JOINT VALUE-DISTRIBUTION THEOREMS ON LERCH ZETA-FUNCTIONS. II

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1 Liet. Matem. Rink.,, No., 24, JOIN VALUE-DISRIBUION HEOREMS ON LERCH ZEA-FUNCIONS. II A. Laurinčikas Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, L-3225 Vilnius ( Kohji Matsumoto Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, , Japan ( Abstract. We give corrected statements of some theorems from [5] and [6] on joint value distribution of Lerch zeta-functions (limit theorems, universality, functional independence). We also present a new direct proof of a joint limit theorem in the space of analytic functions and an extension of a joint universality theorem. Key words: Lerch zeta-function, limit theorem, probability measure, space of analytic functions, support, universality, weak convergence.. INRODUCION Let, as usual, s = σ + it, denote a complex variable, and let N, N, Z, R and C be the set of all positive integers, non-negative integers, integers, real and complex numbers, respectively. he Lerch zeta-function L(λ, α, s) with fixed parameters α, λ R, < α, for σ >, is defined by L(λ, α, s) = e 2πiλm (m + α) s. If λ Z, then the Lerch zeta-function becomes the Hurwitz zeta-function ζ(s, α) = (m + α) s, σ >, and ζ(s, α), for α =, reduces to the Riemann zeta-function. he paper is conditioned by [5] and [6], where the joint value distribution of Lerch zeta-functions was considered and some inaccuracies in the statements of some theorems in these papers were remained. he aim of this paper is to correct and comment the results of [5] and [6], and to give for some of them new proofs

2 2 A. Laurinčikas, K. Matsumoto as well as certain their extensions. For this, first we recall the results of [5]. Denote by B(S) the class of Borel sets of the space S, and let, for >, ν t (...) = meas{t [, ] :...}, where meas{a} is the Lebesgue measure of a measurable set A R, in place of dots a condition satisfied by t is to be written, and the sign t in ν t means that the measure is taken over t [, ]. Let r N \ {}, and let L(λ, α, σ + it),..., L(λ r, α r, σ r + it) be a collection of Lerch zeta-functions. hroughout the paper, as in [5], we suppose that λ j Z, j =,..., r. heorem of [5] remains without any changes. It asserts that, for min σ j > j r 2, on (Cr, B(C r )), there exists a probability measure P such that the probability measure ν t ((L(λ, α, σ + it),..., L(λ r, α r, σ r + it)) A), A B(C r ), converges weakly to P as. Now we will present the corrected statement of heorem 2 from [5]. his theorem deals with joint distribution of Lerch zetafunctions in the space of analytic functions, and differently from heorem, gives the explicit form of the limit measure. For its statement, we need some additional notation. For D = {s C : σ > 2 }, denote by H(D) the space of analytic on D functions equipped with the topology of uniform convergence on compacta, and let H r (D) = H(D)... H(D). Let γ = {s C : s = }, and define }{{} r Ω = γ m, where γ m = γ for all m N. By the ikhonov theorem the infinite-dimensional torus Ω is a compact topological Abelian group, therefore, the probability Haar measure m H on (Ω, B(Ω)) can be defined. his gives a probability space (Ω, B(Ω), m H ). Denote by ω(m) the projection of ω Ω to the coordinate space γ m. Moreover, define Ω r = Ω... Ω r, where Ω j = Ω for j =,..., r. hen Ω r is also a compact topological group. Denote by m Hr the probability Haar measure on (Ω r, B(Ω r )), and on the probability space (Ω r, B(Ω r ), m Hr ) define an H r (D)-valued random element L(s, ω) by L(s, ω) = (L(λ, α, s, ω ),..., L(λ r, α r, s, ω r )), where L(λ j, α j, s, ω j ) = e 2πiλ jm ω j (m) (m + α j ) s,

