universality of various zeta-functions

Universality of Various Zeta-Functions

Roma Kacinskaite 1,2

Department of MathematicsSiauliai UniversitySiauliai, Lithuania

Abstract

This is a survey of results on joint universality in Voronin’s sense of various zeta-functions, when in the collection of these functions some of them have the Eulerproduct and the others have not.

Keywords: analytic function, joint approximation, periodic sequence, universality,zeta-function.

In the number theory, the investigation of universality of zeta-functions isinteresting and important subject of the studies.

In 1975, S. M. Voronin proved that every analytic non-vanishing functionon compact subsets can be approximated by the shifts of the Riemann zeta-function ζ(s), s = σ + it, which, for σ > 1, is defined by

ζ(s) =∞∑

m=1

1

ms=

∏p

(1− 1

ps

)−1

1 Partially supported by the European Commission within the 7th Framework Prog-ramme FP/2011–2014 project INTEGER (INstitutional Transformation for Effecting Gen-der Equality in Research), Grant Agreement No. 266638.2 Email:[email protected]

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(here p denotes a prime number). Now this property we call as universality.

Theorem 1.1 ([12]) Let 0 < r < 14, and let f(s) be any non-vanishing con-

tinuous function on the disc |s| ≤ r which is analytic in the interior of thisdisc. Then, for every ε > 0, there exists a number τ = τ(ε) ∈ R such that

max|s|≤r

∣∣∣ζ(s +

3

4+ iτ

)− f(s)

∣∣∣ < ε.

We can state it in more general form. Let D = {s ∈ C : 12

< σ < 1}.Theorem 1.2 ([6]) Let K be a compact subset of the strip D with connectedcomplement. Let f(s) be a continuous non-vanishing function on K which isanalytic in the interior of K. Then, for every ε > 0,

lim infT→∞

1

Tmeas

(τ ∈ [0, T ] : sup

s∈K|ζ(s + iτ)− f(s)| < ε

)> 0.

Last inequality shows that the set of translations of the Riemann zeta-function which approximate a given analytic function f(s) has positive lowerdensity.

Many authors generalized the Voronin theorem for other zeta- and L-functions. We can mentioned B. Bagchi, S. M. Gonek, A. Reich, A. Lau-rincikas, K. Matsumoto, H. Nagoshi, J. Steuding, D. Siauciunas, the authorand other (for history and results, see, for example, [3], [7], [10]). There existsa conjecture of Linnik-Ibragimov that all functions in some half-plane definedby Dirichlet series, analytically continuable to the left of absolute conver-gence half-plane and satisfying some natural growth conditions are universalin Voronin sense.

The first result on joint approximation of a given collection of analyticfunctions by a collection of shifts of zeta-functions belongs to S. M. Voronin[11]. He investigated the collection of Dirichlet L-functions L(s, χ). We recallthat the function L(s, χ) attached to a character χ mod d, d ∈ N, for σ > 1,is given

L(s, χ) =∞∑

m=1

χ(m)

ms=

∏p

(1− χ(p)

ps

)−1

.

Theorem 1.3 ([11]) Let χ1, . . . , χn be pairwise non-equivalent Dirichlet cha-racters, and L(s, χ1), . . . , L(s, χn) are the corresponding Dirichlet L-functions.For j = 1, . . . , n, let Kj denote a compact subset of the strip D with connectedcomplement, and fj(s) be a continuous non-vanishing function on Kj and

R. Kacinskaite / Electronic Notes in Discrete Mathematics 43 (2013) 129–135130

analytic in the interior of Kj. Then, for every ε > 0,

lim infT→∞

1

Tmeas

(τ ∈ [0, T ] : sup

1≤j≤nsups∈Kj

|L(s + iτ, χj)− fj(s)| < ε

)> 0.

More complicated situation we have in the two-dimensional or multidi-mensional case when in the same collection part of the zeta-functions haveEuler product but the other do not have. In 2007, H. Mishou proved the jointuniversality theorem for the Riemann zeta-function ζ(s) and Hurwitz zeta-function ζ(s, α) with a transcendental parameter α [8]. It is well-known thatthe function ζ(s, α), 0 < α ≤ 1, for σ > 1, is defined by

ζ(s, α) =∞∑

m=0

1

(m + α)s.

