A discrete limit theorem for the periodic Hurwitz zeta-function

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1 Lietuvos matematikos rinkinys ISSN Proc. of the Lithuanian Mathematical Society, Ser. A Vol. 56, 205 DOI: /LMR.A pages A discrete it theorem for the periodic Hurwitz zeta-function Audronė Rimkevičienė,2 Faculty of Mathematics and Informatics, Vilnius University Naugarduko 4, LT Vilnius 2 Šiauliai State College Aušros 40, LT-7624 Šiauliai audronerim@gmail.com Abstract. In the paper, we prove a it theorem of discrete type on the weak convergence of probability measures on the complex plane for the periodic Hurwitz zeta-function. Keywords: Hurwitz zeta-function, it theorem, probability measure, weak convergence. Let s σ + it be a complex number, α, 0 < α, be a fixed parameter, and let a {a m : m N 0 N {0}} be a periodic sequence of complex numbers with minimal period k N. The periodic Hurwitz zeta-function ζ(s, α; a) is defined, for σ >, by the Dirichlet series and, by using the equality, ζ(s,α;a) l0 a m (m+α) s, ζ(s,α;a) k ( k s a l ζ s, α+l ), k where ζ(s, α) is the classical Hurwitz zeta-function, can be meromorphically continued to the whole complex plane with unique simple pole at the point s with residue a def k a l. k l0 If a 0, then the function ζ(s,α;a) is entire one. In [2, 4, 6] and [7], it theorem on the weak convergence of probability measures on the complex plane C for the function ζ(s,α;a) with parameter α of various arithmetical types were obtained. In these works, the weak convergence for T meas{ t [0,T]: ζ(σ +it,α;a) A }, A B(C), where B(S) denotes the Borel σ-field of the space S, and measa is the Lebesgue measure of a measurable set A R, was considered.

2 A discrete it theorem for the periodic Hurwitz zeta-function 9 The theorems obtained are of continuous type because the imaginary part t can take arbitrary real values. The aim of this note is to prove a discrete it theorem for the function ζ(s,α;a) when t takes values from the set {h m : m N 0 }, where h > 0 is a fixed number. Define the set { } (log(m+α): ) π L(α,h,π) m N0,, h and the torus Ω m N 0 γ m, where γ m is the unit circle {s C: s } for all m N 0. The torus Ω is a compact topological group, therefore, on (Ω,B(Ω)), the probability Haar measure m H can be defined. This gives the probability space (Ω,B(Ω),m H ). Denote by ω(m) the projection of an element ω Ω to the coordinate space γ m, m N 0, and, on the probability space (Ω,B(Ω),m H ), define the complex-valued random element ζ(σ,α,ω;a) by the formula ζ(σ,α,ω;a) a m ω(m) (m+α) σ, σ > 2, and denote by P ζ the distribution of ζ(σ,α,ω;a), i.e., P ζ (A) m H ( ω Ω: ζ(σ,α,ω;a) A ), A B(C). Theorem. Suppose that the set L(α,h,π) is linearly independent over the field of rational numbers Q, and that σ > 2. Then the probability measure P N (A) def N + #{ 0 m N: ζ(σ +imh,α;a) A }, A B(C), converges weakly to the measure P ζ as N. For the proof of Theorem, the following two lemmas involving the set L(α,h,π) are applied. Let Q N (A) def N + #{ 0 m N: ( (m+α) imh : m N 0 ) A }, A B(Ω). Lemma. Suppose that the set L(α,h,π) is linearly independent over Q. Then Q N converges weakly to the Haar measure m H as N. Proof of the lemma is given in [3, Lemma 2.3]. For a h ((m + α) ih : m N 0 ), on the probability space (Ω,B(Ω),m H ), define the transformation ψ h by ψ h (ω) a h ω, ω Ω. Then ψ h is a measurable measure preserving transformation. Lemma 2. Suppose that the set L(α,h,π) is linearly independent over Q. Then the transformation ψ h is ergodic, i.e., if A B(Ω) and A h ψ h (A) differ one from other at most by m H -measure zero, then m H (A) 0 or m H (A). Liet. matem. rink. Proc. LMS, Ser. A, 56, 205,

3 92 A. Rimkevičienė Proof of the lemma is given in [3, Lemma 2.8]. The further proof of Theorem can be divided into following parts: it theorems for absolutely convergent Dirichlet series, approximation of the function ζ(σ, α; a) in the mean by absolutely convergent Dirichlet series, it theorems for ζ(σ, α; a) and ζ(σ,α,ω;a), and identification of the it measure. For m N 0 and n N, define ν n (m,α) exp { } ( m+α n+α, where )σ σ > 2 is a fixed number, and set a m ν n (m,α) ζ n (s,α;a) (m+α) s and ζ n (s,α,ω;a) a m ω(m)ν n (m,α) (m+α) s. Then the latter series both are absolutely convergent for σ > 2. A B(C), let P N,h (A) def N + #{ 0 m N: ζ n (σ +imh,α;a) A }, Moreover, for and P N,h,ω (A) def N + #{ 0 m N: ζ n (σ +imh,α,ω;a) A }. Lemma 3. Suppose that the set L(α,h,π) is linearly independent over Q and that σ > 2. Then P N,h and P N,h,ω both converge weakly to the same probability measure P n on (C,B(C)) as N. Proof. The lemma is a result of the application of Lemma, Theorem 5. of [], and the invariance of the Haar measure. Lemma 2 is applied to show that, for almost all ω Ω, the estimate T 0 ζ(σ +it,ω;a) 2 O(T), T, is valid for σ > 2. From this, using the Gallagher lemma, Lemma.4 of [5], we deduce that, for almost all ω Ω, the estimate N + ζ(σ +imh,α,ω;a) 2 O(), T, is valid for σ > 2. Analogically, we obtain, for σ > 2, the bound N + ζ(σ +imh,α;a) 2 O(), T. Using the latter estimates and contour integration, we arrive to the following assertion.

