A SURVEY ON THE THEORY OF UNIVERSALITYFOR ZETA AND L-FUNCTIONS
KOHJI MATSUMOTO

Abstract. We survey the results and the methods in the theory of universality for various zeta and L-functions, obtained inthese forty years after the first discovery of the universality for the
Riemann zeta-function by Voronin.

Contents
1.
2.
3.
4.

Voronin’s universality theorem
A rough sketch of the history
Generalization to zeta and L-functions with Euler products
The joint universality for zeta and L-functions with Euler
products
5. The strong universality
6. The joint strong universality
7. The universality for multiple zeta-functions
8. The mixed universality
9. The strong recurrence
10. The weighted universality
11. The discrete universality
12. The χ-universality
13. Other applications
14. The general notion of universality
15. The hybrid universality
16. Quantitative results
17. The universality for derived functions
18. Ergodicity and the universality
References

1
5
6
10
13
17
20
21
22
24
25
28
29
30
32
34
35
37
38

1. Voronin’s universality theorem
Let N be the set of positive integers, N0 = N ∪ {0}, Z the ring ofrational integers, Q the field of rational numbers, R the field of real
1

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KOHJI MATSUMOTO

numbers, and C the field of complex numbers. In the present article,
the letter p denotes a prime number.
For any open region D ⊂ C, denote by H(D) the space of C-valued
holomorphic functions defined on D, equipped with the topology of
uniform convergence on compact sets.
For any subset K ⊂ C, let H c (K) be the set of continuous functions
defined on K, and are holomorphic in the interior of K, and let H0c (K)be the set of all elements of H c (K) which are non-vanishing on K. By
meas S we mean the usual Lebesgue measure of the set S, and by #S
the cardinality of S.
√
Let s = σ + it ∈ C (where σ = ℜs, t = ℑs, and i = −1),and ζ(s) the Riemann∑zeta-function. This function is defined by the−sinfinite series ζ(s) = ∞in the half-plane σ > 1, and can ben=1 n
continued meromorphically to the whole of C. It is well known thatthe investigation of ζ(s) in the critical strip 0 < σ < 1 is extremely
important in number theory, but its behaviour there still remains quite
mysterious. A typical example of expressing one of such mysterious
features of ζ(s) is Voronin’s universality theorem.
Consider the closed disc K(r) with center 3/4 and radius r, where0 < r < 1/4. Then Voronin [205] proved that for any f ∈ H0c (K(r))
and any ε > 0, there exists a positive number τ for which(1.1)

max |ζ(s + iτ ) − f (s)| < εs∈K(r)

holds. Roughly speaking, any non-vanishing holomorphic function canbe approximated uniformly by a certain shift of ζ(s).
Actually, Voronin’s proof essentially includes the fact that the set ofsuch τ has a positive lower density. Moreover, now it is known thatthe disc can be replaced by more general type of sets. The modern
statement of Voronin’s theorem is as follows. Let D(a, b) (a < b)indicate a vertical strip
D(a, b) = {s ∈ C | a < σ < b}.
Theorem 1. (Voronin’s universality theorem) Let K be a compact
subset of D(1/2, 1) with connected complement, and f ∈ H0c (K). Then,for any ε > 0,}{1(1.2) lim inf meas τ ∈ [0, T ] sup |ζ(s + iτ ) − f (s)| < ε > 0
T →∞ Ts∈K
holds.

THE THEORY OF UNIVERSALITY FOR ZETA AND L-FUNCTIONS

3

Let φ(s) be a Dirichlet series, and let K be a compact subset ofD(a, b) with connected complement. If{}
1(1.3) lim inf meas τ ∈ [0, T ] sup |φ(s + iτ ) − f (s)| < ε > 0
T →∞ Ts∈K
holds for any f ∈ H0c (K) and any ε > 0, then we say that the universality holds for φ(s) in the region D(a, b). Theorem 1 implies that the universality holds for the Riemann zeta-function in the region D(1/2, 1).
zeta-function has the Euler product expression ζ(s) =∏ The Riemann
−s −1
(1
−
p
)
,
where
p runs over all prime numbers. This is valid onlypin the region σ > 1, but even in the region D(1/2, 1), it is possible
to show that a finite truncation of the Euler product “approximates”
ζ(s) in a certain mean-value sense. On the other hand, since {log p}p
is linearly independent over Q, we can apply the Kronecker-Weyl approximation theorem to obtain that any target function f (s) can beapproximated by the above finite truncation. This is the basic structure of the proof of Theorem 1.
Remark 1. Here we recall the statement of the Kronecker-Weyl theorem. For x ∈ R, the symbol ||x|| stands for the distance from x to thenearest integer. Let α1 , . . . , αm be real numbers, linearly independent
over Q. Then, for any real numbers θ1 , . . . , θm and any ε > 0,
(1.4)

1meas {τ ∈ [0, T ] | ||τ αk − θk || < ε (1 ≤ k ≤ m) } > 0
T →∞ Tlim

holds.
So far, three proofs are known for Theorem 1. Needless to say, one ofthem is Voronin’s original proof, which is also reproduced in [64]. This
proof is based on, besides the above facts, Pecherskiı̆’s rearrangement
theorem [185] in Hilbert spaces. The second proof is given by Good [41],which will be discussed in Section 16. Gonek was inspired by the idea
of Good to write his thesis [40], in which he gave a modified version
of Good’s argument. The third is a more probabilistic proof due toBagchi [3] [4]. Bagchi [3] is an unpublished thesis, but its contents
are carefully expounded in [76]. A common feature of the work of
Gonek and Bagchi is that they both used the approximation theoremof Mergelyan [141] [142].
Remark 2. Mergelyan’s theorem asserts that, when K is a compact
subset of C, any f ∈ H c (K) can be approximated by polynomials uniformly on K. This is a complex analogue of the classical approximation
theorem of Weierstrass.

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KOHJI MATSUMOTO

Here we mention some pre-history. In 1914, Bohr and Courant [13]proved that, for any σ satisfying 1/2 < σ ≤ 1, the set{ζ(σ + iτ ) | τ ∈ R}
is dense in C. In the next year, Bohr [11] proved that the same result
holds for log ζ(σ + iτ ). These results are called denseness theorems.
Before obtaining his universality theorem, Voronin [204] discovered
the following multi-dimensional analogue of the theorem of Bohr and
Courant.
Theorem 2. (Voronin [204]) For any σ satisfying 1/2 < σ ≤ 1, theset{(ζ(σ + iτ ), ζ ′ (σ + iτ ), . . . , ζ (m−1) (σ + iτ )) | τ ∈ R}
(where ζ (k) denotes the k-th derivative) is dense in Cm .
Remark 3. Actually Voronin proved a stronger result. For any s = σ+itwith 1/2 < σ ≤ 1 and for any h > 0, the set{(ζ(s + inh), ζ ′ (s + inh), . . . , ζ (m−1) (s + inh)) | n ∈ N}
is dense in Cm .
The universality theorem of Voronin may be regarded as a natural
next step, bacause it is a kind of infinite-dimensional analogue of thetheorem of Bohr and Courant, or a denseness theorem in a function
space.
Another refinement of the denseness theorem of Bohr and Courant
is the limit theorem on C, due to Bohr and Jessen [14]. Recently,
this theorem is usually formulated by probabilistic terminology. Let
σ > 1/2. For any Borel subset A ⊂ C, put
1meas {τ ∈ [0, T ] | ζ(σ + iτ ) ∈ A} .
T
This PT,σ is a probability measure on C. Then the modern formulationof the limit theorem of Bohr and Jessen is that there exists a probability
measure Pσ , to which PT,σ is convergent weakly as T → ∞ (see [76,Chapter 4]).
Bagchi [3] proved an analogue of the above theorem of Bohr andJessen on a certain function space, and used it in his alternative proofof the universality theorem. Therefore, to prove some universality-type
theorem by Bagchi’s method, it is necessary to obtain some functional
limit theorem similar to that of Bagchi. There are indeed a lot of papers
devoted to the proofs of various functional limit theorems, for the purpose of showing various universality theorems. However in the following
PT,σ (A) =THE THEORY OF UNIVERSALITY FOR ZETA AND L-FUNCTIONS

5

sections we do not explicitly mention this closely related topic of functional limit theorems, and for the details we refer to Laurinčikas [76] or
J. Steuding [201]. The connection between the theory of universality
and functional limit theorems is also discussed in the author’s survey
articles [136] [137].
After the publication of Voronin’s theorem in 1975, now almost forty
years have passed. Voronin’s theorem attracted a lot of mathematicians, and hence, after Voronin, quite many papers on universality
theory have been published. The aim of the present article is to survey
the developments in this theory in these forty years. The developmentsin these years can be divided into three stages.
(I) The first stage: 1975 ∼ 1987.
(II) The second stage: 1996 ∼ 2007.
(III) The third stage: 2007 ∼ present.
In the next section we will give a brief discussion what were the main
topics in each of these stages.
The author expresses his sincere gratitude to Professors R. Garunkštis,
R. Kačinskaitė, S. Kanemitsu, E. Karikovas, A. Laurinčikas, R. Macaitienė, H. Mishou, H. Nagoshi, T. Nakamura, L. Pańkowski, D. Šiaučiūnas,
and J. Steuding for their valuable comments and/or sending relevant
articles.
2. A rough sketch of the history(I) The first stage.
This is the first decade after Voronin’s paper. The original impact ofVoronin’s discovery was still fresh. Mathematicians who were inspired
by Voronin’s paper tried to discuss various generalizations, analogies,refinements and so on. Here is the list of main results obtained in thisdecade.
• Alternative proofs (mentioned in Section 1).
• Generalizations to the case of Dirichlet L-functions, Dedekind zetafunctions etc.
• The joint universality.
• The strong universality (for Hurwitz zeta-functions).
• The strong recurrence.
• The discrete universality.
• The χ-universality.
• The hybrid universality.
• A quantitative result.
It is really amazing that many important aspects in universality theory,developed extensively in later decades, had already been introduced

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KOHJI MATSUMOTO

and studied in this first decade. Unfortunately, however, many of thoseresults were written only in the theses of Voronin [207], Gonek [40] andBagchi [3], all of them remain unpublished. This situation is probably
one of the reasons why in the next several years there were so few
publications on universality.
(II) The second stage.
During several years around 1990, the number of publications concerning universality is very small. Of course it is not completely empty.
For example, the book of Karatsuba and Voronin [64] was publishedin this period. But the author prefers to choose the year 1996 as thestarting point of the second stage, because in this year the importantbook [76] of Laurinčikas was published. This is the first textbook which
is mainly devoted to the theory of universality and related topics, andespecially, provides the details of unpublished work of Bagchi. Thanks
to the existence of this book, many mathematicians of younger generation can now easily go into the theory of universality. In fact, in thissecond stage, a lot of students of Laurinčikas started to publish theirpapers, and they formed the strong Lithuanian school.
The main topic in this decade was probably the attempt to extend
the class of zeta and L-functions for which the universality property
holds. It is now known that the universality property is valid for arather wide class of zeta-functions. J. Steuding’s lecture note [201]includes the exposition of this result, and also of many other results
obtained after the publication of the book of Laurinčikas [76]. Therefore the publication year of this lecture note is appropriate to the endof the second stage.
(III) The third stage.
Now comes the third, present stage. The theory of universality is
now developing into several new directions. The notions of• the mixed universality,• the composite universality,• the ergodic universality,
were introduced recently. Other topics in universality theory have also
been discussed extensively.
In the following sections, we will discuss more closely each topic inuniversality theory.
3. Generalization to zeta and L-functions with Euler
products
Is it possible to prove the universality property for other zeta andL-functions? This is surely one of the most fundamental question. In