3 Joint value-distribution theorems on Lerch zeta-functions. II 3 ω j Ω, j =,..., r, and ω = (ω,..., ω r ). Denote by P L the distribution of the random element L(s, ω), that is P L (A) = m Hr (ω Ω r : L(s, ω) A), A B(H r (D)), and define the probability measure P by P (A) = ν τ ((L(λ, α, s + iτ),..., L(λ r, α r, s + iτ)) A), A B(H r (D)). We recall that the numbers a,..., a r are algebraically independent over the field of rational numbers Q if the coefficients of every polynomial p with rational coefficients satisfying p(a,..., a r ) = are equal to zero. HEOREM. Suppose that α,..., α r are algebraically independent over Q and that λ j Z, j =,..., r. hen the probability measure P converges weakly to P L as. heorem is the corrected statement of heorem 2 from [5]. In heorem 2 of [5] it is required that the numbers α,..., α r should be transcendental, however, this is not sufficient for the proof of Lemma 9 of [5]. Since, for σ >, the shifting parameters of the functions L(λ, α, s),..., L(λ r, α r, s) are different, for the proof of Lemma 9 of [5] we need a limit theorem on the torus Ω r (see Lemma 4 below). Moreover, in this theorem, the limit measure must be the Haar measure. herefore, we have to use a condition that the set r j= {log(m + α j )} should be linearly independent over Q. he latter requirement is satisfied if the numbers α,..., α r are algebraically independent over Q. his is also used in application of elements of ergodic theory for the identification of the limit measure. So, except for the above change in the proof of Lemma 9 of [5], the proof of heorem 2 from [5] remains the same, and only in its statement the condition of the transcendence of α,..., α r is changed by the algebraic independence over Q. Below we will give a new direct proof of heorem which does not use the modified Cramér-Wald criterion (a statement of the type of Lemma 9 in [5]). We note that heorem 3 of [5] is true. In the case of rational λ, the Dirichlet series for L(λ, α, s) is reduced to ordinary Dirichlet series (with exponents log m), therefore, even in the multidimensional case, we can use a limit theorem on the torus ˆΩ = γ p, p where γ p = γ for all primes p, and the linear independence over Q of logarithms of prime numbers. he further proof runs in a standard way. he same changes must be done also in heorems and 2 from [6], on the joint universality and functional independence of Lerch zeta-functions, since their

4 4 A. Laurinčikas, K. Matsumoto proof is based on heorem 2 of [5]. hus, in heorems and 2 of [6] the condition that α,..., α r are transcendental must be changed by a more strong requirement that the numbers α,..., α r are algebraically independent over Q. We also note that the same changes must be done in heorems 5.3., 5.3.2, 6.3. and 7.2. of [4]. For example, the transcendental numbers α = e and α 2 = e 2 are not algebraically independent over Q, therefore they do not satisfy the hypotheses of heorem. On the other hand, it is known that the numbers e π and π are algebraically independent over Q. So, we can take, for example, α = e π /2 and α 2 = π/4 in heorem. Now we state one generalization of heorems and 2 (after the above correction) from [6]. Let λ,..., λ r be arbitrary rational numbers with denominators q,..., q r, respectively. Denote by k = [q,..., q r ] the least common multiple, and define A = e 2πiλ e 2πiλ 2 e 2πiλr e 4πiλ e 4πiλ 2 e 4πiλr e 2πkiλ e 2πkiλ 2 e 2πkiλr HEOREM 2. Suppose that α,..., α r are algebraically independent over Q and that rank(a) = r. Let K j be a compact subset of the strip D = { s C : 2 < σ < } with connected complement, and let f j (s) be a continuous on K j function which is analytic in the interior of K j, j =,..., r. hen, for every ε > lim inf ντ ( sup j r. sup L(λ j, α j, s + iτ) f j (s) < ε s K j ) >. HEOREM 3. Suppose that α,..., α r are algebraically independent over Q and that rank(a) = r. Let F j be a continuous on C Nr function, j =,..., l, and l s j F j (L(λ, α, s),..., L(λ r, α r, s), L (λ, α, s),..., j= L (λ r, α r, s),..., L (N ) (λ, α, s),..., L (N ) (λ r, α r, s)) = identically for s C. hen F j, j =,..., l. heorem 3 is deduced from heorem 2 in the same way as in [6] where heorem 2 of [6] is obtained from heorem of [6]. 2. PROOF OF HEOREM 2.. Joint limit theorems for Dirichlet polynomials. First we will prove a joint limit theorem on the torus Ω r for the probability measure Q,r (A) = ν τ ((( (m + α ) iτ ) ( : m N,..., (m + αr ) iτ )) ) : m N A, A B(Ω r ).

