dmju-5.tex November 22, 2018 ON MIXED JOINT DISCRETE UNIVERSALITY FOR A CLASS OF ZETA-FUNCTIONS ROMA KACINSKAITˇ E˙ AND KOHJI MATSUMOTO ABSTRACT. We prove a mixed joint discrete universality theorem for a Matsumoto zeta- function ϕ(s) (belonging to the Steuding subclass) and a periodic Hurwitz zeta-function ζ(s,α;B). For this purpose, certain independence condition for the parameter α and the minimal step of discrete shifts of these functions is assumed. This paper is a continuation of authors’ works [12] and [13]. Keywords: discrete shift, joint approximation, linear independence, periodic Hurwitz zeta- function, Matsumoto zeta-function, universality. AMS classification: 11M06, 11M41, 11M36. 1. INTRODUCTION In analytic number theory, the problem of so-called mixed joint universality in Vo- ronin’s sense is a very interesting problem since it solves a problem on simultaneous approximation of certain tuples of analytic functions by shifts of tuples consisting of zeta-functions having an Euler product expansion over the set of primes and other zeta- functions without such product. For such type of universality, a very important role is played by the parameters which occur in the definitions of zeta-functions. The first result on mixed joint universality was obtained by H. Mishou in [21]. He proved that the Riemann zeta-function ζ (s) and the Hurwitz zeta-function ζ (s,α) with transcendental parameter α are jointly universal. Let P, N, N0, Z, Q, R and C be the sets of all primes, positive integers, non-negative integers, integers, rational numbers, real numbers and complex numbers, respectively. Denote by s = σ + it a complex variable. Recall that the functions ζ (s) and ζ (s,α), 0 < α ≤ 1, for σ > 1, are defined by ∞ 1 1 −1 ∞ 1 arXiv:1611.08126v1 [math.NT] 24 Nov 2016 ζ s ∑ ∏ 1 and ζ s,α ∑ , ( ) = s = − s ( ) = α s m=1 m p∈P p m=0 (m + ) respectively. Both of them are analytically continued to the whole complex plane, except for a simple pole at the point s = 1 with residue 1. Note that the Riemann zeta-function has the Euler product expansion, while in general the Hurwitz zeta-function does not have α 1 (except the cases = 2 , 1). For Mishou’s result and further statements, we introduce some notations. Let D(a,b) = {s ∈ C : a < σ < b} for any a < b. For every compact set K ⊂ C, denote by Hc(K) the 1 2 ROMAKACINSKAITˇ E˙ AND KOHJI MATSUMOTO set of all C-valued continuous functions defined on K and holomorphic in the interior c c of K. By H0(K) we denote the subset of H (K), consisting of all elements which are non-vanishing on K. Theorem 1 ([21]). Suppose that α is a transcendental number. Let K1 and K2 be compact 1 c subsets of thestrip D( 2 ,1) with connected complements. Suppose that f1(s) ∈ H0(K1) and c f2(s) ∈ H (K2). Then, for every ε > 0, 1 liminf meas τ ∈ [0,T] : sup |ζ (s+iτ)− f1(s)| < ε, sup |ζ (s+iτ,α)− f2(s)| < ε > 0. T ∞ → T s∈K1 s∈K2 Here, as usual, meas{A} denotes the Lebesgue measure of the measurable set A ⊂ R. Note that J. Sander and J. Steuding [23] proved the same type of universality, but for rational α, by a quite different method. In [12], we consider the mixed joint universality property for a wide class of zeta- functions consisting of Matsumoto zeta-functions ϕ(s) belonging to the Steuding class S and periodic Hurwitz zeta-functions ζ s,α;B . ( ) e Recall the definition of the polynomial Euler products ϕ(s) or so-called Matsumoto zeta-functions. (Remark: The function ϕ(s) was introduced by the second author in [18].) e For m ∈ N, let g(m) be a positive integer and pm the mth prime number. Moreover, let ( j) e am ∈ C, and f ( j,m) ∈ N,1 ≤ j ≤ g(m). The function ϕ(s) is defined by the polynomial Euler product ∞ g(m) e ( j) −sf ( j,m) −1 ϕ(s) = ∏ ∏ 1 − am pm . (1) m=1 j=1 It is assumed that e α ( j) β g(m) ≤ C1 pm and |am | ≤ pm (2) with a positive constant C1 and non-negative constants α and β. In view of (2), the function ϕ(s) converges absolutely for σ > α + β + 1, and hence, in this region, it can be given by the absolutely convergent Dirichlet series e ∞ c ϕ(s) = ∑ k . (3) ks k=1 e The shifted function ϕ(s) is given