THE THEORY OF UNIVERSALITY FOR ZETA AND L-FUNCTIONS

7

Section 1 we explained that a key point in the proof of Theorem 1 isthe Euler product expression. Therefore we can expect the universality
property for other zeta and L-functions which have Euler products.
The universality of the following zeta and L-functions were provedin the first decade.
∑∞−s• Dirichlet L-functions L(s, χ) =(χ is a Dirichlet
n=1 χ(n)ncharacter; see Voronin [205]) in D(1/2, 1),• certain Dirichlet series with multiplicative coeﬃcients (see Reich[188], Laurinčikas [68] [69] [70] [71] [72]),∑• Dedekind zeta-functions ζF (s) = a N (a)−s , where F is a number field, a denotes a non-zero integral ideal, and N (a) its norm (seeVoronin [207] [208], Gonek [40], Reich [189] [190]). Here, the universality can be proved in D(1/2, 1) if F is an Abelian extension of Q
(Gonek [40]), but for general F (̸= Q), the proof is valid only in the
narrower region D(1 − d−1
F , 1), where dF = [F : Q]. The reason is thatthe mean value estimate for ζF (s), applicable to the proof of universality, is known at present only in D(1 − d−1
F , 1).
Later, Laurinčikas also obtained universality theorems for Matsumoto
zeta-functions1 ( [78] under a strong assumption), and for the zetafunction attached to Abelian groups ([82] [83]). Laurinčikas and Šiaučiūnas[122] proved the universality for the periodic zeta-function ζ(s, A) =∑∞−s(where A = {an }∞n=1 is a multiplicative periodic sequence
n=1 an nof complex numbers) when the technical condition
(3.1)

∞
∑

|apm |p−m/2 ≤ c

m=1(with a certain constant c < 1) holds for any prime p. Schwarz, R.
Steuding and J. Steuding [192] proved another universality theorem
on certain general Euler products with conditions on the asymptotic
behaviour of coeﬃcients.
However, there was an obstacle when we try to generalize further. A
typical class of L-functions
with Euler products is that of automorphic∑∞
L-functions. Let g(z) = n=1 a(n)e2πinz be a holomorphic normalized
Hecke-eigen
of weight κ with respect to SL(2, Z) and let∑∞cusp form−s
L(s, g) = n=1 a(n)n be the associated L-function. The universalityof L(s, g) was first discussed by Kačėnas and Laurinčikas [53], but they
showed the universality only under a very strong assumption.
1This notion was first introduced by the author [133], to which the limit theoremof Bohr and Jessen was generalized. See also Kačinskaitė’s survey article [56].

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KOHJI MATSUMOTO

What was the obstacle? The asymptotic formula
∑1
√
(3.2)
= log log x + a1 + O(exp(−a2 log x))pp≤x(where a1 , a2 are constants) is classically well known, and is used in
the proof of Theorem
the corresponding asymptotic for∑ 1. However−1mula for the sum p≤x |a(p)|p is not known. To avoid this obstacle,
Laurinčikas and the author [111] invented a new method of using (3.2),
combined with the known asymptotic formula
∑
x
(3.3)
|e
a(p)|2 =(1 + o(1)),
log x
p≤x
where ea(p) = a(p)p−(κ−1)/2 . This method is called the positive density
method. Modifying Bagchi’s argument by virtue of this positive density
method, one can show the following unconditional result.
Theorem 3. (Laurinčikas and Matsumoto [111]) The universality holds
for L(s, g) in the region D(κ/2, (κ + 1)/2).
The positive density method was then applied to prove the universality for more general class of L-functions; certain Dirichlet series
with multiplicative coeﬃcients (Laurinčikas and Šleževičienė [126]), Lfunctions attached to new forms with respect to congruence subgroups(Laurinčikas, Matsumoto and J. Steuding [116]), L-functions attached
to a cusp form with character (Laurinčikas and Macaitienė [105]), anda certain subclass of the Selberg class2 (J. Steuding [197]). J. Steuding
extended his result further in his lecture note [201]. He introduced awide class Se of L-functions defined axiomatically and proved the unie The class S,
e now sometimes called theversality for elements of S.
Steuding class, is not included in the Selberg class, but is a subclass ofthe class of Matsumoto zeta-functions.
Since the Shimura-Taniyama conjecture has been established, we
now know that the L-function L(s, E) attached to a non-singular elliptic curve E over Q is an L-function attached to a new form. Therefore
the universality for L(s, E) is included in [116]. The universality ofpositive powers of L(s, E) was studied in Garbaliauskienė and Laurinčikas [31].
Mishou [144] [145] used a variant of the positive density method toshow the universality for Hecke L-functions of algebraic number fields
2The notion of Selberg class was introduced by Selberg [193]. For basic definitions

and results in this theory, consult a survey [63] of Kaczorowski and Perelli, or J.
Steuding [201].

THE THEORY OF UNIVERSALITY FOR ZETA AND L-FUNCTIONS

9

in the region D(1−d−1
F , 1). Lee [127] showed that, under the assumptionof a certain density estimate of the number of zeros, it is possible toprove the universality for Hecke L-functions in the region D(1/2, 1).
The universality for Artin L-functions was proved by Bauer [7] by adiﬀerent method, based on Voronin’s original idea.
Let g1 and g2 be cusp forms. The universality for the RankinSelberg L-function L(s, g1 ⊗ g1 ) was shown by the author [134], andfor L(s, g1 ⊗ g2 ) (g1 ̸= g2 ) was by Nagoshi [165] (both in the narrower
region D(3/4, 1)). The latter proof is based on the above general resultof J. Steuding [197] [201]. The universality of symmetric m-th power
L-functions (m ≤ 4) and their Rankin-Selberg L-functions was studied
by Li and Wu [129].
Another general result obtained by the positive density method is thefollowing theorem, which is an extension of the result of J. Steuding[197].
∑−s
Theorem 4. (Nagoshi and J. Steuding [166]) Let φ(s) = ∞n=1 a(n)nbe a Dirichlet series belonging to the Selberg class. Denote the degreeof φ(s) by dφ , and put σφ = max{1/2, 1 − d−1φ }. Assume that∑x|a(p)|2 =(3.4)(λ + o(1))
log
x
p≤x
holds with a certain positive constant λ. Then the universality holds
for φ(s) in the region D(σφ , 1).
Remark 4. When φ is the Riemann zeta-function, the formula (3.4)(with λ = 1) is nothing but the prime number theorem. Therefore
we may say that the positive density method enables us to prove theuniversality for zeta-functions with Euler products, provided an asymptotic formula of the prime-number-theorem type is known.
Let h be a Hecke-eigen Maass form, and let L(s, h) be the associatedL-function. The universality for L(s, h) was proved by Nagoshi [163][164]. In the proof of Theorem 3 in [111], Deligne’s estimate (Ramanujan’s conjecture) |ea(p)| ≤ 2 is essentially used. Assumptions of thesame type are required in Steuding’s general result [197] [201] and alsoin Theorem 4. Since Ramanujan’s conjecture for Maass forms has not
yet been proved, Nagoshi in his first paper [163] assumed the validityof Ramanujan’s conjecture for h to show the universality. Then in thesecond paper [164] he succeeded to remove this assumption, by invoking
the asymptotic formula for the fourth power mean of the coeﬃcients
due to M. Ram Murty.

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KOHJI MATSUMOTO

Theorem 5. (Nagoshi [164]) The universality holds for L(s, h) in theregion D(1/2, 1).
Another important class of zeta-functions which have Euler products
is the class of Selberg zeta-functions. In this case, instead of the primenumber-theorem type of results, the prime geodesic theorem plays animportant role. Let
D = {d ∈ N | d ≡ 0 or 1 (mod 4), d is not a square}.
For each d ∈ D, let h+ (d) be the number of inequivalent√ primitive
quadratic forms of discriminant d, and ε(d) = (u(d)+v(d) d)/2, where(u(d), v(d)) is the fundamental solution of the Pell equation u2 − v 2 d =4. Then, the prime geodesic theorem for SL(2, Z) implies
∫ x
∑
dt
+
(3.5)
h (d) =+ O(xα )
log
t
0
d∈D
ε(d)2 ≤xwith a certain α < 1.
Theorem 6. (Drungilas, Garunkštis and Kačėnas [23]) Let Z(s) be
the Selberg zeta-function attached to SL(2, Z). If (3.5) holds, then theuniversality holds for Z(s) in the region D(1/2 + α/2, 1).
As for the value of α, it is known that one can take α = 71/102+ε forany ε > 0 (Cai [19]). It is conjectured that one could take α = 1/2 + ε.
If the conjecture is true, then Theorem 6 implies that Z(s) has theuniversality property in D(3/4, 1). The paper [23] includes a discussion
which suggests that D(3/4, 1) is the widest possible region where theuniversality for Z(s) holds.
4. The joint universality for zeta and L-functions with
Euler products
The results presented in the previous sections give approximation
properties of some single zeta or L-function. Here we discuss simultaneous approximations by several zeta or L-functions.
Let K1 , . . . , Kr be compact subsets of D(a, b) with connected complements, and fj ∈ H0c (Kj ) (1 ≤ j ≤ r). If Dirichlet series φ1 (s), . . . , φr (s)satisfy{1(4.1)
lim inf meas τ ∈ [0, T ] sup |φj (s + iτ ) − fj (s)| < ε
T →∞ Ts∈Kj}(1 ≤ j ≤ r) > 0

