Annales Univ. Sci. Budapest., Sect. Comp. 4 (23) 73 85 JOIN LIMI HEOREMS FOR PERIODIC HURWIZ ZEA-FUNCION. II G. Misevičius (Vilnius Gediminas echnical University, Lithuania) A. Rimkevičienė (Šiauliai State College, Lithuania) Dedicated to Professors Zoltán Daróczy Imre Kátai on their 75th birthday Communicated by Bui Minh Phong (Received March 27, 23ccepted April 7, 23) Abstract. In the paper, we prove a joint limit theorem for a collection of periodic Hurwitz zeta-functions with transcendental rational parameters.. Introduction Let s σ + it be a complex variable, < α, be a fixed parameter, a {a m : m N N {} be a periodic sequence of complex numbers with minimal period k N. he periodic Hurwitz zeta-function ζ(s) is defined, for σ >, by the series ζ(s, a) a m (m + α) s, continues analytically to the whole complex plane, except, maybe, for a simple pole at the point s with residue a def k a m. k Key words phrases: Distribution, limit theorem, periodic Hurwitz zeta-function, probability measure. 2 Mathematics Subject Classification: M4.
74 G. Misevičius A. Rimkevičienė If a, then the function ζ(s) is entire. his easily follows from the equality ζ(s, a) k ( k s a m ζ s + m ), σ >, k where ζ(s) is the classical Hurwitz zeta-function. In [5], two joint limit theorems on the weak convergence of probability measures on the complex plane for periodic Hurwitz zeta-functions were proved. For j,..., r, let ζ(s j, a j ) be a periodic Hurwitz zeta-function with parameter α j, < α j, periodic sequence of complex numbers a j {a mj : m N } with minimal period k j N. For brevity, we use the notation σ (σ,..., σ r ), σ + it (σ + it,..., σ r + it) (α,... r ), a (a,..., a r ) ζ(s) ( ζ(s, ),... ζ(s r r ) ). Denote by B(S) the class of Borel sets of the space S, by measa the Lebesgue measure of a measurable set A R. hen in [5], the weak convergence as of the probability measure P (A) def meas{ t [, ] : ζ(σ + it) A }, A B(C r )), was discussed. he cases of algebraically independent rational parameters α,... r were considered. For statements of the mentional results, we need some notation definitions. Denote by γ {s C : s } the unit circle on the complex pane, define Ω γ p, γ m Ω 2 p where γ m γ for all m N, γ p γ for all primes p, respectively. he tori Ω Ω 2 are compact topological Abelian groups with respect to the product topology the operation of pointwise multiplication. Moreover, let Ω r Ω j, where Ω j Ω for j,..., r. hen Ω is also a compact topological group. his gives two probability spaces (Ω, B(Ω ), m H ) (Ω 2, B(Ω 2 ), m 2H ), where m H m 2H are the probability Haar measures on (Ω, B(Ω )) (Ω 2, B(Ω 2 )), respectively. Denote by ω j (m) ω 2 (p) the projections of ω j Ω j to γ m, of ω 2 Ω 2 to γ p, respectively. Let ω (ω.... r ) be the elements of Ω. On the probability space (Ω, B(Ω ), m H ) define the C r -valued rom element ζ(σ) by the formula (σ) ( ζ(σ j ),...,