It has an analytic continuation to the whole complex plane except a simplepole at s = 1 with residue 1. If α = 1, then the Hurwitz zeta-function ζ(s, 1)reduces to the Riemann zeta-function ζ(s). Then the following statementholds.

Theorem 1.4 ([8]) Suppose that α is a transcendental number such that 0 <α < 1. Let K1 and K2 be compact subsets of the strip 1

2< σ < 1 with connected

complements. Assume that functions fj(s) are continuous on Kj and analyticin the interior of Kj for each j = 1, 2. In addition, we suppose that f1(s) doesnot vanish on K1. Then, for all positive ε,

lim infT→∞

1

Tmeas(τ ∈ [0, T ] : max

s∈K1

|ζ(s + iτ)− f1(s)| < ε,

maxs∈K2

|ζ(s + iτ, α)− f2(s)| < ε) > 0.

The joint approximation of a given collection of analytic functions by acollection of shifts of periodic zeta-function and periodic Hurwitz zeta-functionis obtained by A. Laurincikas and the author in [4].

Let a = {am : m ∈ N} be a periodic sequence of complex numbers with theleast period k ∈ N. The periodic zeta-function ζ(s; a), for σ > 1, is defined bythe series

ζ(s; a) =∞∑

m=1

am

ms,

and by analytic continuation elsewhere. From the periodicity of sequence a

R. Kacinskaite / Electronic Notes in Discrete Mathematics 43 (2013) 129–135 131

follows that, for σ > 1,

ζ(s; a) =1

ks

k∑m=1

amζ(s,

m

k

).

Last equality gives an analytic continuation to the whole complex plane forthe function ζ(s; a), except, maybe for the point s = 1 with residue a =1k

∑km=1 am. If a = 0, then ζ(s; a) is an entire function.

If the sequence a is completely multiplicative, then the periodic zeta-function ζ(s; a) coincides with the Dirichlet L-function L(s, χ).

The periodic Hurwitz zeta-function ζ(s, α; b) with a fixed parameter α,0 < α ≤ 1, is defined, for σ > 1, by

ζ(s, α; b) =∞∑

m=0

bm

(m + α)s,

where b = {bm : m ∈ N ∪ {0}} is a periodic sequence of complex numbers bm

with a minimal period l ∈ N. From the periodicity of b, for σ > 1, we have

ζ(s, α; b) =1

ls

l−1∑m=0

bmζ

(s,

m + α

l

).

This gives an analytic continuation of the function ζ(s, α; b) to the wholecomplex plane, except, for a simple pole at s = 1 with residue b = 1

l

∑l−1m=0 bm.

If b = 0, then periodic Hurwitz zeta-function is an entire function.

Theorem 1.5 ([4]) Suppose that α is a transcendental number. Let K1 andK2 be a compact subsets of the strip D with connected complements, f1(s) bea continuous non-vanishing function on K1 which is analytic in the interiorof K1, and let f2(s) be a continuous function on K2 which is analytic in theinterior of K2. Then, for every ε > 0,

lim infT→∞

1

Tmeas

(τ ∈ [0, T ] : sup

s∈K1

|ζ(s + iτ ; a)− f1(s)| < ε,

sups∈K2

|ζ(s + iτ, α; b)− f2(s)| < ε)

> 0.

Multidimensional version of Theorem 1.5 is obtained by A. Laurincikas in[5].


Theorem 1.6 ([5]) Suppose that the sequences a1, ..., ar1 are multiplicative,and, for all prime p, holds the inequality

∞∑j=1

|ajpg |pg/2

< 1, j = 1, ..., r1.

Let α1, ..., αr2 be algebraically independent over Q. Suppose that K1, ..., Kr1

and K1, ..., Kr2 are compact subsets of the strip D, their complements are con-nected. Suppose that f1(s), ..., fr1(s) are continuous non-vanishing functionsin K1, ..., Kr1 and analytic in interior K1, ..., Kr1, and f1(s), ..., fr2(s) are con-tinuous in K1, ..., Kr2 and analytic in interior K1, ..., Kr2, respectively. Then,for every ε > 0,

lim infT→∞

1

Tmeas

(τ ∈ [0, T ] : sup

1≤j≤r1

sups∈Kj

|ζ(s + iτ ; aj)− fj(s)| < ε,

sup1≤j≤r2

sups∈Kj

|ζ(s + iτ, αj; bj)− fj(s)| < ε)

> 0.