THE THEORY OF UNIVERSALITY FOR ZETA AND L-FUNCTIONS

11

for any ε > 0, we call the joint universality holds for φ1 (s), . . . , φr (s)in the region D(a, b). The joint universality for Dirichlet L-functions
was already obtained in the first decade by Voronin [206] [207] [208],
Gonek [40], and Bagchi [3] [4], independently of each other:
Theorem 7. (Voronin, Gonek, Bagchi) Let χ1 , . . . , χr be pairwise nonequivalent Dirichlet characters, and L(s, χ1 ), . . . , L(s, χr ) the corresponding Dirichlet L-functions. Then the joint universality holds forL(s, χ1 ), . . . , L(s, χr ) in the region D(1/2, 1).
To prove such a theorem of simultaneous approximations, it is obviously necessary that the behaviour of L-functions appearing in thetheorem should be “independent” of each other. In the situation ofTheorem 7, this is embodied by the orthogonality relation of Dirichlet
characters, which is essentially used in the proof.
The joint universality for Dedekind zeta-functions was studied by
Voronin [207] [208]. Bauer’s work [7] mentioned in the preceding section
actually proves a joint universality theorem on Artin L-functions, in theregion D(1 − (2dF )−1 , 1). Lee [128] extended the region to D(1/2, 1)
under the assumption of a certain density estimate.
Let gj (1 ≤ j ≤ r) be multiplicative arithmetic functions. Laurinčikas [74]the joint universality of the associated Dirich∑∞considered−s
let series n=1 gj (n)n (1 ≤ j ≤ r). In this case, the “independence”
condition is given by the following matrix condition. Let P1 , . . . , Pk(k∑ ≥ r) be−1certain sets of prime numbers, with the condition thatsatisfies a good asymptotic formula, and assume thatp≤x,p∈Pi p
gj (n) is a constant gjl on the set Pl (1 ≤ j ≤ r, 1 ≤ l ≤ k). Laurinčikas [74] proved a joint universality theorem under the condition
that the rank of the matrix (gjl )1≤j≤r,1≤l≤k is equal to r.
Laurinčikas frequently used various matrix conditions to obtain joint
universality theorems. A joint universality theorem on Matsumoto
zeta-functions under a certain matrix condition was proved in [79]. A
joint universality for automorphic L-functions under a certain matrix
condition was discussed in [112].
A matrix condition naturally appears in the joint universality theoryof periodic zeta-functions (see [107] [103]). Let Aj = {ajn }∞n=1 be amultiplicative periodic sequence (whose least period we denote by kj )of complex numbers, and ζ(s, Aj ) the associated periodic zeta-function(1 ≤ j ≤ r). Let k be the least common multiple of k1 , . . . , kr . Define
the matrix A = (ajl )j,l , where 1 ≤ j ≤ r and 1 ≤ l ≤ k, (l, k) = 1.
Then Laurinčikas and Macaitienė [103] proved the joint universality forζ(s, A1 ), . . . , ζ(s, Ar ) in the region D(1/2, 1), if we assume rank(A) = r
and a technical condition similar to (3.1).

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KOHJI MATSUMOTO

Using the positive density method, it is possible to prove a joint
universality
theorem for twisted automorphic L-functions. Let g(z) =∑∞2πinz
be a holomorphic normalized Hecke-eigen cusp form,
n=1 a(n)e
χ
,
.
.
.
,
χ
be
pairwise
non-equivalent Dirichlet characters, and L(s, g, χj ) =
r
∑1 ∞
−s
the associated χj -twisted L-function.
n=1 a(n)χj (n)nTheorem 8. (Laurinčikas and Matsumoto [113]) The joint universality
holds for L(s, g, χ1 ), . . . , L(s, g, χr ) in the region D(κ/2, (κ + 1)/2).
To prove this result, we need a prime number theorem for a(p) inarithmetic progressions, that is
∑
1
x
(4.2)
|e
a(p)|2 =(1 + o(1)),
φ(q)
log
x
p≤x
p≡h(mod q)

where (h, q) = 1 and φ(q) is Euler’s totient function.
J. Steuding [201, Theorem 12.8] generalized Theorem 8 to the Steude A joint version of [126] was given by Šleževičienė [194].
ing class S.
A joint universality theorem on L-functions of elliptic curves, undera certain matrix condition, was given by Garbaliauskienė, Kačinskaitė
and Laurinčikas∑
[29].
∞
−s
Let φj (s) =(j = 1, 2) be elements of the Selberg
n=1 aj (n)nclass. The following orthogonality conjecture of Selberg [193] is wellknown: if φ1 (s), φ2 (s) are primitive, then
∑ a1 (p)a2 (p) { log log x + O(1) if φ = φ ,
1
2
(4.3)
=
O(1)
otherwise.
pp≤x

Inspired by this conjecture, J. Steuding [201, Section 12.5] proposed:
Conjecture 1. (J. Steuding) Any finite collection of distinct primitive
functions in the Selberg class is jointly universal 3.
Towards this conjecture, recent progress has been mainly due toMishou. In [153], Mishou proved the following. Consider two strips
D1 = D(1/2, 3/4) and D2 = D(3/4, 1). Let Kj be a compact subset of
Dj and fj ∈ H0c (Kj ) (j = 1, 2). Then
{
1
(4.4)
lim inf meas τ ∈ [0, T ] sup |ζ(s + iτ ) − f1 (s)| < ε,T →∞ Ts∈K}|L(s + iτ, g) − f2 (s)| < ε > 0
3J. Steuding also mentioned the expectation that any two functions φ , φ

∑

1
2 in
a1 (p)a2 (p)= O(1).
p≤x
p

the Selberg class would be jointly universal if and only if
However H. Nagoshi, and then H. Mishou, pointed out that there are counterexamples to this statement.

THE THEORY OF UNIVERSALITY FOR ZETA AND L-FUNCTIONS

13

holds, where L(s, g) is the automorphic L-functions attached to a certain cusp form g. In Mishou [154], this result was generalized to thecase of several L-functions belonging to the Selberg class.
The result (4.4) is weaker than the joint universality, because K1 andK2 are in the strips disjoint to each other. In [156], Mishou succeeded inremoving this restriction to obtain the following theorem. Let g, g1 , g2
be holomorphic normalized Hecke-eigen cusp forms.
Theorem 9. (Mishou [156]) (i) ζ(s) and L(s, g) are jointly universalin D(1/2, 1).
(ii) If g1 and g2 are distinct, then L(s, g1 ) and L(s, g2 ) are jointly
universal in D(1/2, 1).
(iii) ζ(s) and L(s, sym2 g) are jointly universal in D(2/3, 1).
(iv) If g1 and g2 are distinct, then ζ(s) and L(s, g1 ⊗ g2 ) are jointly
universal in D(3/4, 1).
(v) L(s, g1 ) and L(s, g1 ⊗ g2 ) are jointly universal in D(3/4, 1).
Remark 5. The universality theorem for L(s, g ⊗ g) by the author [134](mentioned in Section 3) was proved in D(3/4, 1), but the above theorem of Mishou especially implies that the universality for L(s, g ⊗ g)
is valid in the wider region D(2/3, 1).
A remarkable feature of Mishou’s method is that it does not dependon any periodicity of coeﬃcients. His proof is based on orthogonality
relations of Fourier coeﬃcients. Theorem 9 is a strong support toConjecture 1.
5. The strong universality
So far we have talked about universality only for zeta and L-functions
with Euler products. However already in the first decade, the universality for zeta-functions without Euler products was also studied. Let
0 < α ≤ 1. The
with the parameter α is defined∑ Hurwitz zeta-function−sby ζ(s, α) = ∞
(n
+
α)
,
and
does
not
have the Euler product (exn=0
cept for the special cases α = 1, 1/2). The known universality theoremfor ζ(s, α) is as follows.
Theorem 10. (Bagchi [3], Gonek [40]) Let K be a compact subset ofD(1/2, 1) with connected complement, and f ∈ H c (K). Then for any
ε > 0,}{
1(5.1) lim inf meas τ ∈ [0, T ] sup |ζ(s + iτ, α) − f (s)| < ε > 0
T →∞ Ts∈K
holds, provided α is transcendental or rational (̸= 1, 1/2).

14

KOHJI MATSUMOTO

To prove this theorem, when α is transcendental, we use the factthat the elements of the set{log(n + α) | n ∈ N0 }are linearly independent over Q. On the other hand, when α = a/b isrational, then in view of the formula
∑
bs
(5.2)
ζ(s, a/b) =χ(a)L(s, χ),φ(b)χ(mod b)we can reduce the problem to the joint universality of Dirichlet Lfunctions, so we can apply Theorem 7. The case of algebraic irrational
α is still open.
A remarkable point is that, in the statement of Theorem 10, we
do not assume that the target function f (s) is non-vanishing on K.
This is a big diﬀerence from Theorem 1, and when a universality-type
theorem holds without the non-vanishing assumption, we call it a strong
universality theorem.
Strong universality has an important application to the theory ofzero-distribution. Let a < σ1 < σ2 < b, and let
N (t; σ1 , σ2 ; φ) = #{ρ ∈ C | σ1 ≤ ℜρ ≤ σ2 , 0 ≤ ℑρ ≤ T, φ(ρ) = 0}
for a function φ. (In the above definition, zeros are counted withmultiplicity.) Then we have the following consequence.
Theorem 11. If φ(s) is strongly universal in the region D(a, b), thenthere exists a positive constant C for which
N (T ; σ1 , σ2 ; φ) ≥ CT

(5.3)

holds for any σ1 , σ2 satisfying a < σ1 < σ2 < b.
Proof. Let δ be a small positive number, 0 < ε < δ, σ1 < σ0 < σ2 , andK = {s ∈ C | |s − σ0 | < δ}. We choose δ so small that K ⊂ D(a, b).
We apply the strong universality to this K, f (s) = s − σ0 and ε toobtain that the set of real numbers τ such thatsup |φ(s + iτ ) − f (s)| < ε|s−σ0 |≤δis of positive lower density. Then, for such τ ,sup |φ(s + iτ ) − f (s)| < δ =|s−σ0 |≤δinf

|s−σ0 |=δ|f (s)|.

Therefore by Rouché’s theorem we see that f (s) + (φ(s + iτ ) − f (s)) =φ(s + iτ ) has the same number of zeros as that of f (s) in the region|s − σ0 | < δ, but the latter is obviously 1. That is, for each τ in theabove set, φ(s) has one zero in |s − (σ0 + iτ )| < δ.
□THE THEORY OF UNIVERSALITY FOR ZETA AND L-FUNCTIONS

15

Corollary 1. If α is transcendental or rational (̸= 1, 1/2), thenC1 T ≤ N (T ; σ1 , σ2 ; ζ(s, α)) ≤ C2 T
holds for any 1/2 < σ1 < σ2 < 1.
As for the upper bound part of this corollary, see [102, Chapter 8,
Theorem 4.10].
Further topics on the application of universality to the distributionof zeros will be discussed in Section 9 and Section 15.
Now strong unversality theorems are known for many other zetafunctions. The Estermann zeta-function is defined by
(
) ∑
)
(
∞
k
k
E s, , α =(5.4)
σα (n) exp 2πi n n−s ,
l
l
n=1
∑
where k and l are coprime integers and σα (n) = d|n dα . The strong
universality for E(s, k/l, α) was studied in Garunkštis, Laurinčikas,Šleževičienė and J. Steuding [36]. The∑method is to write E(s, k/l, α) as−sa linear combination of E(s, χ, α) = ∞n=1 σα (n)χ(n)n , and apply ajoint universality theorem for E(s, χ, α) which follows from Šleževičienė
[194].
The Lerch zeta-function is defined byζ(s; α, λ) =∞∑e2πiλn (n + α)−s ,n=0

where 0 < α ≤ 1 and λ is real. When λ is an integer, then ζ(s; α, λ)reduces to the Hurwitz zeta-function, so we may assume 0 < λ < 1.
The strong universality for ζ(s; α, λ) was proved by Laurinčikas [77]when α is transcendental. The case when α is rational was discussed
by Laurinčikas [80]. See also the textbook [102] of Laurinčikas andGarunkštis.
Let B = {bn } is a periodic sequence, not necessarily ∑multiplica−s
tive. The universality for periodic zeta-functions ζ(s, B) = ∞n=1 bn n
was first studied by J. Steuding [198] (see [201, Chapter 11]). Kaczorowski [60] proved that there exists a constant c0 = c0 (B) such thatthe universality holds for ζ(s, B), provided thatmax ℑ(s) − min ℑ(s) ≤ c0 .
s∈K

s∈K

This result is a consequence of the hybrid joint universality theoremof Kaczorowski and Kulas [61] (see Section 15). Javtokas and Laurinčikas [49] [50] studied the strong universality for periodic Hurwitz