5 Joint value-distribution theorems on Lerch zeta-functions. II 5 LEMMA 4. he probability measure Q,r converges weakly to the Haar measure m Hr on (Ω r, B(Ω r )) as. Proof. he dual group of Ω r is r j= Z mj, where Z mj = Z for all m N and j =,..., r. he element (k,..., k r ) = (k, k,..., k r, k r,...) r Z mj, where only a finite number of integers j= k mj, m N, j =,..., r, are distinct from zero, acts on Ω r by (x,..., x r ) (x k,..., xk r r ) = r j= x k mj mj, x j = (x j, x 2j,...), x mj γ, m N, j =,..., r. herefore, the Fourier transform g,r (k,..., k r ) of the measure Q,r is g,r (k,..., k r ) = = = r exp iτ Ω r j= r j= x k mj mj dq,r = e iτk mj log(m+α j ) dτ = j= m= k mj log(m + α j ) dτ. Since α,..., α r are algebraically independent over Q, we find that if (k,..., k r ) = (,..., ), exp{i g,r (k,..., k r ) = j= i k mj log(m+α j ) j= k mj log(m+α j )} if (k,..., k r ) (,..., ). Consequently, we have { if lim g (k,..., k,r(k,..., k r ) = r ) = (,..., ), if (k,..., k r ) (,..., ). Hence by continuity theorems for probability measures on locally compact groups ([2], heorem.4.2), we obtain that the measure Q,r converges weakly to m Hr as.

6 6 A. Laurinčikas, K. Matsumoto Now let σ j > 2, and, for m, n N, { v j (m, n) = exp Define, for N j N, ˆω j Ω, s D, L Nj,j,n(λ j, α j, s) = L Nj,j,n(λ j, α j, s, ˆω j ) = N j ( m + αj n + α j N j ) σj }, j =,..., r. e 2πiλ jm v j (m, n) (m + α j ) s, e 2πiλ jm v j (m, n)ˆω j (m) (m + α j ) s, j =,..., r, and consider the weak convergence of the probability measures and P,N,...,N r,n(a) = ν τ ((L N,,n(λ, α, s + iτ),..., L Nr,r,n(λ r, α r, s + iτ)) A) ˆP,N,...,N r,n(a) = ν τ ((L N,,n(λ, α, s + iτ, ˆω ),..., L Nr,r,n(λ r, α r, s + iτ, ˆω r )) A), where A B(H r (D)). HEOREM 5. he probability measures P,N,...,N r,n and ˆP,N,...,N r,n both converge weakly to the same probability measure on (H r (D), B(H r (D))) as. Proof. Let a function h : Ω r H r (D) be defined by h(ω,..., ω r ) = ( N e 2πiλ m v (m, n)ω (m) (m + α ) s,..., N r (ω,..., ω r ) Ω r. hen the function h is continuous, and e 2πiλrm v r (m, n)ωr (m + α r ) s h ( ((m + α ) iτ : m N ),..., ((m + α r ) iτ : m N ) ) = ((L N,,n(λ, α, s + iτ),..., L Nr,r,n(λ r, α r, s + iτ)). ) (m), Hence P,N,...,N,n = Q,r h, therefore by Lemma 4 and heorem 5. of [] we obtain that the measure P,N,...,N r,n converges weakly to m Hr h as. Now let h : Ω r Ω r be given by h (ω,..., ω r ) = (ω ˆω,..., ω r ˆω r ). hen (L N,,n(λ, α, s + iτ, ˆω ),..., L Nr,r,n(λ r, α r, s + iτ, ˆω r )) =

