PREHISTORY OF THE VORONIN UNIVERSALITY THEOREM 1 ...

’iauliai Mathematical Seminar
1(9), 2006, 4153
PREHISTORY OF THE VORONIN
UNIVERSALITY THEOREM
Antanas LAURINƒIKAS
Vilnius University, Naugarduko 24, LT03225, Vilnius,
’iauliai University, P. Vi²inskio 19, LT-77156 ’iauliai, Lithuania;
e-mail: antanas.laurincikas@maf.vu.lt
Abstract. The results of H. Bohr, R. Courant, B. Jessen and S. M. Voronin
which inuenced the Voronin universality theorem for the Riemann zetafunction
are surveyed and discussed.
Key words and phrases: denseness, Kronecker theorem, permutation of terms,
Riemann zeta-function, universality.
Mathematics Subject Classication: 11M06.
1. Introduction
The theory of the Riemann zeta-function ζ(s), s = σ + it, occupies one of
central places in analytic number theory. We recall that ζ(s) is dened, for
σ > 1, by
ζ(s) =
∞∑
m=1
1
m s ,
and by analytic continuation elsewhere. Many problems of analytic number
theory and, in general, of mathematics are inuenced by value distribution
of the function ζ(s). In particular, the zero distribution of ζ(s) is closely
related to the prime number theorem. For the rst time, this was observed
by B. Riemann in 1859 [9]. Ch. de la Vallée Poussin [10] and J. Hadamard [5]
successfully developed Riemann's ideas and proved independently the prime
number distribution law
π(x) = ∑ px
1 ∼ lix =
∫ x
2
du
, x → ∞. (1)
log u

42 A. Laurin£ikas
Here and in the sequel, p denotes a prime number, and f(x) ∼ g(x), x → a,
means that lim f(x)/g(x) =
x→a
1. The equality (1) follows from the fact that
ζ(σ + it) ≠ 0 for σ 1. The famous Riemann hypothesis (RH), non-proved
and non-disproved till our days, asserts that all non-trivial zeros of ζ(s) lie
on the critical line σ = 1 2, i. e., ζ(σ + it) ≠ 0 for σ > 2. 1 H. von Koch proved
[7] that RH is equivalent to the estimate
π(x) = lix + O(x 1 2 log x).
Similarly ζ(σ + it) ≠ 0, for σ > α, 1 2
< α < 1, if and only if, for any ε > 0,
π(x) = lix + O(x α+ε ).
All these results show the importance of the value distribution of the function
ζ(s).
In 1975, a Russian mathematician S. M. Voronin (19471997) discovered
that the set of values of ζ(s) in the strip {s ∈ C : 1 2
< σ < 1} is very rich,
every analytic function can be approximated by shifts ζ(s + iτ). He called
the latter property of ζ(s) the universality. More precisely, the Voronin
universality theorem is the following statement [6], [14].
Theorem V. Let 0 < r < 4. 1 Let f(s) be a continuous non-vanishing function
on the disc |s| r which is analytic in the interior of this disc. Then,
for every ε > 0, there exists a real number τ = τ(ε) such that
(
max
∣ ζ s + 3 ) 4 + iτ − f(s)
∣ < ε.
|s|r
Theorem V is a result of development of investigations of H. Bohr, R. Courant,
B. Jessen and S. M. Voronin on the denseness of some sets of values of
the function ζ(s). The aim of this note is to recall the mentioned results.
2. Bohr's denseness theorem
H. Bohr, probably, was the rst who began to study the value distribution of
the Riemann zeta-function. Among his other results, the following theorem
obtained jointly with R. Courant occupies a particular place, see [1], [2], [3].
Theorem B 1 . Let σ, 1 2
< σ 1, be xed. Then the set {ζ(σ + it) : t ∈
(−∞, +∞)} is dense in C.
First H. Bohr proved Theorem B 1 under RH.