16

KOHJI MATSUMOTO

zeta-function
ζ(s, α, B) =∞∑bn (n + α)−s .

n=0They proved that the strong universality holds for ζ(s, α, B), when α
is transcendental.
A more general situation was considered by Laurinčikas, Schwarz and
J. Steuding [121]. Let {λn }∞n=1 be an increasing sequence of real numbers, linearly independent over Q,∑∞and λn → ∞ as n → ∞. Define thegeneral Dirichlet series f (s) = n=1 an exp(−λn s), which is assumed∑
to be convergent absolutely in the region σ > σa . Put r(x) = λn ≤x 1
and cn = an exp(−λn σa ). We suppose(i) f (s) cannot be represented as an Euler product,(ii) f (s) can be continued meromorphically to σ > σ1 , and holomorphic in D(σ1 , σa ),(iii) For σ > σ1 it holds that f (s) = O(|t|α ) with some α > 0,(iv) For σ > σ1 it holds that∫ T
|f (σ + it)|2 dt = O(T ),−Tκ(v) r(x) = Cx + O(1) with∑ a κ > 1,(vi) |cn | is bounded and λn ≤x |cn |2 = θr(x)(1 + o(1)) with a θ > 0.
Then we have
Theorem 12. (Laurinčikas, Schwarz and J. Steuding [121]) If f (s)satisfies all the above conditions, then the strong universality holds forf (s) in the region D(σ1 , σa ).
In Section 3 and Section 5, we have seen a lot of examples of zeta
and L-functions, with or without Euler products, for which the universality property holds. How general is this property expected to hold?
The following conjecture predicts that any “reasonable” Dirichlet series
would satisfy the universality property.
Conjecture 2. (Yu. V. Linnik and I. A. Ibragimov) All functions
given by Dirichlet series and meromorphically continuable to the leftof the half-plane of absolute convergence are universal in some suitableregion.
Remark 6. Actually this conjecture has trivial counter-examples. For
example,= 1 if n is a power of 2 and an = 0 otherwise. The∑ let an −sancan be continued to (2s − 1)−1 , which is obviously
series ∞n=1 n
not universal. Therefore some additional condition should be added tomake the rigorous statement of the above conjecture.

THE THEORY OF UNIVERSALITY FOR ZETA AND L-FUNCTIONS

17

6. The joint strong universality
The joint universality property is also possible to be valid amongzeta-functions without Euler products. The first attempt to this direction is a series of papers of Laurinčikas and the author [110] [114] [115]on the joint universality for Lerch zeta-functions. Here, a matrix
condition again appears. Let λ1 , . . . , λr be rational numbers. Write
λj = aj /qj , (aj , qj ) = 1, and let k be the least common multiple ofq1 , . . . , qr . Define the matrix L = (exp(2lπiλj ))1≤l≤k,1≤j≤r . Then, by
virtue of a variant of the positive density method, we have
Theorem 13. (Laurinčikas and Matsumoto [110] [114]) Suppose that
α1 , . . . , αr are algebraically independent over Q, and that rank(L) = r.
Then the joint strong universality holds for ζ(s, α1 , λ1 ), . . . , ζ(s, αr , λr )in the region D(1/2, 1)4 .
The Lerch zeta-function ζ(s, α, λ) with rational λ is a special caseof periodic Hurwitz zeta-functions. The joint strong universality of
periodic Hurwitz zeta-functions was first studied by Laurinčikas [88][89]. Let Bj = {anj }∞n=0 be periodic sequences with period kj , k be theleast common multiple of k1 , . . . , kr , and define B = (ajl )1≤j≤r,1≤l≤k .
Theorem 14. (Laurinčikas [89]) If α is transcendental and rank(B) =r, then the joint strong universality holds for ζ(s, α, B1 ), . . . , ζ(s, α, Br )in the region D(1/2, 1).
Next in [51] [124], the joint universality for ζ(s, αj , Bj ) (1 ≤ j ≤r) was discussed. Some matrix conditions were still assumed in [51],but finally in [124], a joint universality theorem free from any matrix
condition was obtained.
Theorem 15. (Laurinčikas and Skerstonaitė [124]) Assume that theelements of the set(6.1){log(n + αj ) | 1 ≤ j ≤ r, n ∈ N0 }are linearly independent over Q. Then the joint strong universality
holds for ζ(s, α1 , B1 ), . . . , ζ(s, αr , Br ) in the region D(1/2, 1).
In the case Bj = {1}∞n=0 for 1 ≤ j ≤ r (that is, the case of Hurwitz
zeta-functions), the above result was already given in Laurinčikas [91].
In [90] [125], a more general joint strong universality for ζ(s, αj , Bjl )(1 ≤ j ≤ r, 1 ≤ l ≤ lj with lj ∈ N) was discussed under certain matrix
conditions.
4In [110], the theorem is stated under the weaker assumption that α , . . . , α1r aretranscendental, but (as is pointed out in [114]) this assumption should be replacedby the algebraic independence over Q.

18

KOHJI MATSUMOTO

Laurinčikas [84] [85] and [38] (with Genys) studied the joint strong
universality for general Dirichlet series, under the same assumptions asin Theorem 12 and a certain matrix condition.
Now return to the problem of the joint universality for Lerch zetafunctions. Theorem 15 implies, especially, that the assumptions ofTheorem 13 can now be replaced by just the linear independence of(6.1).
Is the assumption (6.1) indeed weaker than the assumptions of Theorem 13? The answer is yes, and the following result of Mishou [152]gives an example: Let α1 , α2 be two transcendental numbers, 0 <α1 , α2 < 1, α1 ̸= α2 , and α2 ∈ Q(α1 ). Then Mishou [152] provedthat the joint strong universality holds for ζ(s, α1 ) and ζ(s, α2 ). Dubickas [24] extended Mishou’s result to the case of r transcendental
numbers, which is also an extension of [91].
Let m1 , m2 , m3 be relatively prime positive integers (≥ 2), and λ0 =n3 /m3 (with another integer n3 ). Nakamura [167] proved the joint
strong universality for
{ (
)
}
n1
n2
ζ s, α, λ0 ++0 ≤ n1 < m1 , 0 ≤ n2 < m2
(6.2)
m1 m2
when α is transcendental. He pointed out that various other types ofjoint universality can be deduced from the above.
In [115] and in Laurinčikas [92], the joint universality of ζ(s, αj , λjµj )(1 ≤ j ≤ r, 1 ≤ µj ≤ mj , where mj is some positive integer) is
discussed. Write λjµj = ajµj /qjµj , (ajµj , qjµj ) = 1, and let kj be
the least common multiple of qjµj (1 ≤ µj ≤ mj ). Define Lj =(exp(2lπiλjµj )1≤l≤kj ,1≤µj ≤mj ). Then in [92] it is shown that if the elements of the set (6.1) are linearly independent over Q, and rank(Lj ) =kj (1 ≤ j ≤ r), then the joint strong universality holds for ζ(s, αj , λjµj ).
How about the joint universality for Lerch zeta-functions when theparameter λ is not rational? Nakamura [167] noted that the joint strong
universality for (6.2) also holds if we replace λ0 by any non-rational real
number. Also, Nakamura [168] extended the idea in [114] to obtain thefollowing more general result.
Theorem 16. (Nakamura [168]) If α1 , . . . , αr are algebraically independent over Q, then for any real numbers λ1 , . . . , λr the joint strong
universality holds for ζ(s, α1 , λ1 ), . . . , ζ(s, αr , λr ) in the region D(1/2, 1).
On the other hand, Mishou [155] proved the following measuretheoretic result.
Theorem 17. (Mishou [155]) There exists a subset Λ ⊂ [0, 1)r whoser-dimensional Lebesgue measure is 1, and for any transcendental realTHE THEORY OF UNIVERSALITY FOR ZETA AND L-FUNCTIONS

19

number α and (λ1 , . . . , λr ) ∈ Λ, the joint strong universality holds forζ(s, α, λ1 ), . . . , ζ(s, α, λr ) in the region D(1/2, 1).
Moreover in the same paper Mishou gives the following two explicit
descriptions of λ1 , . . . , λr (0 ≤ λj < 1) for which the above joint universality holds;(i) λ1 , . . . , λr are algebraic irrational and 1, λ1 , . . . , λr are linearly
independent over Q,(ii) λ1 = exp(u1 ), . . . , λr = exp(ur ) where u1 , . . . , ur are distinct
rational numbers.
Mishou’s proof is based on two classical discrepancy estimates due
to W. M. Schmidt and H. Niederreiter. These results lead Mishou topropose the following conjecture.
Conjecture 3. (Mishou [155]) The joint strong universality holds forζ(s, α, λ1 ), . . . , ζ(s, α, λr ) in the region D(1/2, 1), for any transcendental real number α (0 < α < 1) and any distinct real numbers λ1 , . . . , λr
(0 ≤ λj < 1).
For single zeta or L-functions, there is Conjecture 2, which assertsthat universality would hold for any “reasonable” Dirichlet series. As
for the joint universality, the situation is much more complicated. If
there is some relation among several Dirichlet series, then the behaviourof those Dirichlet series cannot be independent of each other, so thejoint universality among them cannot be expected. Nakamura [168]pointed out that some collections of Lerch zeta-functions cannot be
jointly universal, because of the inversion formula among Lerch zetafunctions. In the same paper Nakamura introduced the generalized
Lerch zeta-function of the form∞∑
e2πiλnζ(s, α, β, γ, λ) =(6.3),
s−γ (n + β)γ
(n
+
α)
n=0
and showed that, under suitable choices of parameters the joint strong
universality sometimes holds, and sometimes does not hold.
In [169], Nakamura considered more general series(6.4)

ζ(s, α, C) =

∞
∑

c(n)
,
s
(n
+
α)
n=0

where C = {c(n)}∞n=0 is a bounded sequence of complex numbers, andusing it, constructed some counter-examples to the joint universality.
The proof in [169] is based on a non-denseness property and a limit
theorem on a certain function space.

