A mixed joint universality theorem for zeta-functions Renata Macaitien˙e
Siauliai University, Lithuaniaˇ
In 1975, S. M. Voronin observed a very interesting property of the Riemann zeta-functionζ(s),s=σ+it. He proved that every analytic non-vanishing function can be approximated with a given accuracy uniformly on compact subsets of the strip D={s ∈C : 12 < σ < 1} by shifts ζ(s+iτ). The last version of the Voronin theorem is the following statement.
Theorem 1[2]. Suppose thatKbe a compact subset of the stripDwith connected complement, and that f(s) is a continuous non-vanishing function on K which is analytic in the interior of K. Then, for every ε >0,
lim inf
T→∞
1 Tmeas
{
τ ∈[0;T] : sup
s∈K|ζ(s+iτ)−f(s)|< ε }
>0.
Where meas{A} denotes the Lebesgue measure of a measurable set A⊂R.
Later, many authors obtained that some other zeta- andL-functions also have the above universality property. Among them, the Hurwitz zeta-function ζ(s, α) with transcendental parameterα, 0< α≤1 [3]. However, in this case, the approximated function f(s) is not necessarily non-vanishing on K.
Y. V. Linnik and I. A. Ibragimov conjectured that all functions given by Dirichlet series and satisfying some natural conditions are universal.
The first result on the joint universality also belongs to S. M. Voronin. In 1975, he proved [7] the joint universality of DirichletL-functions with pairwise non-equivalent characters. In 2007, 2008, H. Mishou obtained [5], [6] the joint universality of the functionsζ(s) andζ(s, α). This case is very interesting because the functionζ(s) has the Euler product over primes while ζ(s, α), in general, has not the Euler product.
In 2006, A. Javtokas and A. Laurinˇcikas began to study the periodic Hurwitz zeta-function which is a generalization of the function ζ(s, α). Let A = {am : m ∈ N0 = N∪ {0}} be a periodic sequence of complex numbers with minimal period k ∈ N, and α, 0 < α ≤ 1, be a fixed parameter. Then, the periodic Hurwitz zeta-function ζ(s, α;A) is defined, for σ >1, by
ζ(s, α;A) =
∑∞ m=0
am (m+α)s,
and is meromorphically continued to the whole complex plane. The mentioned authors obtained [1] the universality of the function ζ(s, α;A) with transcendental parameter α.
A. Laurinˇcikas and his students also investigated the joint universality of periodic Hurwitz zeta-functions. The most general result is contained in [4]. For j = 1, ..., r, let αj, 0 < αj ≤ 1, be a fixed parameter, lj ∈ N, and, for j = 1, ..., r, l = 1, ..., lj, let Ajl ={amjl :m ∈N0} be a periodic sequence of complex numbers with minimal period kjl ∈ N, and ζ(s, αj;Ajl) denote the corresponding periodic Hurwitz zeta- function. Denote by kj be the least common multiple of the periods kj1, ..., kjlj,
j = 1, ..., r, and define
Bj =
a1j1 a1j2 . . . a1jlj a2j1 a2j2 . . . a2jlj . . . . . . . . . . . . akjj1 akjj2 . . . akjjlj
, j = 1, ..., r.
Theorem 2 [4]. Suppose that the system {log(m +αj) : m ∈ N0, j = 1, ..., r} is linearly independent over the field of rational numbers Q, and that rank(Bj) = lj, j = 1, ..., r. For every j = 1, ..., r and l = 1, ..., lj, letKjl be a compact subset of the strip D with connected complement, and let fjl(s) be a continuous on Kjl function which is analytic in the interior of Kjl. Then, for every ϵ >0,
lim inf
T→∞
1 Tmeas
{
τ ∈[0;T] : sup
1≤j≤r
sup
1≤l≤lj
sup
s∈Kjl
|ζ(s+iτ, αj;Ajl)−fjl(s)|< ϵ }
>0.
In the report, we present a mixed joint universality theorem for the Riemann zeta-function and periodic Hurwitz zeta-functions ζ(s, αj;Ajl), j = 1, ..., r, l = 1, ..., lj.
Theorem 3. Suppose that α1, ..., αr are algebraically independent over Q, and that all hypotheses on Kjl and fjl of Theorem 2 are satisfied. Moreover, let K be a compact subset of the strip D with connected complement, and let f(s) be a continuous non-vanishing on K function which is analytic in the interior of K. Then, for every ϵ >0,
lim inf
T→∞
1 Tmeas
{
τ ∈[0;T] : sup
1≤j≤r
sup
1≤l≤lj
sup
s∈Kjl|ζ(s+iτ, αj;Ajl)−fjl(s)|< ϵ, sup
s∈K|ζ(s+iτ)−f(s)< ϵ }
>0.
References
[1] A. Javtokas, A. Laurinˇcikas, The universality of the periodic Hurwitz zeta- function, Integral Transforms and Spec. Funct. 17(10) (2006), 711–722.
[2] A. Laurinˇcikas, Limit Theorems for the Riemann Zeta-Function, Kluwer, Dor- drecht, 1996.
[3] A. Laurinˇcikas, R. Garunkˇstis, The Lerch Zeta-Function, Kluwer, Dordrecht, 2002.
[4] A. Laurinˇcikas, S. Skerstonait˙e, Joint universality for periodic Hurwitz zeta- functions. II, in: New Directions in Value-Distribution Theory of Zeta and L-functions, R. Steuding and J. Steuding (Eds.), Shaker Verlag, Aachen, 2009, 161–170.
[5] H. Mishou, The joint value distribution of the Riemann zeta function and Hur- witz zeta functions, Lith. Math. J.47 (2007), 32–47.
[6] H. Mishou, The joint value distribution of the Riemann zeta function and Hur- witz zeta functions. II, Arch. Math.(Basel) 90(3) (2008), 230–238.
[7] S. M. Voronin, On the functional independence of Dirichlet L-functions, Acta Arith. 27 (1975), 493–503 (in Russian).