Joint limit theorems for periodic Hurwitz zeta-function. II 75 ζ(σ r r r r ) ), where, for σ j > 2, ζ(σ j j j, a j ) a mj ω j (m), j,..., r. σj (m + α j ) Let P,ζ be the distribution of the rom element ζ(σ). he first joint theorem of [5] is the following statement, heorem.. Suppose that min j > jr 2, that the numbers α,... r are algebraically independent over Q. hen P converges weakly to P,ζ as. Now let α j aj q j, < a j < q j, a j, q j N, (a j, q j ), j,..., r. On the probability space (Ω 2, B(Ω 2 ), m 2H ), define the C r -valued rom element ζ(σ 2 ) by the formula ζ(σ 2 ) ( ζ(σ 2 ),..., ζ(σ r r 2 ; a r ) ), where, for σ j > 2, ζ(σ j j j j ) ω 2 (q j )q σj j a (m aj)/q j,jω 2 (m) m σj, j,..., r, Let P 2,ζ be the distribution of the rom element ζ(σ 2 ). he second joint theorem of [5] is of the following form. heorem.2. For j,..., r, suppose that α j aj q j, < α j < q j, a j, q j N, (α j, q j ), that σ j > 2. hen P converges weakly to P 2,ζ as. he aim of this note is to consider the weak convergence of the probability measure P (A) meas{ t [, ] : ζ(σ +it, a ) A }, A B(C r+r ), where σ (σ,..., σ r, σ,..., σ r ) (α,... r, α,..., α r ), a (a,..., a r, â,..., â r ), ζ(s ) (ζ(s ),..., ζ(a r r ), ζ(s, α ; â ),..., ζ(s, α r ; â r ). Here the parameters α,... r are algebraically independent over Q, while the parameters α,..., α r are rational. For j,..., r, â j {â mj : m N} is a periodic sequence of complex numbers with minimal period k j N. Define Ω Ω Ω 2. hen again Ω is a topological compact group, we have a new probability space (Ω, B(Ω), m H ), where m H is the probability Haar measure on (Ω, B(Ω)). Denote by ω (ω,... r 2 ) the elements of Ω,
76 G. Misevičius A. Rimkevičienė on the probability space (Ω, B(Ω), m H ), define the C r+r -valued rom element ζ(σ ) by the formula where, for σ j > 2,, for σ j > 2, ζ(σ ) ( ζ(σ )...., ζ(σ r r r r ), ζ( σ, α, ω 2 ; â ),..., ζ( σ r, α r 2 ; â r ) ), ζ(σ j j j j ) ζ( σ j, α j 2 ; â j ) ω 2 (q j )q σj j a mj ω j (m), j,..., r, σj (m + α j ) â (m aj)/q j,jω 2 (m) m σj, j,..., r. Let P ζ be the distribution of the rom element ζ(σ ). Now we state the main result of the paper. heorem.3. Suppose that min ( min σ j, min σ j ) > jr jr 2, the numbers α,... r are algebraically independent over Q, that, for j,..., r, α j aj q j, < a j < q j, a j, q j N, (a j, q j ). hen P converges weakly to P ζ as. 2. A limit theorem on Ω Denote by P the set of all prime numbers. Lemma 2.. Suppose that the numbers α,... r are algebraically independent over Q. hen Q (A) def meas{ t [, ] : ( ((m + α ) it : m N ),..., ((m + α r ) it : m N ), (p it : p P) ) A }, A B(Ω), converges weakly to the Haar measure m H as. Proof of the lemma is given in [3, heorem 3].
Joint limit theorems for periodic Hurwitz zeta-function. II 77 3. Limit theorems for absolutely convergent series Let σ > 2 be a fixed number, { ( ) σ } m + αj u n (m j ) exp, m N, n N, j,..., r, n + α j { ( m ) } σ v n (m) exp, m, n N. n For j,..., r, the sequence a j is bounded. herefore, a stard application of the Mellin formula contour integration imply the absolute convergence for σ > 2 of the series ζ n (s j j ) ζ n (s j j j ) For j,..., r, define f(s, α j ) q s j hen we have that f n (s, α j ; â j ) a mj u n (m j ) (m + α j ) s, j,..., r, a mj ω j (m)u n (m j ) (m + α j ) s, j,..., r. â (m aj)/q j,jv n (m) m s. ζ(s, α j ; â j ) f(s, α j )f(s, α j ; â j ), j,..., r. Also, for j,..., r, define f( σ j, α j 2 ) ω 2 (q j )q σj j f n ( σ j, α j 2 ; â j ) â (m aj)/q j,jω 2 (m)v n (m) m σj. hen, similarly as above, we have that the series for f n (s, α j ; â j ) f n ( σ j, α j 2 ; â j ) converge absolutely for σ > 2. Let, for brevity, F n (σ ) ( ζ n (σ ),..., ζ n (σ r r r ), f( σ, α ), f n ( σ, α ; â ),...,..., f( σ r, α r ), f n ( σ r, α r ; â r ) )