20

KOHJI MATSUMOTO

The above results of Nakamura suggest that it is not easy to find asuitable joint version of Conjecture 2.
Remark 7. It is to be noted that there is the following simple principle
of producing the joint universality. Let φ ∈ H(D(σ1 , σ2 )), and assume
that φ is universal. Let K be a compact subset of D(σ1 , σ2 ) withconnected complement, λ1 , . . . , λr be complex numbers, and Kj = {s +λj | s ∈ K} (1 ≤ j ≤ r). Assume these Kj ’s are disjoint. Thenφj (s) = φ(s + λj ) (1 ≤ j ≤ r) are jointly universal. If φ is strongly
universal, then φj ’s are jointly strongly universal. This is the shifts
universality principle of Kaczorowski, Laurinčikas and J. Steuding [62].

7. The universality for multiple zeta-functions
An important generalization of the notion of zeta-functions is multiple zeta-functions, defined by certain multiple sums. The history of
the theory of multiple zeta-functions goes back to the days of Euler,
but extensive studies started only in 1990s.
The problem of searching for universality theorems on multiple zetafunctions was first proposed by the author [135]. In this paper theauthor wrote that one accessible problem would be the universality ofBarnes multiple zeta-functions
∞
∑
n1 =0

···

∞
∑

(w1 n1 + · · · + wr nr + α)−snr =0(where w1 , . . . , wr , α are parameters). Nakamura [168] pointed out thathis ζ(s, α, β, γ, λ) (see (6.3)) includes the twisted Barnes double zetafunction of the form
∞ ∑∞∑e2πiλ(n1 +n2 )(n1 + n2 + α)sn =0 n =0
1

2

as a special case. Therefore [168] includes a study of universality for
Barnes double zeta-functions. More generally, the series ζ(s, α, C) (see(6.4)) studied in Nakamura [169] includes twisted Barnes r-ple zetafunctions (for any r).
The Euler-Zagier r-ple sum is defined by
∑
1 −s2
r
(7.1)
n−s
· · · n−s
1 n2
r ,n1 >n2 >···nr ≥1

THE THEORY OF UNIVERSALITY FOR ZETA AND L-FUNCTIONS

21

where s1 , . . . , sr are complex variables. Nakamura [170] considered theuniversality of the following generalization of (7.1) of Hurwitz-type:(7.2)

ζr (s1 , . . . , sr ; α1 , . . . , αr ) =∑(n1 + α1 )−s1 (n2 + α2 )−s2 · · · (nr + αr )−sr ,n1 >n2 >···nr ≥0

where 0 < αj ≤ 1 (1 ≤ j ≤ r). Nakamura’s results suggest that
universality for the multiple zeta-function (7.2) is connected with thezero-free region. One of his main results is as follows.
Theorem 18. (Nakamura [170]) Let ℜs2 > 3/2, ℜsj ≥ 1 (3 ≤ j ≤ r).
Assume α1 is transcendental, and ζr−1 (s2 , . . . , sr ; α2 , . . . , αr ) ̸= 0. Thenthe strong universality holds for ζr (s1 , . . . , sr ; α1 , . . . , αr ) as a function
in s1 in the region D(1/2, 1).
In [173], Nakamura considered a generalization of Tornheim’s double
sum of Hurwitz-type and proved the strong universality for it.
8. The mixed universality
In Section 4 we discussed the joint universality among zeta or Lfunctions with Euler products. Then in Section 6 we considered thejoint universality for those without Euler products. Is it possible tocombine these two directions to obtain certain joint universality results
between two (or more) zeta-functions, one of which has Euler products
and the other does not? The first aﬃrmative answers are due to Sander
and J. Steuding [191], and to Mishou [149]. The work of Sander and J.
Steuding will be discussed later in Section 15. Here we state Mishou’s
theorem.
Theorem 19. (Mishou [149]) Let K1 , K2 be compact subsets of D(1/2, 1)
with connected complements, and f1 ∈ H0c (K1 ), f2 ∈ H c (K2 ). Then,for any ε > 0, we have{1
lim inf meas τ ∈ [0, T ] sup |ζ(s + iτ ) − f1 (s)| < ε,(8.1)
T →∞ Ts∈K1}
sup |ζ(s + iτ, α) − f2 (s)| < ε > 0,s∈K2

provided α is transcendental.
This type of universality is now called the mixed universality. The
essential point of the proof of this theorem is the fact that the elementsof the set{log(n + α) | n ∈ N0 } ∪ {log p | p : prime}

22

KOHJI MATSUMOTO

are linearly independent over Q.
Mishou’s theorem was generalized to the periodic case by Kačinskaitė
and Laurinčikas [58]5. Let A be a multiplicative periodic sequence satisfying (3.1) and B a (not necessarily multiplicative) periodic sequence.
They proved that if α is transcendental, then the mixed universality
holds for ζ(s, A) and ζ(s, α, B) in the region D(1/2, 1).
Laurinčikas [93] proved a further generalization. Let A1 , . . . , Ar1 be
multiplicative periodic sequences (with inequalities similar to (3.1))
and B1 , . . . , Br2 be periodic sequences. Then, under certain matrix
conditions, the mixed universality for ζ(s, Aj1 ) (1 ≤ j1 ≤ r1 ) andζ(s, αj2 , Bj2 ) (1 ≤ j2 ≤ r2 ) holds, provided α1 , . . . , αr2 are algebraicallyindependent over Q.
Now mixed universality theorems are known for many pairs of zeta
or L-functions.
• The Riemann zeta-function and several periodic Hurwitz zetafunctions (Genys, Macaitienė, Račkauskienė and Šiaučiūnas [39]),• The Riemann zeta-function and several Lerch zeta-functions (Laurinčikas and Macaitienė [106]),• Several Dirichlet L-functions and several Hurwitz zeta-functions(Janulis and Laurinčikas [47]),• Several Dirichlet L-functions and several periodic Hurwitz zetafunctions (Janulis, Laurinčikas, Macaitienė and Šiaučiūnas [48]),• An automorphic L-function and several periodic Hurwitz zetafunctions (Laurinčikas, Macaitienė and Šiaučiūnas [109], Macaitienė
[131], Pocevičienė and Šiaučiūnas [186], Laurinčikas and Šiaučiūnas[123]).
9. The strong recurrence
In Section 5 we mentioned that the strong universality implies theexistence of many zeros in the region where universality is valid (Theorem 11). This immediately gives the following corollary:
Corollary 2. The Riemann zeta-function ζ(s) cannot be strongly universal in the region D(1/2, 1).
Because if ζ(s) is strongly universal, then by Theorem 11 we have
N (T ; σ1 , σ2 ; ζ) ≥ CT for 1/2 < σ1 < σ2 < 1, which contradicts withthe known zero-density estimate N (T ; σ1 , σ2 ; ζ) = o(T ).
The same conclusion can be shown for many other zeta or L-functions,for which some suitable zero-density estimate is known; or, under theassumption of the analogue of the Riemann hypothesis.
5This paper was published in 2011, but was already completed in 2007.

THE THEORY OF UNIVERSALITY FOR ZETA AND L-FUNCTIONS

23

On the other hand, if the Riemann hypothesis is true, then ζ(s) has
no zero in the region D(1/2, 1). Therefore we can choose f (s) = ζ(s)in Theorem 1 to obtain{}1(9.1) lim inf meas τ ∈ [0, T ] sup |ζ(s + iτ ) − ζ(s)| < ε > 0.
T →∞ Ts∈K
This is called the strong recurrence property of ζ(s). Bagchi discovered that the converse implication is also true.
Theorem 20. (Bagchi [3]) The Riemann hypothesis for ζ(s) is true if
and only if (9.1) holds in the region D(1/2, 1).
Bagchi himself extended this result to the case of Dirichlet L-functionsin [4] [5]. The same type of result in terms of Beurling zeta-functions
was given by R. Steuding [203].
Remark 8. It is obvious that the notion of the strong recurrence isclosely connected with the notion of almost periodicity. Bohr [12]proved that the Riemann hypothesis for L(s, χ) with a non-principal
character χ is equivalent to the almost periodicity of L(s, χ) in the region ℜs > 1/2. Recently Mauclaire [138] [139] studied the universality
in a general framework from the viewpoint of almost periodicity.
Nakamura [171] proved that, if d1 = 1, d2 , . . . , dr are algebraic real
numbers which are linearly independent over Q, then the joint universality of the form{1
lim inf meas τ ∈ [0, T ] sup |ζ(s + idj τ ) − fj (s)| < ε(9.2)
T →∞ Ts∈Kj}(1 ≤ j ≤ r)>0

holds, where Kj are compact subsets of D(1/2, 1) with connected complement and fj ∈ H0c (Kj ). A key point of Nakamura’s proof is thefact that the elements of the set {log pdj | p : prime, 1 ≤ j ≤ r} arelinearly independent over Q, which follows from Baker’s theorem intranscendental number theory. From the above result it is immediate
that
(9.3)
}
{
1
lim inf meas τ ∈ [0, T ] sup |ζ(s + iτ ) − ζ(s + idτ )| < ε > 0
T →∞ Ts∈K
holds if d is algebraic irrational. Nakamura also proved in the same
paper that (9.3) is valid for almost all d ∈ R. (Note that (9.3) for
d = 0 is (9.1), hence the Riemann hypothesis.)

24

KOHJI MATSUMOTO

Nakamura’s paper sparked oﬀ the interest in this direction of research; Pańkowski [182] proved that (9.3) holds for all (algebraic andtranscendental) irrational d, using the six exponentials theorem in transcendental number theory. On the other hand, Garunkštis [34] andNakamura [172], independently, claimed that (9.3) holds for all nonzero rational. However their arguments included a gap, which was partially filled by Nakamura and Pańkowski [176]. The present situation
is:Theorem 21. (Garunkštis, Nakamura, Pańkowski) The inequality (9.3)
holds if d is irrational, or d = a/b is non-zero rational with (a, b) = 1,
|a − b| ̸= 1.
See also Mauclaire [140], and Nakamura and Pańkowski [178]. It is
to be noted that the argument of Garunkštis [34] and Nakamura [172]is correct for log ζ(s), so
(9.4)

{
1
lim inf meas τ ∈ [0, T ]T →∞ T}
sup | log ζ(s + iτ ) − log ζ(s + idτ )| < ε

>0

s∈K

can be shown for any non-zero d ∈ R. If (9.4) would be valid for d = 0,
it would imply the Riemann hypothesis.
The strong recurrence property can be shown for more general zeta
and L-functions. Some of the aforementioned papers actually consider
not only the Riemann zeta-function, but also Dirichlet L-functions.
A generalization to a subclass of the Selberg class was discussed byNakamura [174]. The case of Hurwitz zeta-functions has been studied
by Garunkštis and Karikovas [35] and Karikovas and Pańkowski [65].
10. The weighted universality
Laurinčikas [75] considered a weighted version of the universalityfor ζ(s). Let w(t) be a positive-valued function of bounded variation
defined on [T0 , ∞) (where T0 > 0), satisfying that the variation on[a, b] does not exceed cw(a) with a certain c > 0 for any subinterval[a, b] ⊂ [T0 , ∞). Define∫ T
U (T, w) =w(t)dt,T0

and assume that U (T, w) → ∞ as T → ∞.
We further assume the following property of w(t), connected withergodic theory. Let X(τ, ω) be any ergodic process defined on a certain
probability space Ω, τ ∈ R, ω ∈ Ω, E(|X(τ, ω)|) < ∞, and sample