7 Joint value-distribution theorems on Lerch zeta-functions. II 7 = h ( h ( ((m + α ) iτ : m N ),..., ((m + α r ) iτ : m N ) )). Similarly as above we find that ˆP,N,...,N r,n converges weakly to the measure m Hr (hh ) as. However, the invariance of the measure m Hr yields and the theorem follows. m Hr (hh ) = (m Hr h )h = m Hr h, 2.2. Limit theorems for absolutely convergent series. For j =,..., r, let ω j Ω, and e 2πiλjm v j (m, n) L n,j (λ j, α j, s) = (m + α j ) s and L n,j (λ j, α j, s, ω j ) = e 2πiλ jm ω j (m)v j (m, n) (m + α j ) s. Note that these two series both converge absolutely for σ > 2. Define the probability measures and P,n (A) = ν τ ((L n, (λ, α, s + iτ),..., L n,r (λ r, α r, s + iτ)) A) ˆP,n (A) = ν τ ((L n, (λ, α, s + iτ, ω ),..., L n,r (λ r, α r, s + iτ, ω r )) A), where A B(H r (D)). HEOREM 6. On (H r (D), B(H r (D))) there exists a probability measure P n such that the measures P,n and ˆP,n both converge weakly to P n as. def Proof. We apply heorem 5 with N =... = N r = N. hen by heorem 5 def the measures P,N,...,N r,n = P,N,n and ˆP def,n,...,n r,n = ˆP,N,n both converge to the same measure P N,n as. First we observe that, for any fixed n, the family {P N,n : N N } is tight. Let η be a random variable defined on a certain probability space (ˆΩ, B(ˆΩ), P) and uniformly distributed on [, ]. Define, for j =,..., r, X,N,j,n = X,N,j,n (s) = L N,j,n (λ j, α j, s + i η), which is an H(D) - valued random element defined on (ˆΩ, B(ˆΩ), P). hen by heorem 5 def D X,N,n = (X,N,,n,..., X,N,r,n ) X N,n, () where X N,n = (X N,,n,..., X N,r,n ) is an H r (D)-valued random element with the D distribution P N,n, and means the convergence in distribution.

8 8 A. Laurinčikas, K. Matsumoto n= For the investigation of the weak convergence of the measures P,n and ˆP,n we need a metric on H r (D) which induces its topology. Let {K n } be a sequence of compact subsets of D such that K n = D, K n K n+, and if K is a compact of D, then K K n for some n. hen ρ(f, g) = n= 2 n ρ n (f, g), f, g H(D), + ρ n (f, g) ρ n (f, g) = sup s K n f(s) g(s), is a metric on H(D) which induces its topology. hen ρ r (f, g) = max j r ρ(f j, g j ), where f = (f,..., f n ) H r (D), g = (g,..., g n ) H r (D), is a desired metric on H r (D). he series for L n,j (λ j, α j, s), j =,..., r, converge absolutely for σ > 2. herefore, for M lj >, j =,..., r, l N, and some σ l > /2 lim sup P j= ( sup s K l X,N,j,n (s) > M lj l j= l j= j= lim sup P ( for at least one j =,..., r sup s K l X,N,j,n (s) > M lj sup lim sup sup L N,j,n (λ j, α j, s + iτ) dτ l M lj N s K l sup lim sup M lj N ( sup lim sup M lj N 2 C l l j= M j= lj 2 2 ( N sup M lj N ( (m + α j ) 2σ l ) L N,j,n (λ j, α j, σ l + it) dt l ) ) /2 L N,j,n (λ j, α j, σ l + it) 2 dt vj 2 (m, n) (m + α j ) 2σ l ) /2 ) /2 def R lj = C l M lj j=

