δ. However, in order to fulfill the first inequality of (12), Theorem 4
yields only a large value for τ which leads to a big M (τ ) so that δ has
to be chosen rather small. Here we see an indirect link between uniform
approximation and the order of growth of the zeta-function on verticallines.
It should be noticed that Gauthier and Clouâtre [3] obtained a noneffective variant of Theorem 4; they proved that every function holomorphic on a compact set K ⊂ C having connected complement can beapproximated by translates of Taylor polynomials of the zeta-function.
Here K is not necessarily contained in the right half of the critical strip.
Their proof depends on Voronin’s theorem.

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3. Linear combinations
The approach of the previous section also applies to any derivative ζ (j) (s) of the zeta-function in place of ζ(s). Moreover, we can evenallow linear combinations.
Theorem 6. Let λ0 , λ1 , . . . , λJ be any complex numbers, not all equal
to zero, and defineJX
λj ζ (j) (s).
ℓ(s) =
j=0

Suppose that σ0 ∈ ( 12 , 1), g : K → C is continuous, g(s0 ) 6= 0, andanalytic for |s| < r. Then, for any ǫ ∈ (0, |g(s0 )|) there exist real numbers τ ∈ [T, 2T ] and δ = δ(ǫ, g, τ ) > 0 defined by (9) such thatmax

|s−s0 |≤δr|ℓ(s + iτ ) − g(s)| < ǫ

for all T > C = C(g, ǫ, σ0 , λ0 , . . . , λJ ), where δ and C are effectively
computable positive constants.
Proof: The key idea for the proof of this theorem is also to use Theorem 3. Since the case J = 0 was already treated by Theorem 4, we
assume J ≥ 1 in the following proof.
Let τ be real. Using the local power series expansion
ζ(s + iτ ) =

∞
X
1
k=0

k!

ζ (k) (s0 + iτ )(s − s0 )k ,we findℓ(s + iτ ) =

∞
J
X
1 X
k=0

k! j=0λj ζ (k+j) (s0 + iτ )(s − s0 )k .

Hence, different from the previous section, we now have to find τ which
is a solution of the system of inequalities
(13)

J
X

λj ζ (k+j) (s0 + iτ ) − g (k) (s0 ) <

j=0

ǫ
3

for 0 ≤ k < n. For this purpose we solve the system of linear equations
(14)

J
X
j=0

λj xk+j = g (k) (s0 )for 0 ≤ k < nEffective Uniform Approximation217

in the unknowns x0 , x1 , . . . , xn+J−1 . Since this system consists of n equations and n + J variables, it follows that the space of solutions has apositive dimension.
We claim that, amongst those infinitely many solutions, we can finda solution such that x0 6= 0. In fact, if λ0 = 0 this is clear. If λ0 6= 0 we
have
x0 =

1
(g(s0 ) − (λ1 x1 + · · · + λJ xJ )).
λ0

If all λj = 0 (j ≥ 1), then x0 = λ−10 g(s0 ) 6= 0. If some λj (j ≥ 1)
does not vanish, then let c be any fixed non-zero real number. We cansolve the system (14) with the additional condition x0 = c because of
the above positivity of the dimension. Hence the claim follows.
Now we use Theorem 3 to find a positive real number τ such that(15)

|λj ζ (k+j) (s0 + iτ ) − λj xk+j | <ǫ
3(J + 1)for 0 ≤ k + j < n + J. Summing (15) up on j = 0, 1, . . . , J and combining with (14), we obtain (13). This corresponds to (4) in the proofof Theorem 4. The rest of the proof follows as in the previous section;
the constant C can be made effective by solving the system of linear
equations explicitly. Thus the proof of Theorem 6 is complete.
In the above proof, the assumption g(s0 ) 6= 0 is used only in the casewhen all λj = 0 (j ≥ 1). Therefore this assumption is actually not necessary unless all λj = 0 (j ≥ 1). However the latter case is that we treatedin Theorem 4, and we have already seen that the condition g(s0 ) 6= 0
can be removed in the sense of Corollary 5. Therefore we can conclude
Corollary 7. The condition g(s0 ) 6= 0 can be removed from the statement of Theorem 6.
Acknowledgements. The idea for this paper originates from a workshop held at the Department of Mathematics of Universidad Autónoma
de Madrid in December 2005 which was financially supported by the‘Rámon y Cajal’-programme of the Spanish Ministry for Science and
Education. The authors are very grateful for this support and the kind
hospitality at UA Madrid. They are also indebted to the referee formany useful comments and suggestions; especially, corollaries were firstpointed out by the referee.

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[2] R. Garunkštis, On the Voronin’s universality theorem for the Riemann zeta-function, Fiz. Mat. Fak. Moksl. Semin. Darb. 6 (2003),
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[3] P. M. Gauthier and R. Clouâtre, Approximation by translates of Taylor polynomials of the Riemann zeta function, Comput.
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Ramūnas Garunkštis and Antanas Laurinčikas:
Faculty of Mathematics and Informatics
Vilnius University
Naugarduko 2403225 Vilnius
Lithuania
E-mail address: ramunas.garunkstis@maf.vu.lt
E-mail address: antanas.laurincikas@maf.vu.lt
Kohji Matsumoto:
Graduate School of Mathematics
Nagoya University
Furocho, Chikusaku
Nagoya 464-8602
Japan
E-mail address: kohjimat@math.nagoya-u.ac.jp
Jörn and Rasa Steuding:
Department of MathematicsWürzburg University
Am Hubland, 97
218 Würzburg
Germany
E-mail address: steuding@mathematik.uni-wuerzburg.dePrimera versió rebuda el 6 de novembre de 2008,
darrera versió rebuda el 31 de març de 2009.