THE THEORY OF UNIVERSALITY FOR ZETA AND L-FUNCTIONS

25

paths are Riemann integrable almost surely on any finite interval. Assume that∫ T1
(10.1)w(τ )X(t + τ, ω)dτ = E(X(0, ω)) + o((1 + |t|)α )U (T, w) T0
almost surely for any t ∈ R, with an α > 0, as T → ∞. Denote by
I(A) the indicator function of the set A.
Theorem 22. (Laurinčikas [75]) Suppose that w(t) satisfies all theabove conditions. Let K be a compact subset of D(1/2, 1) with connected complement, f ∈ H0c (K). Then
∫ T
1
(10.2) lim infw(τ )T →∞ U (T, w) T
0
({
})
×I
τ ∈ [T0 , T ] sup |ζ(s + iτ ) − f (s)| < εdτ > 0s∈Kholds for any ε > 0.
In the course of Bagchi’s proof of the universality theorem, there isa point where the Birkhoﬀ-Khinchin theorem
∫
1 T
lim
(10.3)X(τ, ω)dτ = E(X(0, ω))T →∞ T 0
in ergodic theory is used. This is the motivation of Laurinčikas [75];clearly (10.1) is a generalization of (10.3).
Laurinčikas [78] generalized Theorem 22 to the case of Matsumoto
zeta-functions. Weighted universality theorems for L-functions L(s, E)of elliptic curves over Q were reported by Garbaliauskienė [26] [27].
11. The discrete universality
In the previous sections, we discussed the behaviour of zeta or Lfunctions when the imaginary part τ of the variable is moving continuously. However, we can also obtain a kind of universality theorems
when τ only moves discretely. We already mentioned in Section 1 that
Voronin’s multi-dimensional denseness theorem is valid in this sense(see Remark 3).
The first discrete universality theorem is due to Reich [189] on Dedekind
zeta-functions. Let F be a number field and ζF (s) the associatedDedekind zeta-function.
Theorem 23. (Reich [189]) Let K be a compact subset of the region
D(1 − max{2, dF }−1 , 1) with connected complement, and f ∈ H0c (K).

26

KOHJI MATSUMOTO

Then, for any real h ̸= 0 and any ε > 0,{}1
(11.1)
lim inf # n ≤ N sup |ζF (s + ihn) − f (s)| < ε > 0.
N →∞ Ns∈K
The joint discrete universality theorem for Dirichlet L-functions wasgiven in Bagchi [3]. He also obtained the discrete universality for ζ(s, α)when α is rational, for which Sander and J. Steuding [191] gave adiﬀerent approach.
Kačinskaitė [54] (see also [56]) proved a discrete universality theorem
for Matsumoto zeta-functions under the condition that exp(2πk/h) is
irrational for any non-zero integer k. Ignatavičiūtė [46] reported thatcertain discrete universality and certain joint discrete universality holdfor Lerch zeta-functions, provided exp(2π/h) is rational.
As can be seen in the above examples, an interesting point on discrete
universality is that the arithmetic nature of the parameter h plays arole.
The discrete universality for L-functions L(s, E) of elliptic curves was
first studied in [30] under the assumption that exp(2πk/h) is irrational
for any non-zero integer k. However this condition was then removed:
Theorem 24. (Garbaliauskienė, Genys and Laurinčikas [28]) The discrete universality holds for L(s, E) for any real h ̸= 0 in the region
D(1, 3/2).
The same type of result can be shown, more generally, for L-functions
attached to new forms. When exp(2πk/h) is irrational for any non-zero
integer k, this was done in Laurinčikas, Matsumoto and J. Steuding[117].
The discrete universality for periodic zeta-functions was studied in
Kačinskaitė, Javtokas and Šiaučiūnas [57] and Laurinčikas, Macaitienėand Šiaučiūnas [108], while the case of periodic Hurwitz zeta-functions
was discussed by Laurinčikas and Macaitienė [104]. The result in[104] especially includes the discrete universality of the Hurwitz zetafunction ζ(s, α) when α is transcendental. Laurinčikas [100] further
proved that if the set{log(m + α) | m ∈ N0 } ∪ {2π/h}
is linearly independent over Q, then the discrete universality holds forζ(s, α). See also [18]. A joint version is studied in Laurinčikas [101].
Macaitienė [130] obtained a discrete universality theorem for general
Dirichlet series.
Theorem 25. (Macaitienė [130]) Let f (s) be general Dirichlet series
as in Theorem 12, and further suppose that λn are algebraic numbers

THE THEORY OF UNIVERSALITY FOR ZETA AND L-FUNCTIONS

27

and exp(2π/h) ∈ Q. Then the discrete universality holds for f (s) in
the region D(σ1 , σa ).
The discrete analogue of mixed universality can also be considered.
This direction was first studied by Kačinskaitė [55]. Consider the caseexp(2π/h) ∈ Q. Write exp(2π/h) = a/b with a, b ∈ Z and (a, b) = 1.
Denote by Ph the set of all prime numbers appearing as a prime factor
of a or b. Define the modified Dirichlet L-function Lh (s, χ) by removing
all Euler factors corresponding to primes in Ph , that is
)−1
∏(
χ(p)
Lh (s, χ) =
1− s
.
p
p∈P
/ h

Theorem 26. (Kačinskaitė [55]) Let K1 , K2 be compact subsets ofD(1/2, 1) with connected complements, and f1 ∈ H0c (K1 ), f2 ∈ H c (K2 ).
If α is transcendental, B is a periodic sequence and exp(2π/h) ∈ Q as
above, then{1
lim inf # n ≤ N sup |Lh (s + ihn, χ) − f1 (s)| < ε,(11.2)
N →∞ Ns∈K1}
sup |ζ(s + ihn, α, B) − f2 (s)| < ε > 0s∈K2

for any ε > 0

6.

Buivytas and Laurinčikas [16] proved that if the set{log p | p : prime} ∪ {log(m + α) | m ∈ N0 } ∪ {2π/h}
is linearly independent over Q, then the discrete mixed universality
holds for ζ(s) and ζ(s, α), that is,{1
lim inf # n ≤ N sup |ζ(s + ihn) − f1 (s)| < ε,(11.3)
N →∞ Ns∈K1}
sup |ζ(s + ihn, α) − f2 (s)| < ε > 0.
s∈K2

In (11.2) and (11.3), the shifting parameter h is common to the bothof relevant zeta (or L)-functions. Buivytas and Laurinčikas [17] studied
the case when the parameter for ζ(s) is diﬀerent from the parameterfor ζ(s, α).
We will encounter a rather diﬀerent type of discrete universality theorems in Section 18.
6The statement in [55] is given for L(s, χ), but her argument is valid not forL(s, χ), but for Lh (s, χ).

28

KOHJI MATSUMOTO

12. The χ-universality
The main point of Voronin’s universality theorem is the existenceof τ , the imaginary part of the complex variable s, satisfying a certain
approximation condition. This is the common feature of all universality
theorems mentioned in the previous sections. However there is anothertype of universality, which is concerned with the existence of a charactersatisfying certain approximation conditions.
The first theorem in this direction is as follows.
Theorem 27. (Bagchi [3], Gonek [40], Eminyan [25]) Let K be a compact subset of D(1/2, 1) with connected complement, and f ∈ H0c (K).
Let Q be an infinite set of positive integers. Then for any ε > 0,
(12.1){1# χ(mod q)
lim
inf
q→∞
φ(q)
q∈Q

}
sup |L(s, χ) − f (s)| < ε

>0

s∈K

holds, provided Q is one of the following7:(i) Q is the set of all prime numbers;(ii) Q is the set of positive integers of the form q = pa11 · · · par r(a1 , . . . , ar ∈ N ∪ {0}), where {p1 , . . . , pr } is a fixed finite set of prime
numbers.
This type of results is called the χ-universality, or the universalityin χ-aspect. The universality for Hecke L-functions of number fields inχ-aspect was discussed by Mishou and Koyama [157], and by Mishou[147] [148].
Let χd be a real Dirichlet character with discriminant d. Anotherinteresting direction of research is the universality for L(s, χd ) in daspect. This direction was studied in a series of papers by Mishou andNagoshi [158] [159] [160] [161]. Let Λ+ (resp. Λ− ) be the set of all
positive (resp. negative) discriminants, and Λ+ (X) (resp. Λ− (X)) be
the set of discriminants d satisfying 0 < d ≤ X (resp. −X ≤ d < 0).
Theorem 28. (Mishou and Nagoshi [158]) Let Ω be a simply connecteddomain in D(1/2, 1) which is symmetric with respect to the real axis.
Let f (s) be holomorphic and non-vanishing on Ω, and positive-valued
7Bagchi [3, Theorem 5.3.11] proved the case (i).

In the statement of Gonek[40, Theorem 5.1] there is no restriction on Q, but Mishou [146] pointed out thatcondition (ii) is necessary to verify Gonek’s proof. Eminyan [25] studied the special
case r = 1 of (ii).