9 with a certain C l > and Joint value-distribution theorems on Lerch zeta-functions. II 9 ( ) /2 R lj = (m + α j ) 2σ <. l aking M lj = C l R lj 2 l r/ε, hence we find that lim sup P ( sup s K l X,N,j,n (s) > M lj Now () and (2) show that, for all l N, Define for at least one j =,..., r ) ε 2 l. (2) P( sup s K l X N,j,n (s) > M lj for at least one j =,..., r) ε 2 l. (3) H r ε = {(f,..., f r ) H r (D) : sup s K l f j (s) M lj, j =,..., r, l N}. hen the set H r ε is compact, and by (3) or, by the definition of X N,n, P(X N,n (s) H r ε ) ε, P N,n (H r ε ) ε for all N N. his proves the tightness of the family {P N,n : N N }. By the definition, for j =,..., r, lim L N,j,n(λ j, α j, s) = L n,j (λ j, α j, s), N the convergence being uniform on compact subsets of D. herefore, for every ε >, lim lim sup ν τ (ρ r (L N,n (s + iτ), L n (s + iτ)) ε) N Here lim lim sup ρ r (L N ε N,n (s + iτ), L n (s + iτ))dτ lim lim sup N ε j= ρ(l N,j,n (s + iτ), L n,j (s + iτ))dτ =. (4) L N,n (s) = (L N,,n (λ, α, s),..., L N,r,n (λ r, α r, s))

10 A. Laurinčikas, K. Matsumoto and L n (s) = (L n, (λ, α, s),..., L n,r (λ r, α r, s)). Now define, for j =,..., r, X,j,n = X,j,n (s) = L n,j (λ j, α j, s + i η), and put hen in view of (4) X,n = (X,,n,..., X,r,n ). lim N lim sup P(ρ r (X,N,n, X,n ) ε) =. (5) Since the family {P N,n : N N } is tight, by the Prokhorov theorem it is relatively compact. Let {P N,n} {P N,n } be such that P N,n converges weakly to some measure P n as N. hen we have that X N,n D N P n. (6) he space H r (D) is separable, and (), (5) and (6) show that the hypothesis of heorem 4.2 from [] are satisfied. Consequently, X,n D P n, (7) and this is equivalent to the weak convergence of the measure P,n to P n as. Formula (7) shows that the measure P n is independent of the sequence N. herefore, we have D X N,n P n. (8) N Now, repeating the above arguments for the random elements X,N,n = (L N,,n (λ, α, s + i η, ω),..., L N,r,n (λ r, α r, s + i η, ω)) and X,n = (L n, (λ, α, s + i η, ω),..., L n,r (λ r, α r, s + i η, ω)), and using (8) we similarly obtain that the measure P,n also converges weakly to P n as. he theorem is proved Proof of heorem. We start with approximation in mean of the vectors (L(λ, α, s),..., L(α, λ, s)) and (L(λ, α, s, ω ),..., L(λ r, α r, s, ω r )) by the vectors (L n, (λ, α, s),..., L n,r (λ r, α r, s, )) and (L n, (λ, α, s, ω ),..., L n,r (λ r, α r, s, ω r )), respectively. Let L(s) = (L(λ, α, s),..., L(λ r, α r, s)), and L(s, ω) = (L(λ, α, s, ω ),..., L(λ r, α r, s, ω r )) L n (s, ω) = (L n, (λ, α, s, ω ),..., L n,r (λ r, α r, s, ω r )).