Prehistory of the Voronin universality theorem 43
We will give a sketch of the proof. Consider the set of values of the
function
F x (σ; θ 2 , θ 3 , ...) = − ∑ px
log
( )
1 − e−2πiθp
p σ ,
where θ 2 , θ 3 , θ 5 , ... are real variables indexed by prime numbers. It turns
out that this set contains a disc which radius tends to innity as x → ∞.
Therefore, for every a ∈ C, there exist x and a vector (θ2 0, θ0 3 , θ0 5 , ...) such that
F x (σ; θ2, 0 θ3, 0 θ5, 0 ...) = a. (2)
It is well known that the set {log p : p is prime} is linearly independent
over the eld of rational numbers. Hence, by the Kronecker theorem, for
every ε > 0, there exists a dense set {τ j } of real numbers such that
∥
τ j log p
2π
− θp
0 ∥ < ε.
Here ‖x‖ is the distance of x from the closest integer. This together with
(2) shows that, for every ε > 0, there exists a dense set {t j } of real numbers
such that ∣ (
)
∣∣∣ log 2
F x σ; t j
2π , t log 3
j
2π , ... − a
∣ < ε 1,
or, by the denition of the function F x , for s j = σ + it j ,
∣ log ∏ px
(
1 − 1
p s j
) −1 − a∣
∣ < ε 1. (3)
Next, it is proved that there exists x such that
∣
) −1 ∣ ∣∣∣
< ε 1 . (4)
∣ log ζ(s j) − log ∏ px
(
1 − 1
p s j
Now by (3) and (4), for suciently large x, we can nd a subset A ⊂ {t j }
such that
| log ζ(σ + it j ) − a| < 2ε 1
if t j ∈ A. This proves the theorem.
Using similar arguments, H. Bohr and B. Jessen obtained [4] the following
important theorem.
Theorem B 2 . Let a ∈ C \ {0}. Then, for every σ 1 , σ 2 , 1 2 < σ 1 < σ 2 1,
there exists a constant c = c(a, σ 1 , σ 2 ) > 0 such that the number of solutions
of the equation ζ(s) = a in the rectangle σ 1 < σ < σ 2 , |t| < T is ∼ cT as
T → ∞.

44 A. Laurin£ikas
3. Generalizations of Theorem B 1
In 1972, S. M. Voronin obtained [11], [12] a multidimensional analogue of
Theorem B 1 . He considered the value distribution of the vectors
and
(ζ(s 1 + iτ), ζ(s 2 + iτ), ..., ζ(s n + iτ))
(ζ(s + iτ), ζ ′ (s + iτ), ..., ζ (n−1) (s + iτ)),
where s, s 1 , ..., s n are xed complex numbers, and τ ∈ (−∞, ∞).
Let n > 1, C n = C × ... × C, and let C denote the extended complex
plane, C n = C × ... × C.
} {{ }
n
} {{ }
n
Theorem V 1 . Let h > 0, (s 1 , s 2 , ..., s n ) ∈ C n , 1 2 < Res j 1, j = 1, ..., n,
and s j ≠ s k for j ≠ k. Then the sequence of points from C n
η m = (ζ(s 1 + imh), ζ(s 2 + imh), ..., ζ(s n + imh)), m ∈ N,
is dense in C n .
The proof of Theorem V 1 is based on the following statement (main
Lemma).
Lemma V 1 . Suppose that (s 1 , s 2 , ..., s n ) ∈ C n , s j ≠ s k , k ≠ j, 1 2 < Re s j 1
and Im s j > 2, j = 1, . . . , n. Then, for every square matrix (a kl ) ∈ C and
n2
arbitrary y > 0, ε > 0, there exists a nite set of prime numbers M such
that:
1 0 M contains all prime numbers less than y;
2 0 For k, l = 1, ..., n,
∣ (log ζM (s k , θ 0 )) (l−1) ∣
− a k,l < ε,
where, for θ = (θ 2 , θ 3 , ...),
ζ M (s, θ) = ∏
p∈M
(
1 − 1 ) −1
p s e−2πiθ p
and θ 0 = ( 1
2 , 2 2 , 3 2 , 4 2 , ...) .
Proof. S. M. Voronin, dierently than H. Bohr, applies a theorem of
E. Steinitz on the permutation of the terms of series in R n . Suppose that