by e ∞ c ∞ c ϕ(s) = ∑ k = ∑ k (4) ks+α+β ks k=1 e k=1 −α−β with ck = k ck. For σ > 1, the last series in (4) converges absolutely too. Also, suppose that, for the function ϕ(s), the following assumptions hold (for the de- e tails, see [18]): ϕ σ σ 1 σ (a) (s) can be continued meromorphically to ≥ 0, where 2 ≤ 0 < 1, and all poles in this region are included in a compact set which has no intersection with the line σ = σ0, MIXEDJOINTDISCRETEUNIVERSALITY 3 C (b) ϕ(σ + it) = O(|t| 2) for σ ≥ σ0, where C2 is a positive constant, (c) the mean-value estimate T 2 |ϕ(σ0 + it)| dt = O(T). (5) Z0 It is possible to discuss functional limit theorems for Matsumoto zeta-functions (see Section 2 below), but this framework is too wide to consider the universality property. To investigate the universality, we introduce the Steuding subclass S, for which the following ϕ slightly more restrictive conditions are required. We say that thee function (s) belongs to the class S, if the following conditions are fulfilled: (i) theree exists a Dirichlet series expansion ∞ ϕ a(m) (s) = ∑ s m=1 m with a(m) = O(mε ) for every ε > 0; (ii) there exists σϕ < 1 such that ϕ(s) can be meromorphically continued to the half- plane σ > σϕ , and holomorphic except for at most a pole at s = 1; (iii) there exists a constant c ≥ 0 such that ε ϕ(σ + it) = O(|t|c+ ) for every fixed σ > σϕ and ε > 0; (iv) there exists the Euler product expansion over prime numbers, i.e., l −1 ϕ a j(p) (s) = ∏ ∏ 1 − s ; p∈P j=1 p (v) there exists a constant κ > 0 such that 1 lim ∑ |a(p)|2 = κ, x→∞ π (x) p≤x where π(x) denotes the number of primes up to x, i.e., p ≤ x. ∗ For ϕ ∈ S, let σ be the infimum of all σ1 for which e T ∞ 2 1 ϕ σ 2 |a(m)| | ( + it)| dt ∼ ∑ 2σ 2T Z−T m=1 m σ σ 1 σ ∗ holds for every ≥ 1. Then it is known that 2 ≤ < 1. (Remark: The class S was introduced by J. Steuding in [24].) e Now we recall the definition of the periodic Hurwitz zeta-function ζ (s,α;B) with a fixed parameter α, 0 < α ≤ 1. (Remark: The function ζ (s,α;B) was introduced by A. Javtokas and A. Laurincikasˇ in [8].) Let B = {bm : m ∈ N0} be a periodic sequence 4 ROMAKACINSKAITˇ E˙ AND KOHJI MATSUMOTO of complex numbers (not all zero) with minimal period k ∈ N. For σ > 1, the function ζ (s,α;B) is defined by ∞ b ζ s,α;B ∑ m . ( ) = α s m=0 (m + ) It is known that k−1 α ζ α B 1 ζ l + σ (s, ; ) = s ∑ bl s, , > 1. (6) k l=0 k The last equality gives an analytic continuation of the function ζ (s,α;B) to the whole complex plane, except for a possible simple pole at the point s = 1 with residue 1 k−1 b := ∑ bl. k l=0 If b = 0, then ζ (s,α;B) is an entire function. In [12], we prove the mixed joint universality property of the functions ϕ(s) and ζ (s,α;B). Theorem 2 ([12]). Suppose that ϕ(s) ∈ S, and α is a transcendental number. Let K1 be a compact subset of D σ ∗,1 , and K be a compact subset of D 1,1 , both with connected ( ) 2 e ( 2 ) c c ε complements. Suppose that f1 ∈ H0(K1) and f2 ∈ H (K2). Then, for every > 0, 1 liminf meas τ ∈ [0,T] : sup |ϕ(s + iτ) − f1(s)| < ε, T ∞ → T s∈K1 sup |ζ (s + iτ,α;B) − f2(s)| < ε > 0. s∈K2 In [13], we obtain a generalization of Theorem 2, in which several periodic Hurwitz zeta-functions are involved. More interesting and convenient in practical applications is so-called discrete univer- sality of zeta-functions (for example, see [2]). This pushes us to extend our investigations of mixed joint universality for a class of zeta-functions to the discrete case. Recall that, in this case, the pair of analytic functions is approximated by discrete shifts of tuple ϕ(s+ikh),ζ (s+ikh,α;B) , k ∈ N0, where h > 0 is the minimal step of given arithmeti- cal progression. The aim of this paper is to prove a mixed joint discrete universality theorem for the collection of the functions (ϕ(s),ζ (s,α;B)), i.e., the discrete version of Theorem 2. For h > 0, let 2π L(P,α,h) = (log p : p ∈ P),(log(m + α) : m ∈ N ), . 0 h MIXEDJOINTDISCRETEUNIVERSALITY 5 Theorem 3. Let ϕ(s) ∈ S, K1, K2, f1(s) and f2(s) satisfy the conditions as in Theorem 2. P,α, Q ε > Suppose that the set L( e h) is linearly independent over .