the joint universality of zeta-functions attached to

58

Proc. Sci. Seminar ˇ Faculty of Physics and Mathematics, Siauliai University, 5, 2002, 58–75

THE JOINT UNIVERSALITY OF ZETA-FUNCTIONS ATTACHED TO CERTAIN CUSP FORMS ˇ Antanas LAURINCIKAS, Kohji MATSUMOTO Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, 2600, Vilnius, Lithuania. ˇ ˇ Faculty of Physics and Mathematics, Siauliai University, Vytauto 84, 5400, Siauliai, Lithuania. e-mail: [email protected] Graduate School of Mathematics, Nagoya Univeristy, Chikusa–ku, Nagoya, 464–8602, Japan. e-mail: [email protected] Abstract.

In the paper a joint universality theorem for zeta-functions of normalized

eigenforms is proved. For this some orthogonality conditions (A)–(C) are used. Key words: cusp form, eigenform, limit theorem, probability measure, random element, support of random element, universality, weak convergence.

1. Introduction The notion of the universality appears in various fields of mathematics. A general definition of the universality is given in [6]. Let X and Y be topological spaces and Tj : X → Y , j ∈ I, be continuous mappings. Then an element x ∈ X is called universal with respect to the family {Tj , j ∈ I} if the set {Tj x : j ∈ I} is dense in Y. Many universal objects of various nature are known, see an excellent survey paper [6]. We recall a result of Birkhoff [4]. He proved that there exists an entire function f such that for every entire function g there exists a sequence of complex numbers {an } such that f (z + an ) −→ g(z) n→∞

locally uniformly in the complex plane C. In the latter theorem universal translates appear. However, the most known universal objects, as translates in the Bikhoff theorem, are not explicitly given: they are either constructed in some countable inductive process, or their existence is based on the Baire category theorem. Only in 1975 S. M. Voronin [26] found a concrete universal object, the Riemann zetafunction ζ(s), s = σ + it, which enjoys some kind of Birkhoff translates universality. To give the modern statement of Voronin’s theorem we use the notation νT (. . .) = T −1 meas{τ ∈ [0, T ] : . . .} for T > 0, where meas{A} denotes the Lebesgue measure of the set A, and in place of dots we write a condition satisfied by τ . Now the Voronin theorem can be

A. Laurinˇ cikas, Kohji Matsumoto

59

stated as follows, see Chapter 6 of [15]. Let K be a compact subset of the strip {s = σ + it ∈ C : 12 < σ < 1} with conected complement. Let f (s) be a non-vanishing continuous function on K which is analytic in the interior of K. Then for any ε > 0 ¡ ¢ lim inf νT sup |ζ(s + iτ ) − f (s)| < ε > 0. T →∞

s∈K

The Voronin theorem was generalized and improved in [1], [5], [8]–[12], [14]–[19], [21], [23]–[26]. By the way, there exists the Linnik–Ibragimov conjecture that all Dirichlet series are universal in the above sense (of course, the case of absolutely convergent series for all s is excluded). An important role because of their application to zero-distribution problems is played by joint universality theorems for Dirichlet series. The joint universality for Dirichlet L-functions was proved in [1], [2], [5], [28]. A theorem of such kind for Dirichlet series with multiplicative coefficients was obtained in [13], while for generalized Euler products it was proved in [19]. The joint universality of Lerch zeta-functions was given in [22]. Let F (z) be a holomorphic cusp form of weight κ for the full modular group SL(2, Z), and we assume that F (z) is a normalized eigenform. Then F (z) has the Fourier series expansion F (z) =

∞ X

c(m)e2πimz ,

c(1) = 1.

m=1

E. Hecke [7] introduced and studied the zeta-function ϕ(s, F ), for σ > by absolutely convergent Dirichlet series with coefficients c(m): ϕ(s, F ) =

∞ X

κ+1 2 ,

given

c(m)m−s .