11 Joint value-distribution theorems on Lerch zeta-functions. II LEMMA 7. We have and, for almost all ω Ω r, lim lim sup ρ r (L(s + iτ), L n n (s + iτ))dτ = lim lim sup ρ r (L(s + iτ, ω), L n n (s + iτ, ω))dτ =. Proof. Since α,..., α r are algebraically independent over Q, they are transcendental. herefore, for each j =,..., r, Lemmas 5.2. and of [4] imply lim lim sup ρ(l(λ j, α j, s + iτ), L n,j (λ j, α j, s + iτ))dτ = n and, for almost all ω j Ω, lim lim sup ρ(l(λ j, α j, s + iτ, ω j ), L n,j (λ j, α j, s + iτ, ω j ))dτ =. n From this and the definition of ρ r the lemma follows. For A B(H r (D)), define P (A) = ν τ ((L(λ, α, s + iτ, ω ),..., L(λ r, α r, s + iτ, ω r )) A). HEOREM 8. On (H r (D), B(H r (D))) there exists a probability measure P such that the measures P and P (for almost all ω) both converge weakly to P as. Proof. We use the same way as that in the proof of heorem 6. First we will show that the family of probability measures {P n : n N } is tight. By heorem 6 D X,n X n, (9) where X n = (X n,,..., X n,r ) is an H r (D) - valued random element with the distribution P n. Since L n,j (λ j, α j, s) is convergent absolutely for σ > /2, j =,..., r, we have lim L n,j (λ j, α j, σ + it) 2 dt = vj 2 (m, n) (m + α j ) 2σ (m + α j ) 2σ

12 2 A. Laurinčikas, K. Matsumoto uniformly in n. Hence it is not difficult to see that, for some σ l > /2, lim sup ( sup L n,j (λ j, α j, s+iτ) dτ C l lim sup s K l 2 with a certain constant C l >, where C l R lj < ( ) /2 R lj = (m + α j ) 2σ. l herefore, for M lj = C l R lj 2 l r/ε, j =,..., r, l N, 2 ) /2 L n,j (λ j, α j, σ l +it) 2 dt lim sup P( sup X,j,n (s) > M lj for at least one j =,..., r) ε s K l 2 l. his and (9) imply Hence we find that P( sup s K l X n,j (s) > M lj for at least one j =,..., r) ε 2 l. P n (H r ε ) ε for all n N, that is the family {P n : n N } is tight. Now let, for j =,..., r, and let X,j = X,j (s) = L(λ j, α j, s + i η), X = (X,,..., X,r ). hen in view of Lemma 7, for every ε >, Hence lim n lim n lim sup ν τ (ρ r (L(s + iτ), L n (s + iτ)) ε) ρ r (L(s + iτ), L ε n (s + iτ))dτ =. lim sup P(ρ r (X (s), X,n (s)) ε) =. () he family {P n : n N } is relatively compact. Let {P n } {P n } be such that P n converges weakly to some measure P on (H r (D), B(H r (D))) as n. hen D X n P. n

13 Joint value-distribution theorems on Lerch zeta-functions. II 3 his, (), (9) and heorem 4.2 of [] show that X D P, () that is P converges weakly to P as. By () the measure P is independent on the sequence n. herefore, X n D P. (2) n By the same way as above, using (2), we obtain that the measure P for almost all ω also converges weakly to P as. o identify the limit measure in heorem 8, we will apply some facts from ergodic theory. Let a τ,j = {(m + α j ) iτ : m N }, τ R, j =,..., r. hen {a τ,j : τ R}, for each j =,..., r, is a one - parameter group. Define the one - parameter family {Φ τ : τ R} = {ϕ τ,,..., ϕ τ,r : τ R} of transformations on Ω r by ϕ τ,j (ω j ) = a τ,j ω j, ω j Ω j, j =,..., r. hen we have a one - parameter group of measurable transformations on Ω r. LEMMA Proof. 9. he one - parameter group {Φ τ : τ R} is ergodic. Let χ : Ω r γ be a character. hen χ(ω) = r j= ω k mj j (m), ω Ω r, where only a finite number of integers k mj are distinct from zero. Suppose that χ is a non - principal character. hen χ(a τ,,..., a τ,r ) = r j= (m + α j ) iτk mj = exp{ iτ j= k mj log(m + α j )}, where only a finite number of integers k mj. Since α,..., α r are algebraically independent, we have that there exists a τ such that χ(a τ,,..., a τ,r). he further proof runs in the same way as than in [3], heorem Proof of heorem. Let A B(H r (D)) be a continuity set of the measure P in heorem 8. hen by heorem 2. of [] and heorem 8 lim ντ ((L(λ, α, s + iτ, ω ),..., L(λ r, α r, s + iτ, ω r )) A) = P (A) (3) for almost all ω Ω r. Now we fix the set A and define a random variable θ on the space (Ω r, B(Ω r ), m Hr ) by the formula { if L(s, ω) A, θ(ω) = if L(s, ω) / A.