Prehistory of the Voronin universality theorem 45
u m ∈ R n , and the series
∞∑
m=1
converges. If, for every unit vector e ∈ R n , the series
∞∑
m=1
u m (5)
(u m , e) (6)
converges relatively, then for each vector a ∈ R n , there exists a permutation
of the terms of series (5) such that the obtained series converges to a.
Let p m be the mth prime number, e ∈ R be an arbitrary unit vector,
2n2
and
u m =
(
p −s 1
m e mπi , − log p m p −s 1
m e mπi , ..., (− log p m ) n−1 p −s 1
m e mπi , ...,
p −s n
m
e mπi , − log p m p −s n
m e mπi , ..., (− log p m ) n−1 p −s n
m e mπi) .
S. M. Voronin proves that there exists a partial series of (6) in R 2n2
∑′
(u m , e)
m
which diverges to +∞. In this case, m is even. Similarly, he constructs a
partial series
∑
m
′′
(u m , e)
which converges to −∞. Hence, it follows that the series (6) converges relatively
in R 2n2 . This implies the relative convergence of the series
where
û m =
(
∞∑
(û m , e),
m=1
log ( 1 − p −s 1
m e mπi) −1
,
(
log ( 1 − p −s 1
m e mπi) −1 )′ , ...,
log ( 1 − p −s n
m e mπi) −1
,
(
log ( 1 − p −sn
m e mπi) −1 )′ , ...,
(
log ( 1 − p −s 1
m e mπi) −1 ) (n−1)
, ...,
(
log ( 1 − p −sn
m e mπi) −1 ) (n−1) ).

46 A. Laurin£ikas
Therefore, by Steinitz's theorem, for every a ∈ C n2 , there exists a permutation
of the series
∑
∞ û m
m=1
in C such that the obtained series converges to a. Now to prove the lemma
n2
it suces to take a rather long partial sum of the obtained series.
Proof of Theorem V 1 . By Lemma V 1 , for every y > 0, the set of
vectors
(
ζM (s 1 , θ 0 ), ..., ζ M (s n , θ 0 ) ) ,
where M runs over all nite sets of prime numbers containing all prime
numbers less than y, is dense in C n . Let θ 1 = (θ p (1)
1
, θ (1)
Then the set of vectors
θ (1)
p m
=
{
0 if p m y,
m
2
if p m > y.
p 2
, ...), where
(
ζM (s 1 , θ 1 ), ..., ζ M (s n , θ 1 ) ) (7)
is also dense in C n . Let (a 1 , ..., a n ) ∈ C r be arbitrary. In view of the denseness
of the set (7), we can nd a set M such that, for k = 1, ..., n,
|ζ M (s k , θ 1 ) − a k | < ε.
Since ζ M (s k , θ) is continuous with respect to θ p , p ∈ M, hence there exists
δ > 0 such that, for k = 1, ..., n,
|ζ M (s k , θ) − a k | < ε (8)
if, for all p ∈ M,
|θ p − θ p (1) | < δ. (9)
In the sequel, S. M. Voronin separates two cases. Let, for prime p,
q p = h log p
2π .
Since the system {log p : p is prime} is linearly independent over the eld of
rational numbers, the system {q p } also has the same property. Therefore, if
the equality
∑
α p q p = 1 (10)
p

Prehistory of the Voronin universality theorem 47
with rational coecients α p , which are non-zeros only for a nite number of
primes p, holds, it is unique. S. M. Voronin considers separately the cases:
1 0 Equality (10) is true.
2 0 Equality (10) is not true.
Denote by P M the space of the variables θ p , p ∈ M, and denote by K a
cube of points from P M satisfying (9). Let P 1 = {p : α p ≠ 0 in (10)}. If (10)
is true, then we take some p ′ ∈ P 1 and nd from (10) that
q p ′ = 1 c 0
(
b 0 +
∑
p∈P 1 \{p ′ }
b p q p
)
with integers c 0 > 0 and b 0 , b p . Then, in this case, in place of {η m } the
sequence η ′ m = η c0 m is considered, and this corresponds h ′ = c 0 h and
η ′ m = (ζ(s 1 + imh ′ ), ..., ζ(s n + imh ′ )).
Furthermore, it is proved that on the hyperplane θ p ′ = 0 there exists a
cube K 1 of positive Jordan measure such that
implies
with
and
θ ′ p =
(θ p1 , θ p2 , ...) ∈ K 1
(θ ′ p 1
, θ ′ p 2
, ...) ∈ K
θ ′ p ′ =
∑
p∈P 1 \{p ′ }
b p θ p
{
c 0 θ p if p ∈ P 1 \ {p ′ },
θ p if p ∉ P 1 .
In the case 2 0 , denote by N ∑ ′
m=1
(mq p1 , mq p2 , ...) ∈ K( mod 1). Let
⎧
⎪⎨ 0 if p = p ′ ,
V p = m qp
c ⎪⎩
0
if p ∈ P 1 \ {p ′ },
mq p if p ∉ P 1 , p ∈ M.
In the case 1 0 ,
N
∑ ′
m=1
the sum over m ∈ [1, N] for which
means the sum over m ∈ [1, N] satisfying
(11)
(V p1 , V p2 , ...) ∈ K 1 ( mod 1). (12)