78 G. Misevičius A. Rimkevičienė F n (σ ) ( ζ n (σ ),..., ζ n (σ r r r r ), f( σ, α 2 ), f n ( σ, α 2 ; â ),..., f( σ r, α r 2 ), f n ( σ r, α r 2 ; â r ) ). Lemma 3.. Suppose that the numbers α,... r are algebraically independent over Q, that min ( min σ ) j, min σ j > jr jr 2. hen the probability measures P,n (A) def meas{ t [, ] : F n (σ +it ) A }, A B(C r+2r ),, for a fixed ω Ω, P,n (A) def meas{ t [, ] : F n (σ +it 2 ) A }, A B(C r+2r ), both converge weakly to the same probability measure P n on (C r+2r, B(C r+2r )) as. Proof. he series defining ζ n (s j j ), j,..., r, f n (s, α j ; â j ), j,..., r, converge absolutely for σ > 2. herefore, the function h n : Ω C r+2r given by the formula h n (ω) F n (σ ) is continuous. Moreover, h n ( (p it : p P ), ((m + α ) it : m N ),..., ((m + α r ) it : m N ) ) F n (σ +it ). hus, we have that P,n Q h n, where Q is the measure of Lemma 2.. his, the continuity of h n, Lemma 2. heorem 5. of [] show that P,n converges weakly to P n m H h n as. Similar arguments give that the measure P,n converges weakly to m H h n as, where the function h n : Ω C r+2r is related to h n by the equality h n (ω ) h n (ω ω ). he invariance of the Haar measure m H with respect to translates by points from Ω leads to the equality m H h n lemma is proved. m H h n. he
Joint limit theorems for periodic Hurwitz zeta-function. II 79 4. Approximation in the mean Let, for j,..., r σ >, f(s, α j ; â j ) â (m aj)/q j,j m s Define f(s, α j 2 ; â j ) â (m aj)/q j,jω 2 (m) m s. F (σ ) ( ζ(σ ),..., ζ(σ r r r ), f( σ, α ), f( σ, α ; â ),..., f( σ r, α r ), f( σ r, α r ; â r ) ), F (σ ) ( ζ(σ ),..., ζ(σ r r r r ), f( σ, α 2 ), f( σ, α 2 ; â ),..., f( σ r, α r 2 ), f( σ r, α r 2 ; â r ) ). In this section, we approximate F (σ ) by F n (σ ), F (σ ) by F n (σ ) in the mean. Denote by ϱ ϱ r+2r the Euclidean metric on C r+2r. Lemma 4.. Suppose that min ( min jr σ j, min jr σ j ) > 2. hen lim lim sup ϱ ( F (σ +it ), F n (σ +it ))dt. n Proof. he lemma follows from one-dimensional results obtained in [4], Lemma 6 equality (3), from the definition of ϱ. Lemma 4.2. Suppose that the numbers α,... r are algebraically independent over Q, that min ( min σ ) j, min σ j > jr jr 2. hen, for almost all ω Ω, lim lim sup ϱ ( F (σ +it ω ), F n (σ +it ω ))dt. n