All universality theorems stated above are of continuous type: we dealwith mathematical objects given by integrals. In discrete case, the trigono-metric and other sums appear. Therefore, discrete universality theorems aremore complicated: there the shifts are taken from certain arithmetic progres-sion with the step h > 0. First result in discrete approximation of analyticfunctions belongs to A. Reich [9].

In 2011, the author obtains joint discrete universality of Dirichlet L-func-tion L(s, χ) and periodic Hurwitz zeta-function ζ(s, α; b) with transcendentalparameter α [1].

Theorem 1.7 ([1]) Suppose that α, K1, K2, f1(s) and f2(s) are the same asin Theorem 1.5. Let h > 0 be a fixed number such that exp{2π

h} is rational.

Then, for every ε > 0,

lim infN→∞

1

N + 1#

(0 < r ≤ N : sup

s∈K1

|L(s + irh, χ)− f1(s)| < ε,

sups∈K2

|ζ(s + irh, α; b)− f2(s)| < ε)

> 0.

It is possible to generalize Theorem 1.7 and obtain the joint universalityof collection of Dirichlet L-functions and periodic Hurwitz zeta-functions withtranscendental parameters.

Theorem 1.8 ([2]) Let h > be a fixed number such that exp{2πh} is ratio-

nal. Suppose that χ1, . . . , χr1 are pairwise non-equivalent Dirichlet characters,


and L(s, χ1), . . . , L(s, χr1) are the corresponding Dirichlet L-functions. Letα1, . . . , αr2 be algebraically independent over Q. Suppose that K1, . . . , Kr1,f1(s), . . . , fr1(s), K1, . . . , Kr2, f1(s), . . . , fr2(s) satisfy the hypothesis of Theo-rem 1.6. Then, for every ε > 0,

lim infN→∞

1

N + 1#

(0 < r ≤ N : sup

1≤j≤r1

sups∈Kj

|L(s + irh, χj)− fj(s)| < ε,

sup1≤j≤r2

sups∈Kj

|ζ(s + irh, αj; bj)− fj(s)| < ε

)> 0

References

[1] Kacinskaite, R., Joint Discrete Universality of Periodic Zeta-Functions, Int.Transf. Spec. Funct. 22(8) (2011), 593–601.

[2] Kacinskaite, R., Joint Discrete Universality of Periodic Zeta-Functions. II,submitted.

[3] Kacinskaite, R., Limit Theorems for Zeta-Functions – with Application inUniversality, Siauliai Math. Semin. 7(15) (2012), 19–40.

[4] Kacinskaite, R., and Laurincikas, A., The Joint Distribution of Periodic Zeta-Functions, Stud. Sci. Math. Hung. 48(2) (2011), 257–279.

[5] Laurincikas, A., Joint Universality of Zeta-Functions with Periodic Coefficients,Izv. Math. 74(3) (2010), 515–539 = Izv. Ross. Akad. Nauk, Ser. Mat. 74(3),(2010) 79–102 (in Russian).

[6] Laurincikas, A., “Limit Theorems for the Riemann Zeta-Function”, Kluwer,Dordrecht, 1996.

[7] Matsumoto, K., Probabilistic Value-Distrubtion Theory of Zeta-Functions,Sugaku Expositions 17(1) (2004), 51–71.

[8] Mishou, H., The Joint Value-Distribution of the Riemann Zeta-Function andHurwitz Zeta-Functions, Liet. Matem. Rink. 47(1) (2007), 62–807 = Lith. Math.J. 47(1)(2007), 32–47.

[9] Reich, A., Zur Universalitat und Hypertranszendenz der DedekindschenZetafunktion, Abh. Braunschweig. Wiss. Ges. 33 (1982), 197–203.

[10] Steuding, J., “Value-Distribution of L-Functions”, Springer-Verlag, Berlin,Heidelberg, New York, 2007.


[11] Voronin, S. M., On the Functional Independence of Dirichlet L-Functions, ActaArith. 27 (1975), 493–503 (in Russian).

[12] Voronin, S. M., Theorem on the “Universality” of the Riemann Zeta-Function,Izv. Akad. Nauk SSSR, Ser. Mat. 39 (1975), 475–486 (in Russian) = Math.USSR Izv. 9 (1975), 443–453.


universality of various zeta-functions

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