48 A. Laurin£ikas
Now let
A =
N
∑ ′
m=1
N∑
k=1
|ζ(s k + imh) − ζ M (s k + imh, 0)| 2 , (13)
where h = h ′ in the case 1 0 . Then it can be proved that there exists N 0 such
that, for N > N 0 ,
A ε 2 Nmeas{K} (or meas{K 1 }). (14)
Case 2 0 . By the modied Kronecker theorem
lim
N→∞
N
∑ ′
1
N
= meas{K}.
m=1
Therefore, there exists N 1 such that, for N > N 1 ,
N
∑ ′
1 1 2 Nmeas{K}.
m=1
Hence and from (14) there exist m satisfying
N∑
|ζ(s k + imh) − ζ M (s k + imh, 0)| 2 < 4ε 2 .
k=1
Thus, for k = 1, ..., n,
|ζ(s k + imh) − ζ M (s k + imh, 0)| < 2ε. (15)
∑
On the other hand, if m is an index in N ′
, then
m=1
(mq p1 , mq p2 , ...) ∈ K(mod1).
Therefore, by the denition of K, (8) and (9), for k = 1, ..., n, yield
However,
|ζ M (s k , mq 1 , mq 2 , ...) − a k | < 2ε. (16)
(
ζ M (s k , mq 1 , mq 2 , ...) = ζ M s k , m h log p 1
, m h log p )
2
, ...
2π 2π

Prehistory of the Voronin universality theorem 49
and thus (15) and (16) give
= ∏ (1 −
p∈M
= ∏
p∈M
(
1 −
log p
e−2πimh 2π
1
p s k+imh
= ζ M (s k + imh, 0),
p s k
) −1
) −1
|ζ(s k + imh) − a k | < 4ε,
and in the case 2 0 the theorem is proved.
∑
Now suppose that (10) is true. Then m in N ′
satises (14). Since the
m=1
set {1, q p1 , q p2 , ...} \ {q p ′} is linearly independent over the eld of rational
numbers, by the modied Kronecker theorem we have again that
Hence, for N > N 1 ,
lim
N→∞
N
∑ ′
m=1
N
∑ ′
m=1
1
N = meas{K 1}.
1 > N 2 meas{K 1}.
This and (14) show that, for N > max(N 1 , N 0 ), there exist m such that
N∑
|ζ(s k + imh) − ζ M (s k + imh, 0)| 2 < 4ε,
k=1
and, for all k = 1, ..., n,
Dene
|ζ(s k + imh) − ζ M (s k + imh, 0)| < 2ε.
⎧ ∑
b p V p if p = p ⎪⎨
′ ,
p∈P 1 \p ′
u p =
⎪⎩ u p = V p if p ∈ M.
u p = c 0 V p if p ∈ G \ p ′ ,

50 A. Laurin£ikas
Since (V p1 , V p2 , ...) ∈ K 1 ( mod 1), by the denition of K 1 we have that
(u p1 , u p2 , ...) ∈ K( mod 1). Then
(
m h log p 1
2π
, m h log p )
2
, ... ∈ K( mod 1),
2π
and by the denition of K, for all k = 1, ..., n,
∣
∣ζ M
(
s k , m h log p 1
2π
Thus, similarly to the case 2 0 ,
, m h log p ) ∣
2 ∣∣
, ... − a k < 2ε.
2π
|ζ(s k + imh) − a k | < ε.
The theorem is proved.
We mention also the following Voronin's results.
Theorem V 2 ([11]). Let 1 2 < Re s 0 1. Then the sequence of points from
C n (
ζ(s 0 + imh), ζ ′ (s 0 + imh), ..., ζ (n−1) (s 0 + imh)
is dense in C n .
)
, m ∈ N,
S. M. Voronin in his talk given at the Vilnius conference presented [13] a
generalization of Theorems V 1 and V 2 in the form of one theorem.
Theorem V 3 . Suppose that s 1 , ..., s n are pairwise dierent complex numbers
such that 1 2 < Re s j 1, j = 1, ..., n. Then the curve
t ∈ R, is dense in C n2 .
(
ζ(s1 + it), ζ ′ (s 1 + it), ..., ζ (n−1) (s 1 + it), ...,
ζ(s n + it), ζ ′ (s n + it), ..., ζ (n−1) (s n + it) ) ,
4. Generalizations of Theorem B 2
S. M. Voronin presented the following multidimensional versions of Theorem
B 2 .