14 4 A. Laurinčikas, K. Matsumoto Denote by E(θ) the expectation of θ. hen we have that E(θ) = Ω r θdm Hr = m Hr (ω Ω r : L(s, ω) A) = P L (A). (4) In virtue of Lemma 9 the random process θ(φ τ (ω)) is ergodic. herefore, by the Birkhoff-Khinchine theorem lim θ(φ τ (ω))dτ = E(θ) (5) for almost all ω Ω r. On the other hand, by the definition of θ and Φ τ we find θ(φ τ (ω))dτ = ν τ ((L(λ, α, s + iτ, ω ),..., L(λ r, α r, s + iτ, ω r )) A). his and (4), (5) show that lim ντ ((L(λ, α, s + iτ, ω ),..., L(λ r, α r, s + iτ, ω r )) A) = P L (A) for almost all ω Ω r. Hence, by (3), P (A) = P L (A) for any continuity set A of the measure P, and this implies the equality P (A) = P L (A) for all A B(H r (D)). he theorem is proved. 3. PROOF OF HEOREM A limit theorem. he proof of heorem 2 is based on a corollary of heorem. Denote by P L the restriction of the distribution P L of the random element L(s, ω) to H r (D ). COROLLARY. Suppose that α,..., α r are algebraically independent over Q. hen the probability measure ν τ ((L(λ, α, s + iτ),..., L(λ r, α r, s + τ)) A), A B(H r (D )), converges weakly to P L as. Proof. he function h : H(D) H r (D ) defined by h(g) = g s D, g H r (D), obviously, is continuous. herefore, the corollary follows from heorem and heorem 5. of [] he support of the measure P L. We recall that the support of the measure P L is a minimal closed set S PL H r (D ) such that P SPL =. he set S PL consists of all g H r (D) such that for every neighborhood G of g the inequality P L (G) > holds.

15 Joint value-distribution theorems on Lerch zeta-functions. II 5 HEOREM. Suppose that α,..., α r are algebraically independent over Q and rank(a) = r. hen the support of P L is the whole of H r (D ). Let where L (s, ω) = (L(λ, α, s, ω),..., L(λ r, α r, s, ω)), s D, L(λ j, α j, s, ω) = ω Ω, j =,..., r. hen, clearly, S PL SP L = H r (D ). e 2πimλ j ω(m) (m + α j ) s, SP L. herefore, it suffices to prove that he support of each ω(m) is the unit circle γ. hus, the support of is the set e 2πimλ j ω(m) (m + α j ) s {g H(D ) : g(s) = e2πimλ j a (m + α j ) s, a γ}, m N, j =,..., r. Since {ω(m) : m N } is a sequence of independent random variables defined on the probability space (Ω, B(Ω), m H ), { e2πimλ j ω(m) (m + α j ) s : m N }, j =,..., r, is a sequence of independent H(D ) - valued random elements defined on the above space. herefore, by Lemma 5 of [7] the support is the closure of the SP L set of all convergent series ( ) e 2πimλ a m (m + α ) s,..., e2πimλr a m (m + α r ) s, (6) where a m γ. hus, we have to show that the latter set is dense in H r (D ). For this we will use Lemma 6 of [7]. Let {b m : b m γ, m N } be a sequence such that the series ( ) e 2πimλ b m (m + α ) s,..., e2πimλr b m (m + α r ) s converges in H r (D ). Such a sequence exists, since ( ) e 2πimλ ω(m) (m + α ) s,..., e2πimλr ω(m) (m + α r ) s