THE THEORY OF UNIVERSALITY FOR ZETA AND L-FUNCTIONS

29

on Ω ∩ R. Let K be a compact subset of Ω. Then, for any ε > 0,
(12.2){
1# d ∈ Λ± (X)lim inf
X→∞ #Λ± (X)}sup |L(s, χd ) − f (s)| < ε

>0

s∈K

holds.
This theorem especially implies that for any s ∈ D(1/2, 1) \ R, theset {L(s, χd ) | d ∈ Λ± } is dense in C, and for any real number σ with
1/2 < σ < 1, the set {L(σ, χd ) | d ∈ Λ± } is dense in the set of positive
real numbers R+ .
In the same paper Mishou and Nagoshi also studied the situation on
the line ℜs = 1, and proved that the set {L(1, χd ) | d ∈ Λ± } is dense
in R+ . Therefore we can deduce denseness results on√class numbersof quadratic fields. Let h(d) be the class number√of Q( d), and whend > 0, let ε(d) be the fundamental unit of Q( d). Then the above
result implies that both the sets{}{}h(d) log ε(d)
h(d)
+
−
√
√
d∈Λ
,
d∈Λ
d
d
are dense in R+ .
In [159] [161], the same type of problem for prime discriminants
are studied. As an application, in [161] Mishou and Nagoshi gave aquantitative result on a problem of Ayoub, Chowla and Walum on
certain character sums [2]. In [160] Mishou and Nagoshi gave someconditions equivalent to the Riemann hypothesis from their viewpoint.
The universality in d-aspect for Hecke L-functions of class groupcharacters for imaginary quadratic fields are studied by Mishou [151].
In √[150], Mishou considered cubic characters associated with the fieldQ( −3), and proved a universality theorem for associated Hecke Lfunctions in cubic character aspect.
13. Other applications
So far we mentioned applications of universality to the theory of
distribution of zeros (Section 5), to the Riemann hypothesis (Section9), and to algebraic number theory (Section 12). Those applications,however, do not exhaust the potentiality of universality theory. In thissection we discuss other applications of universality.
In Section 1 we mentioned that Voronin, before proving his universality theorem, obtained the multi-dimensional denseness theorem(Theorem 2) of ζ(s) and its derivatives. However, now, we can saythat Theorem 2 is just an immediate consequence of the universality

30

KOHJI MATSUMOTO

theorem (see [76, Theorem 6.6.2]). Moreover, from Theorem 2 it is easy
to obtain the following functional-independence property of ζ(s).
Theorem 29. Let fl : Cm → C (0 ≤ l ≤ n) be continuous functions,and assume that the equality(13.1)

n∑

sl fl (ζ(s), ζ ′ (s), . . . , ζ (m−1) (s)) = 0

l=0

holds for all s. Then fl ≡ 0 for 0 ≤ l ≤ n.
This type of result was first noticed by Voronin himself ( [206] [209]);
see also [76, Theorem 6.6.3].
When fl ’s are polynomials, then (13.1) is an algebraic diﬀerential
equation. Therefore Theorem 29 in this case implies that ζ(s) does notsatisfy any non-trivial algebraic diﬀerential equation. This property
was already noticed by Hilbert in his famous address [45] at the 2nd
International Congress of Mathematicians (Paris, 1900). Theorem 29
is a generalization of this algebraic-independence property.
Similarly to the case of ζ(s), if a Dirichlet series φ(s) is universal, it
is easy to prove the theorems analogous to Theorem 2 and Theorem 29
for φ(s).
An application of the universality to the problem on Dirichlet polynomials was done by Andersson [1]. He used the universality theorem
to show that several conjectures proposed by Ramachandra [187] andBalasubramanian and Ramachandra [6], on lower bounds of certain
integrals of Dirichlet polynomials, are false.
The universality property was applied even in physics; see Gutzwiller[44], Bitar, Khuri and Ren [10].
14. The general notion of universality
The main theme of the present article is the universality for zeta andL-functions. However, the notion of universality was first introduced
in mathematics, under a very diﬀerent motivation.
The first discovery of the universality phenomenon is usually attributed to M. Fekete (1914/15,∑reported in [181]), who proved thatnthere exists a real power series ∞n=1 an x such that, for any continuous f : [−1, 1] → R with f (0) = 0 we can choose positive integers
m1 , m2 , . . . for which
mk
∑
n=1

an xn → f (x)(k → ∞)

THE THEORY OF UNIVERSALITY FOR ZETA AND L-FUNCTIONS

31

holds uniformly on [−1, 1]. The proof is based on Weierstrass’ approximation theorem.
G. D. Birkhoﬀ [9] proved that there exists an entire function ψ(z)such that, for any entire function f (z), we can choose complex numbers
a1 , a2 , . . . for which ψ(z + ak ) → f (z) (as k → ∞) uniformly in any
compact subset of C.
The terminology “universality” was first used by Marcinkiewicz [132].
Various functions satisfying some property similar to those discovered by Fekete and Birkhoﬀ are known. However, before the workof Voronin [205], all of those functions were constructed very artificially. So far the class of zeta and L-functions is the only “natural”
class of functions for which the universality property can be proved.
For the more detailed history of this general notion of universality, see
Grosse-Erdmann [43] and Steuding [201, Appendix] 8 .
It is to be noted that the real origin of the whole theory is Riemann’s
theorem that a conditionally convergent series can be convergent (ordivergent) to any value after some suitable rearrangement. In fact,
Fekete’s result may be regarded as an analogue of Riemann’s theorem
for continuous functions, while Pecherskiı̆’s theorem [185] (mentioned
in Section 1 as an essential tool in Voronin’s proof) gives an analogueof Riemann’s theorem in Hilbert spaces.
A very general definition of universality was proposed by GrosseErdmann [42] [43].
Definition 1. Let X, Y be topological spaces, W be a non-empty
closed subset of Y , and Tτ : X → Y (τ ∈ I) be a family of mappings
with the index set I. We call x ∈ X universal with respect to W if theclosure of the set {Tτ (x) | τ ∈ I} contains W .
Let K be as in Theorem 1, X = Y = H(K ◦ ), where K ◦ is theinterior of K. Then obviously H0c (K) ⊂ H(K ◦ ). Put W = H0c (K)(the topological closure of H0c (K) in the space H(K ◦ )). Define Tτ byTτ (f (z)) = f (z + iτ ) for f ∈ H(K ◦ ). Then Theorem 1 implies thatany element of H0c (K) can be approximated by some suitable elementof {Tτ (ζ) | τ ∈ R}. Therefore Theorem 1 asserts that the Riemannzeta-function ζ(s) is universal with respect to H0c (K) in the sense of
Definition 1.
The notion of joint universality can also be formulated in this general
setting.
8In the same Appendix, Steuding mentioned a p-adic version of Fekete’s theorem,

which was originally proved in [196].

32

KOHJI MATSUMOTO(j)Definition 2. Let X, Y1 , . . . , Yr be topological spaces, and Tτ : X →Yj (τ ∈ I, 1 ≤ j ≤ r) be families of mappings. We call x1 , . . . , xr ∈ X(1)(r)
jointly universal if the set {(Tτ (x1 ), . . . , Tτ (xr ) | τ ∈ I} is dense in
Y1 × · · · × Yr .
Remark 9. The case r = 1 of Definition 2 is the case W = Y of
Definition 1.
In Section 1 we mentioned that the Kronecker-Weyl approximation
theorem (see Remark 1) is used in the proof of Theorem 1. We can seethat the Kronecker-Weyl theorem itself implies a certain universality
phenomenon. Let S 1 = {z ∈ C | |z| = 1}, and consider the situation(1)
when X = R, Y1 = · · · = Yr = S 1 in Definition 2. Define Tτ =(r)
· · · = Tτ = Tτ : R → S 1 by Tτ (x) = e2πiτ x . Then the Kronecker-Weyltheorem implies that, if α1 , . . . , αr ∈ R are linearly independent overQ, then the orbit of(Tτ (α1 ), . . . , Tτ (αr )) = (e2πiτ α1 , . . . , e2πiτ αr )is dense in S 1 × · · · × S 1 . Therefore α1 , . . . , αr are jointly universal.
The above observation shows that both the Voronin theorem and theKronecker-Weyl theorem express certain universality properties. Is it
possible to combine these two universality theorems? The answer isyes, and we will discuss this matter in the next section.
15. The hybrid universality
The first aﬃrmative answer to the question raised at the end of
Section 14 was given by Gonek [40], and a slightly general result waslater obtained by Kaczorowski and Kulas [61].
Theorem 30. (Gonek [40], Kaczorowski and Kulas [61]) Let K be acompact subset of D(1/2, 1), f1 , . . . , fr ∈ H0c (K), χ1 , . . . , χr be pairwise
non-equivalent Dirichlet characters, z > 0, and (θp )p≤z be a sequence ofreal numbers indexed by prime numbers up to z. Then, for any ε > 0,
(15.1){
1
lim inf meas τ ∈ [0, T ] max sup |L(s + iτ, χj ) − fj (s)| < ε,1≤j≤r s∈K
T →∞ T
}
log p
max τ
− θp < ε > 0p≤z
2πholds.
The combination of the universality of Voronin type and of KroneckerWeyl type is now called the hybrid universality. The above theorem istherefore an example of the hybrid joint universality.

THE THEORY OF UNIVERSALITY FOR ZETA AND L-FUNCTIONS

33

Pańkowski [183] proved that the second inequality in (15.1) can be replaced by max1≤k≤m ||τ αk −θk || < ε, where α1 , . . . , αm are real numbers
which are linearly independent over Q, and θ1 , . . . , θm are arbitrary real
numbers. This is exactly the same inequality as in the Kronecker-Weyltheorem (see Remark 1).
In the same paper Pańkowski remarked that the same statement
can be shown for more general L-functions which have Euler products.
The hybrid joint universality for some zeta-functions without Euler
products was discussed in Pańkowski [184].
Hybrid universality theorems are quite useful in applications. Gonek[40] used his hybrid universality theorem to show the joint universality
for Dedekind zeta-functions of Abelian number fields (mentioned in
Section 3). The key fact here is that those Dedekind zeta-functions canbe written as products of Dirichlet L-functions. On the other hand, theaim of Kaczorowski and Kulas [61] was∑to study the distribution of zerosof linear combinations of the form
Pj (s)L(s, χj ), where Pj (s) are
Dirichlet polynomials. Kaczorowski and Kulas applied Theorem 30 toshow a theorem9, similar to Theorem 11, for such linear combinations.
Sander and J. Steuding [191] also considered the universality forsums, or products of Dirichlet L-functions. In particular they proved
the joint universality for Hurwitz zeta-functions ζ(s, a/q) (1 ≤ a ≤ q)
under a certain condition on target functions. Hurwitz zeta-functions
usually do not have Euler products, but when a = q (and when q is evenand a = q/2) the corresponding Hurwitz zeta-function is essentially the
Riemann zeta-function and hence has the Euler product. Therefore theresult of Sander and J. Steuding is an example of mixed universality(see Section 8).
In the paper of Kaczorowski and Kulas [61], the coeﬃcients Pj (s) of
linear combinations are Dirichlet polynomials. Nakamura and Pańkowski[177] considered a more general situation when Pj (s) are Dirichlet series. Their general statement is as follows.
Theorem 31. (Nakamura and Pańkowski [177]) Let P1 (s), . . . , Pr (s)(r ≥ 2) be general Dirichlet series, not identically vanishing, absolutely
convergent in ℜs > 1/2. Moreover assume that at least two of thoseare non-vanishing in D(1/2, 1). Let L1 (s), .∑. . , Lr (s) be hybridly jointly
universal in the above sense. Then L(s) = rj=1 Pj (s)Lj (s) is strongly
universal in D(1/2, 1).
As a corollary, by Theorem 11 we find N (T ; σ1 , σ2 ; L) ≥ CT for theabove L(s), for any σ1 , σ2 satisfying 1/2 < σ1 < σ2 < 1.
9This theorem was later sharpened by Ki and Lee [66] by using the method ofmean motions [52] [15].