4 A discrete it theorem for the periodic Hurwitz zeta-function 93 Lemma 4. Suppose that the set L(α,h,π) is linearly independent over Q and that σ > 2. Then sup n N N + and, for almost all ω Ω, sup n N N + Let, for A B(C), ζ(σ +imh,α;a) ζ n (σ +imh,α;a) 0, ζ(σ +imh,α,ω;a) ζn (σ +imh,α,ω;a) 0. P N,ω (A) N + #{ 0 m N: ζ(σ +imh,α,ω;a) A }. Lemma 5. Suppose that the set L(α,h,π) is linearly independent over Q and that σ > 2. Then P N and P N,ω both converge weakly to the same probability measure P on (C,B(C)) as N. Proof. First we show that the family of probability measures {P n : n N} tight. Therefore, by the Prokhorov theorem [], this family is relatively compact. Hence, there exists a sequence {P nk } {P n } such that P nk converges weakly to a certain probability measure P on (C,B(C)) as k. This, Lemmas 3 and 4, and Theorem 4.2 of [] prove the lemma. Proof of Theorem. In virtue of Lemma 5, it suffices to show that the measure P in Lemma 5 coincides with P ζ. Let A be an arbitrary continuity set of the measure P, i.e., P( A) 0, where A is the boundary of A. Then Lemma 5 and the equivalent of weak convergence of probability measures in terms of continuity sets, Theorem 2. of [], imply that n N + #{ 0 m N: ζ(σ +imh,α,ω;a) A } P(A). () Now, on the probability space (Ω,B(Ω),m H ), define the random variable θ by the formula { if ζ(s,α,ω;a) A, θ(ω) 0, otherwise. Obviously, the expectation Eθ of the random variable θ is given by ( ) Eθ θdm H m H ω Ω: ζ(σ,α,ω;a) A Pζ (A). (2) Ω Now we apply Lemma, and obtain by the classical Birkhoff Khintchine ergodicity theorem that N N + θ ( ψh m (ω) ) Eθ (3) Liet. matem. rink. Proc. LMS, Ser. A, 56, 205,

5 94 A. Rimkevičienė for almost all ω Ω. On the other hand, the definitions of the random variable θ and the transformation ψ h yield the equality N + θ ( ψ m h (ω) ) N + #{ 0 m N: ζ(σ +imh,α,ω;a) A }. This together with (2) and (3) shows that, for almost all ω Ω, N N + #{ 0 m N: ζ(σ +imh,α,ω;a) A } P ζ (A). Hence, in view of (), we obtain that P(A) P ζ (A). Since the set A was arbitrary, we have that P(A) P ζ (A) for all continuity sets of the measure P. However, the continuity sets constitute the determining class, therefore, P(A) P ζ (A) for all A B(C). References [] P. Billingsley. Convergence of Probability Measures. John Wiley and Sons, New York, 968. [2] D. Genienė and A. Rimkevičienė. A joint it theorems for the periodic Hurwitz zetafunctions with algebraic irrational parameters. Math. Model. Anal., 8():49 59, 203. [3] A. Laurinčikas. A discrete universality theorem for the Hurwitz zeta-function. J. Number Theory, 43: , 204. [4] G. Misevičius and A. Rimkevičienė. A joint it theorems for the periodic Hurwitz zeta-functions II. Annales Univ. Sci. Budapest., Sect. Comp., 4:73 85, 203. [5] H.L. Montgomery. Topics in Multiplicative Number Theory. Lect. Notes Math., vol Springer-Verlag, Berlin, New York, 97. [6] A. Rimkevičienė. Limit theorems for the periodic Hurwitz zeta-function. Šiauliai Math. Semin., 5(3):55 69, 200. [7] A. Rimkevičienė. Joint it theorems for the periodic Hurwitz zeta-functions. Šiauliai Math. Semin., 6(4):53 68, 20. REZIUMĖ Diskrečioji ribinė teorema periodinei Hurvico dzeta funkcijai A. Rimkevičienė Straipsnyje įrodyta diskretaus tipo ribinė teorema, silpnojo tikimybinių matų konvergavimo prasme, periodinei Hurvico dzeta funkcijai kompleksinėje plokštumoje. Raktiniai žodžiai: Hurvico dzeta funkcija, ribinė teorema, silpnasis konvergavimas, tikimybinis matas.