16 6 A. Laurinčikas, K. Matsumoto is an H r (D ) - valued random element. By the definition of D, for every compact subset K of D, sup <. (m + α j ) 2σ j= s K herefore, it remains to verify only the hypothesis a) of Lemma 6 from [7]. Let µ,..., µ r be complex measures on (C, B(C)) with compact supports contained in D and such that e 2πimλ j (m + α j ) s dµ j(s) <. (7) Since (see [6]) j= C (m + α) s = m s + O(m σ s e O( s ) ), (7) and the properties of the measures µ,..., µ r show that j= C e 2πimλ j m s dµ j (s) <. Hence, by the periodicity of e 2πimλ j, for every l =,..., k, m l(modk) j= C e 2πilλ j m s dµ j (s) <. (8) Define ν l (A) = e 2πilλ j µ j (A), A B(C), l =,..., k. j= hen, clearly, ν,..., ν l are complex measures on (C, B(C)) with compact supports obtained in D, and in view of (8) m l(modk) m s dν l (s) <, l =,..., k. (9) C Now we put hen by (9) ρ l (z) = e sz dν l (s), z C, l =,..., k. C m l(modk) ρ l(log m) <, l =,..., k. (2)

17 Joint value-distribution theorems on Lerch zeta-functions. II 7 Clearly, ρ l (z) is an entire function of exponential type, l =,..., k. hus, by Lemma 6.4. of [3], either ρ l, or lim sup x log ρ l (x) x >, l =,..., k. (2) Suppose that (2) is true. hen Lemma 5 of [6] (which is a version of the positive density method ) shows that m l(modk) ρ l(log m) =, l =,..., k, and this contradicts (2). herefore, we have that, for l =,..., k, ρ l (z). Hence, by the definition of the measures ν,..., ν l, j= e 2πilλ j C e sz dµ j (s), l =,..., k. Since rank(a) = r, this implies e sz dµ j (s), j =,..., r, and we easily deduce that C C s l dµ j (s) for all j =,..., r and l N. Hence we obtain that all hypotheses of Lemma 6 from [7] are satisfied, and we have that the set of all convergent series ( ) e 2πimλ b m a m (m + α ) s,..., e2πimλr b m a m (m + α r ) s with a m γ is dense in H r (D ). Clearly, the set of all convergent series (6) also has the same property. he theorem is proved. Proof of heorem 2. he proof uses heorem and is the same that of heorem from [6]. REFERENCES. P. Billingsley, Convergence of Probability measures, John Wiley, New York (968). 2. H. Heyer, Probability Measures on Locally Compact Groups, Springer - Verlag, Berlin (977). 3. A. Laurinčikas, Limit heorems for the Riemann Zeta-Function, Kluwer, Dordrecht (996). 4. A. Laurinčikas, R. Garunkštis, he Lerch Zeta - Function, Kluwer, Dordrecht (22).

18 8 A. Laurinčikas, K. Matsumoto 5. A. Laurinčikas, K. Matsumoto, Joint value - distribution theorems on Lerch zeta - functions, Lith. Math. J., 38(3), (998). 6. A. Laurinčikas, K. Matsumoto, he joint universality and the functional independence for Lerch zeta - functions, Nagoya Math. J., 57, (2). 7. A. Laurinčikas, K. Matsumoto, he joint universality of zeta - functions attached to certain cusp forms, Fiz. - mat. fak. moksl. seminaro darbai, 5, (22). Jungtinės reikšmi u pasiskirstymo teoremos Lercho dzeta funkcijoms. II A. Laurinčikas, K. Matsumoto. Pateikti ištaisyti kai kuri u teorem u iš [5] ir [6] formulavimai apie jungtini Lercho dzeta funkcij u reikšmi u pasiskirstym a (ribinės teoremos, universalumas, funkcinis nepriklausomumas). Be to, pateiktas naujas tiesioginis jungtinės ribinės teoremos analizini u funkcij u erdvėje i rodymas bei praplėsta jungtinė universalmuo teorema. Rankraštis gautas 26 4