Prehistory of the Voronin universality theorem 51
Theorem V 4 ([11], [12]). Let a region D = {(s 1 , ..., s n ) ∈ C n : 1 2 < Re s k
1, k = 1, ..., n} have the positive Jordan measure. Then, for every a 1 , ..., a n ∈
C \ {0}, the number of solutions (s 1 , ..., s n ) of the system
satisfying
(s 1 , ..., s n ) ∈
⎧
⎪⎨ ζ(s 1 ) = a 1 ,
. . .
⎪⎩
ζ(s n ) = a n ,
N⋃
(D + im(h, ..., h)), h > 0,
m=1
for suciently, large N, is greater that KN, where K = K(D, a 1 , ..., a n , h) >
0.
Theorem V 4 has the following modication in the sense of Theorems V 2
and V 3 .
Theorem V 5 . Let the region D be the same as in Theorem V 4 . Then, for
every a 1 ∈ C \ {0} and a 2 , ..., a n ∈ C, the number of solutions (s 1 , ..., s n ) of
the system ⎧⎪ ⎨
⎪ ⎩
ζ(s 1 ) = a 1 ,
satisfying
(s 1 , ..., s n ) ∈
ζ ′ (s 2 ) = a 2 ,
. . .
ζ (n−1) (s n ) = a n ,
N⋃
(D + im(h, ..., h)), h > 0,
m=1
for suciently large N, is greater that KN, where K = K(D, a 1 , ..., a n , h) >
0.
5. Towards Theorem V
Lemma V 1 suggests a possible approximation of a given analytic function by
the logarithm of a nite Euler product. However, a way used for the proof
of Lemma V 1 requires a generalization of the Steinitz theorem for series in
Hilbert spaces. S. M. Voronin asked his friend D. V. Pechersky to prove
a theorem of such a type. D. V. Pechersky succeded in his research, and

52 A. Laurin£ikas
presented to S. M. Voronin a theorem on permutations of series in Hilbert
spaces. Let H be a real Hilbert space with inner product (a, b) and norm
||a||.
Lemma P 1 ([8]). Suppose that
∞∑
||a m || < ∞, a m ∈ H,
m=1
and for every e ∈ H, ||e|| = 1, the series
∞∑
(a m , e)
m=1
converges relatively for some permutation of its terms. Then, for every a ∈
H, there exists a permutation {m k : k ∈ N} of N such that
∞∑
a mk = a
k=1
in the sense of norm in H.
Lemma P 1 implies an analogue of lemma V 1 which is a key for the proof
of Theorem V.
Lemma V 2 ([14]). Let 0 < r < 1 4, and let f(s) be a continuous function on
|s| r and analytic in the interior of this disc. Then, for every ε > 0 and
y > 0, there exists a nite set M such that:
1 0 {p : p y} ⊂ M;
2 0 ∣
max ∣f(s) − log ζ M (s + 3 4 , θ 0) ∣ < ε.
|s|r
Acknowledgements. The article is a text of a talk given at the Colloquium
on Dirichlet series held at Universidad Autónoma de Madrid. The
author thanks for hospitality the organizers of this Colloquium Professors
Joern Steuding and Rasa Steuding.
References
[1] H. Bohr, Sur la fonction ζ(s) de Riemann, Comptes Rendus 158 (1914),
19861988.

Prehistory of the Voronin universality theorem 53
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Acta Math. 40 (1915), 67100.
[3] H. Bohr, R. Courant, Neue Anwendungen der Theorie der Diophantischen
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Received
4 July 2006