8 G. Misevičius A. Rimkevičienė Proof. he algebraic independence of the numbers α,... r implies their transcendence. herefore, the lemma is a consequence of similar one-dimensional equalities given in [4], Lemma 7 equality (4), of the fact that the Haar measure m H is the product of the Haar measures on (Ω j, B(Ω j )), j,..., r, (Ω 2, B(Ω 2 )). 5. Proof of heorem.3 We start with the following statement. Lemma 5.. Suppose that the numbers α,... r are algebraically independent over Q, that min ( min σ ) j, min σ j > jr jr 2. hen the probability measures P, (A) def { t [, ] : F (σ +it ) A }, A B(C r+2r ), P, (A) def { t [, ] : F (σ +it ω ) A }, A B(C r+2r ), converge weakly to the same probability measure P on (C r+2r, B(C r+2r )) as. Proof. For the proof of the lemma, it suffices to pass from the measures P,n P,n to the measures P, P,, respectively. Let θ be a rom variable defined on a certain probability space ( Ω, B( Ω), P) uniformly distributed on [, ]. Define X,n (σ ) ( X,n, (σ ),..., X,n,r (σ r ), X, ( σ ), X,n, ( σ ),...,..., X,r ( σ r ), X,n,r ( σ r ) ) F n (σ +iθ ). hen, denoting by D the convergence in distribution, we have, in view of Lemma 3., that, for min( min σ j, min σ j ) > jr jr 2, (5.) where D X,n (σ) X n (σ), X n (σ ) ( X n, (σ ),..., X n,r (σ r ), X ( σ ), X n, ( σ ),..., X r ( σ r ) X n,r ( σ r ) )
Joint limit theorems for periodic Hurwitz zeta-function. II 8 is the C r+2r -valued rom element with the distribution P n, P n is the limit measure in Lemma 3.. It is not difficult to see that the family of probability measures {P n : n N} is tight. Indeed, the series for ζ n (s j ) f n (s, α j ; â j ) are convergent absolutely for σ > 2. herefore, we have that, for σ j > 2 σ j > 2, lim ζ n (σ j + it j j ) 2 dt lim f n ( σ j + it, α j ; â j ) 2 dt a mj 2 u 2 n(m j ) (m + α j ) 2σj a mj 2 (m + α j ) 2σj â (m aj)/q j,j 2 vn(m) 2 m 2 σj â (m aj)/q j,j 2 m 2 σj for n N, j,..., r, j,..., r, respectively. Now, denoting ( R j R j (σ j ) a mj 2 ) /2 (m + α j ) 2σj R j R j ( σ j ) ( â (m aj)/q j,j 2 m 2 σj ) /2, we obtain that (5.2) lim sup ζ n (σ j + it j j ) dt R j (σ j ), j,..., r, (5.3) lim sup f n ( σ j + it, α j ; â j ) dt R j ( σ j ), j,..., r.
82 G. Misevičius A. Rimkevičienė Let ε be an arbitrary positive number, M j R j (3r) ε, j,..., r, M j q j (3r ) ε, M 2j R j (3r) ε, j,..., r. hen we deduce from (5.2) (5.3) that (5.4) lim sup P ( ( j : X,n,j (σ j ) > M j ) ( j : X,j ( σ j ) > > M j ( j : X,n,j ( σ j ) > M 2j ) ) r lim sup P( X,n,j (σ j ) > M j )+ + r j r sup n N + + r r lim sup P( X,j (σ j ) > M j ) + lim sup lim sup sup n N M j M j lim sup M 2j r ζ n (σ j + it j j ) dt f( σ j + it, α j ) dt f n ( σ j + it, α j ; â j ) dt ε. lim sup P( X,n,j ( σ j ) > M 2j ) his (5.) show that, for all n N, P ( ( j : X n,j (σ j ) > M j ) ( j : X j ( σ j ) > M j ) ( j : X n,j ( σ j ) > M 2j ) ε. Let ( r M Mj 2 + r M 2 j + r M 2 2j) /2. Define the set K ε {z C 2+2r : ϱ(z, ) M}. hen K ε is a compact subset of C r+2r,, by (4) P(X n (σ ) K ε ) ε for all n N, or equivalently, P n (K ε ) ε for all n N. his means that the family {P n : n N} is tight. Hence, by the Prokhorov theorem, heorem 6. of [], it is relatively compact. herefore, there exists a subsequence {P nk } {P n } such that P nk converges weakly to a certain probability measure P on (C r+2r, B(C r+2r )) as k, that is (5.5) D X nk (σ) P. k