34

KOHJI MATSUMOTO

In the above theorem L(s) is a linear form of Lj ’s, but in [175] [180],
Nakamura and Pańkowski obtained more general statements; they considered polynomials of Lj ’s whose coeﬃcients are general Dirichlet series, and proved results similar to Theorem 31. Note that when coeﬃcients of polynomials are constants, such a result was already given in
Kačinskaitė, J. Steuding, Šiaučiūnas and Šleževičienė [59].
Many important zeta and L-functions have such polynomial expressions. Consequently, Nakamura and Pańkowski succeeded in proving
the inequalities like (5.3) on the distribution of zeros of those zeta
or L-functions, such as zeta-functions attached to symmetric matrices (in the theory of prehomogeneous vector spaces), Estermann zetafunctions, Igusa zeta-functions associated with local Diophantine problems, spectral zeta-functions associated with Laplacians on Riemannian
manifolds, Epstein zeta-functions (see [179]), and also various multiple
zeta-functions (of Euler-Zagier, of Barnes, of Shintani, of Witten andso on).
16. Quantitative results
It is an important question how to obtain quantitative information
related with universality. For example, let(16.1){1d(ζ, f, K, ε) = lim inf meas τ ∈ [0, T ]T →∞ T}
sup |ζ(s + iτ ) − f (s)| < ε.

s∈KTheorem 1 asserts that d(ζ, f, K, ε) is positive; but how to evaluate
this value? Or, how to find the smallest value of τ (which we denoteby τ (ζ, f, K, ε)) satisfying the inequality sups∈K |ζ(s + iτ ) − f (s)| < ε?
Voronin’s proof gives no information on these questions, because in thecourse of the proof Voronin used Pecherskiı̆’s rearrangement theorem,
which is ineﬀective.
The first attempt to get a quantitative version of the universality
theorem is due to Good [41]. The fundamental idea of Good is to combine the argument of Voronin with the method of Montgomery [162],
in which Montgomery studied large values of log ζ(s). Instead of Pecherskiı̆’s theorem, Good used convexity arguments (Hadamard’s threecircles theorem and the Hahn-Banach theorem). Also Koksma’s quantitative version of Weyl’s criterion is invoked. The statements of Good’s
main results are quite complicated, but it includes a quantitative version of the discrete universality theorem for ζ(s).
Remark 10. Actually Good stated the discrete universality for log ζ(s).
From the universality for log ζ(s), the universality for ζ(s) itself can beTHE THEORY OF UNIVERSALITY FOR ZETA AND L-FUNCTIONS

35

immediately deduced by exponentiation. A proof of the universality
for log ζ(s) by Voronin’s original method is presented in the book of
Karatsuba and Voronin [64, Chapter VII].
Good’s idea was further pursued by Garunkštis [33], who obtained amore explicit quantitative result when K is small. His main theorem
is still rather complicated, but as a typical special case, he showed thefollowing inequalities. Let K = K(r) be as in Section 1.
Theorem 32. (Garunkštis [33]) Let 0 < ε ≤ 1/2, f ∈ H(K(0.05)),
and assume maxs∈K(0.06) |f (s)| ≤ 1. Then we haved(log ζ, f, K(0.0001), ε) ≥ exp(−1/ε13 ),τ (log ζ, f, K(0.0001), ε) ≤ exp exp(10/ε13 ).
Besides the above work of Good and Garunkštis, there are various diﬀerent approaches toward quantitative results. Laurinčikas [81]pointed out that quantitative information on the speed of convergenceof a certain functional limit theorem would give a quantitative resulton the universality for Lerch zeta-functions. J. Steuding [199] [200]considered the quantity defined by replacing liminf on (1.3) by limsup,
and discussed its upper bounds.
The author pointed out in [137] that from Theorem 2, by comparing
the Taylor expansions of ζ(s+iτ ) and f (s), it is possible to deduce a certain weaker version of universal approximation. On the other hand, aquantitative version of Theorem 2 was shown by Voronin himself [210].
Combining these two ideas, a quantitative version of weak universal
approximation theorem was obtained in Garunkštis, Laurinčikas, Matsumoto, J. & R. Steuding [37].
A nice survey on the eﬀectivization problem is given in Laurinčikas[99].
17. The universality for derived functions
When some function φ(s) satisfies the universality property, a natural question is to ask whether functions derived from φ(s) by somestandard operations, such as φ′ (s), φ(s)2 , exp(φ(s)) etc, also satisfy
the universality property, or not.
We already mentioned the universality of log ζ(s) (see Remark 10).
Concerning the derivatives, Bagchi [4] proved that mth derivatives ofDirichlet L-functions L(m) (s, χ) (m ∈ N) are strongly universal. Laurinčikas [73] studied the universality for (ζ ′ /ζ)(s), and then considered
the same problem for L(s, g) (for a cusp form g) in [86] [87]. The
universality for derivatives of L-functions of elliptic curves was studied

36

KOHJI MATSUMOTO

by Garbaliauskienė and Laurinčikas [32], and its discrete analogue wasdiscussed by Belovas, Garbaliauskienė and Ivanauskaitė [8].
After these early attempts, Laurinčikas [94] (see also [97]) formulateda more general framework of composite universality. Let F be an operator F : H(D(1/2, 1)) → H(D(1/2, 1)). Laurinčikas [94] consideredwhen F (ζ(s)) has the universality property.
A simple aﬃrmative case is the Lipschitz class Lip(α). We call F
belongs to Lip(α) when
1) for any polynomial q = q(s) and any compact K ⊂ D(1/2, 1),
there exists q0 ∈ F −1 {q} such that q0 (s) ̸= 0 on K, and2) for any compact K ⊂ D(1/2, 1) with connected complement, there
exist c > 0 and a compact K1 ⊂ D(1/2, 1) with connected complement,
for whichsup |F (g1 (s)) − F (g2 (s))| ≤ c sup |g1 (s) − g2 (s)|α
s∈K

s∈K1

holds for all g1 , g2 ∈ H(D(1/2, 1)).
Then it is pointed out in [94] that if F ∈ Lip(α), then F (ζ(s)) has
the universality property. This claim especially includes the proof of
the universality for ζ ′ (s).
To prove other theorems in [94], Laurinčikas applied the method offunctional limit theorems (cf. Mauclaire [140]). Those results imply,
for example, ζ ′ (s)+ζ ′′ (s), ζ(s)m (m-th power), exp(ζ(s)), and sin(ζ(s))
are universal.
Later, Laurinčikas and his colleagues generalized the result in [94] tovarious other situations.
• Hurwitz zeta-functions (Laurinčikas [98] for the single case, Laurinčikas [95] for the joint case, and Laurinčikas and Rašytė [120] forthe discrete case),• Periodic and periodic Hurwitz zeta-functions (Laurinčikas [96], Korsakienė, Pocevičienė and Šiaučiūnas [67]),• Automorphic L-functions (Laurinčikas, Matsumoto and J. Steuding [118]).
A hybrid version of the joint composite universality for Dirichlet Lfunctions was given by Laurinčikas, Matsumoto and J. Steuding [119].
An alternative approach to composite universality was done by Meyrath[143].
Yet another approach is due to Christ, J. Steuding and Vlachou [20].
Let Ω0 ×· · ·×Ωn be an open subset of Cn+1 , and let F : Ω0 ×· · ·×Ωn →C be continuous. In [20], the universality of F (ζ(s), ζ ′ (s), . . . , ζ (n) (s))
was discussed. First, applying the idea in [37], they proved a weaker
form of universal approximation. Then, when F is non-constant andTHE THEORY OF UNIVERSALITY FOR ZETA AND L-FUNCTIONS

37

analytic, they obtained a kind of universality theorem on a certain
small circle. Their proof relies on the implicit function theorem and
the Picard-Lindelöf theorem on certain diﬀerential equations.
18. Ergodicity and the universality
It is quite natural to understand universality from the ergodic viewpoint. In fact, the universality theorem for a certain Dirichlet series
φ(s) implies that the orbit {φ(s + iτ ) | τ ∈ R} is dense in a certain function space, and this orbit comes back to an arbitrarily smallneighbourhood of any target function infinitely often. This is really an
ergodic phenomenon.
Therefore, we can expect that there are some explicit connections
between universality theory and ergodic theory. We mentioned alreadyin Section 10 that the Birkhoﬀ-Khinchin theorem in ergodic theory isused in Bagchi’s proof of the universality.
Recently J. Steuding [202] formulated a kind of universality theorem,
whose statement itself is written in terms of ergodic theory.
Let (X, B, P ) be a probability space, and let T : X → X a measurepreserving transformation. We call T ergodic with respect to P if
A ∈ B satisfies T −1 (A) = A, then either P (A) = 0 or P (A) = 1 holds.
In this case we call (X, B, P, T ) an ergodic dynamical system. Here we
consider the case X = R and B is the standard Borel σ-algebra.
Let D ⊂ C be a domain, K1 , . . . , Kr be compact subsets of D withconnected complements, and fj ∈ H0c (Kj ) (1 ≤ j ≤ r). We call
φ1 , . . . , φr ∈ H(D) is jointly ergodic universal if for any Kj , fj , T ,ε > 0, and for almost all x ∈ R, there exists an n ∈ N for which(18.1)

max sup |φj (s + iT n x) − fj (s)| < ε1≤j≤r s∈Kjholds. Here, T n x means the T -times iteration of T . If the above
statament holds for fj ∈ H c (Kj ) (1 ≤ j ≤ r), then we call φ1 , . . . , φr ∈H(D) jointly strongly ergodic universal.
Theorem 33. (J. Steuding [202]) Let D, Kj , T be as above. Let
φ1 , . . . , φr be a family of L-functions. Then, there exists a real number
τ such that(18.2)

max sup |φj (s + iτ ) − fj (s)| < ε1≤j≤r s∈Kj

for any fj ∈ H0c (Kj ) (resp. H c (Kj )) and any ε > 0, if and only if
φ1 , . . . , φr is jointly (resp. jointly strongly) ergodic universal. And in

38

KOHJI MATSUMOTO

this case, we have(18.3){
1
lim inf # n ∈ N
N →∞ N}max sup |φj (s + iT n x) − fj (s)| < ε1≤j≤r s∈Kj

> 0.

Srichan, R. & J. Steuding [195] discovered the universality produced
by a random walk. They considered a lattice Λ on C and a random
walk (sn )∞n=0 on this lattice, and proved the following result. Let K
be a compact subset of D(1/2, 1) with connected complement (with acondition given in terms of Λ), and f ∈ H0c (K). Then for any ε > 0,{}1
(18.4)
lim inf # n ∈ N sup |ζ(s + sn ) − f (s)| < ε > 0
N →∞ Ns∈K
holds almost surely. They also mentioned a result similar to the above
for two-dimensional Brownian motions.
The results in this section suggest that universality is a kind of ergodic phenomenon and is to be understood from the viewpoint of dynamical systems. Universality theorems imply that the properties ofzeta-functions in the critical strip are quite inaccessible, which is probably the underlying reason of the extreme diﬃculty of the Riemannhypothesis. Moreover in Section 9 we mentioned that the Riemannhypothesis itself can be reformulated in terms of dynamical systems.
Therefore the Riemann hypothesis is perhaps to be understood as aphenomenon with dynamical-system flavour10. In order to pursue thisviewpoint, it is indispensable to study universality more deeply andextensively.

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Graduate School of Mathematics,
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Japan