Joint limit theorems for periodic Hurwitz zeta-function. II 83 Define the C r+2r -valued rom element X (σ ) F (σ +iθ ). using Lemma 4., we find that, for every ε >, hen, lim n lim sup P(ϱ(X (σ), X,n (σ)) ε). his, (5.), (5.5) heorem 4.2 of [] give the relation (5.6) D X (σ) P, we have that P, converges weakly to P as. Moreover, (5.6) shows that the measure P is independent of the sequence {P nk }. Hence, D X n (σ) P. n Similar arguments applied for the C r+2r -valued rom elements X,n (σ ) F n (σ +iθ ) X (σ ) F (σ +iθ ) together with Lemma 4.2 (6) show that the measure P, also converges weakly to P as. Proof of heorem.3. First we identify the limit measure P in Lemma 5.. For this, we apply the ergodicity of the one-parameter group {ϕ t : t R}, where ϕ t (ω ) ( (m + α ) it : m N ),..., (m + α r ) it : m N ), (p it : p P) ) ω Ω, of measurable measure preserving transformations on Ω [3], Lemma 7. We fix a continuity set A of the measure P in Lemma 5.. hen, by heorem 2. of [] Lemma 5., we have that (5.7) lim meas{t [, ] : F (σ +it ) A} P (A). Let the rom variable ξ be defined on (Ω, B(Ω), m H ) by the formula { if F (σ ) A, ξ(ω) otherwise. hen the expectation (5.8) Eξ m H (ω Ω : F (σ ) A) P F (A),
84 G. Misevičius A. Rimkevičienė where P F is the distribution of F. he ergodicity of the group {ϕ t : t R} implies that of thew rom process ξ(ϕ t (ω )). herefore, by the Birkhoff- Khintchine theorem. see, for example, [2], we obtain that, for almost all ω Ω, (5.9) lim ξ(ϕ t (ω ))dt Eξ. However, the definitions of ξ ϕ t show that ξ(ϕ t (ω ))dt meas{t [, ] : F (σ +it ) A}. his together with (5.8) (5.9) leads, for almost all ω Ω, to lim meas{t [, ] : F (σ +it ) A} P F (A). Hence, find that P (A) P F (A) for all continuity sets A of P. Hence, P coincides with P F. It remains to pass from P, to P. Define the function h : C r+2r C r+r by the formula h(z,..., z r, z, z 2,..., z r, z r2 ) (z,..., z r, z, z 2,..., z r, z r2 ). hen h is a continuous function, P P, h. his, the weak convergence of P, to P F heorem 5. of [] show that the measure P converges weakly to P F h as. Moreover, for A B(C r+re ), P F h (A) m H h (ω Ω : F (σ ) A) m H (ω Ω : F (σ ) h A) m H (ω Ω : h(f (σ ) A) m H ( ω Ω : (ζ(σ ),..., ζ(σ r r r r ), f( σ, α 2 ) f( σ, α 2 ; â ),..., f( σ r, α r 2 )f( σ r, α r 2 ; â r )) A ) m H ( ω Ω : (ζ(σ ),..., ζ(σ r r r ), ζ( σ, α 2 ; â ),..., ζ( σ r, α r 2 ; â r )) A ) m H (ω Ω : ζ ) A) P ζ (A). (σ hus, the measure P converges weakly to P ζ proved. as. he theorem is
Joint limit theorems for periodic Hurwitz zeta-function. II 85 References [] Billingsley, P., Convergence of Probability Measures, Wiley, New York, 968. [2] Cramér H., M.R. Leadbetter, Stationary Related Stochastic Processes, Wiley, Nework, 967. [3] Laurinčikas, A., Joint universality of zeta-functions with periodic coefficients, Izv.RAN, Ser.Matem., 74(3) (2), 79 2 (in Russian) Izv.:Math., 74(3) (2), 55 539. [4] Rimkevičienė, A., Limit theorems for the periodic Hurwitz zetafunction, Šiauliai Math. Seminar, 5(3) (2), 55 69. [5] Rimkevičienė, A., Joint limit theorems for periodic Hurwitz zetafunctions, Šiauliai Math. Seminar, 6(4) (2), 53 68. G. Misevičius Vilnius Gediminas echnical University Saulėtekio av. L-233 Vilnius Lithuania gintautas.misevicius@gmail.com A. Rimkevičienė Šiauliai State College Aušros al. 4 L-7624 Šiauliai Lithuania audronerim@gmail.com