m=1

The function ϕ(s, F ) is analytically continuable to an entire function. Let D = {s ∈ C : κ2 < σ < κ+1 2 }. Then the univerality of ϕ(s, F ) is contained in the following theorem [23]: Let F (z) be a normalized eigenform of weight κ for SL(2, Z). Let K be a compact subset of D with connected complement, and let f (s) be a non-vanishing continuous function on K which is analytic in the interior of K. Then for any ε > 0 ¡ ¢ lim inf νT sup |ϕ(s + iτ, F ) − f (s)| < ε > 0. T →∞

s∈K

It is the purpose of the present paper to obtain the joint universality of zetafunctions attached to normalized eigenforms. Let Fj (z) be a holomorphic cusp form of weight κj for the full modular group SL(2, Z), and we assume that Fj (z) is a normalized eigenforms, j = 1, . . . , n, n > 2. Consider the zeta-functions ϕ(s, Fj ) =

∞ X m=1

cj (m)m−s ,

j = 1, . . . , n,

The joint universality of zeta-functions attached to certain cusp forms

60

and their analytic continuation. Here cj (m) denote the coefficients of Fourier series expansion for Fj (z). To obtain the joint universality of the functions ϕ(s, Fj ), j = 1, . . . , n, we will use some restrictions. In the case of Dirichlet L-functions the orthogonality of Dirichlet characters is employed. In our case we will apply the following condition. 1−κj Let cjp = cj (p)p 2 where p denotes a prime number. Suppose that there exist functions ρ1 (p), . . . , ρn (p) and real numbers θ1 , . . . , θn such that X

|ρj (p)|2 =

p6x

¢ θj x ¡ 1 + o(1) , log x

x → ∞,

j = 1, . . . , n,

(A)

and numbers λjk , k, j = 1, . . . , n, satisfying cjp = λjk ρk (p) + bjk (p), where

X |bjk (p)| p

for σ > 12 , and

pσ ¯ ¯ λ11 ¯ ¯ ... ¯ ¯ λn1

Let Dj = {s ∈ C :

κj 2

0.

16j6n s∈Kj

2. A joint limit theorem for the functions ϕ(s, Fj ) For any region G on C, by H(G) denote the space of analytic on G functions equipped with the topology of uniform convergence on compacta. Let N > 0, © κj κj + 1 Dj,N = s ∈ C : j2 , has the Euler product expansion ¶−1 µ ¶−1 Yµ αj (p) βj (p) ϕ(s, Fj ) = 1− 1− ps ps p with cj (p) = αj (p) + βj (p), j = 1, . . . , n. On the probability space (Ω, B(Ω), mH ) define the Hr -valued random element ϕ(s1 , . . . , sn , ω; F1 , . . . , Fn ) by the formula ¡ ¢ ϕ(s1 , . . . , sn , ω; F1 , . . . , Fn ) = ϕ(s1 , ω; F1 ), . . . , ϕ(sn , ω; Fn ) , (1) where

¶−1 µ ¶−1 Yµ αj (p)ω(p) βj (p)ω(p) ϕ(sj , ω; Fj ) = 1− 1− , psj psj p sj ∈ Dj,N ,

j = 1, . . . , n.

Let Pϕ be the distribution of the random element ϕ(s1 , . . . , sn , ω; F1 , . . . , Fn ), i. e. ¡ ¢ Pϕ (A) = mH ω ∈ Ω : ϕ(s1 , . . . , sn , ω; F1 , . . . , Fn ) ∈ A , A ∈ B(Hn ). Lemma 1. The probability measure PT converges weakly to the measure Pϕ as T → ∞. © κ ª Proof. Let Dj,0 = s ∈ C : σ > 2j , j = 1, . . . , n, Hn,0 = H(D1,0 ) × . . . × H(Dn,0 ), and

¡ ¢ PT,0 (A) = νT ϕ(s1 + iτ, F1 ), . . . , ϕ(sn + iτ, Fn ) ∈ A ,

A ∈ B(Hn,0 ).

Denote by ϕ0 (s1 , . . . , sn , ω; F1 , . . . , Fn ) the Hn,0 -valued random element defined by formula (1) with sj ∈ Dj,0 , j = 1, . . . , n. Then in [20] it was proved that the probability measure PT,0 converges weakly to the distribution of the random element ϕ0 as T → ∞. Since the function h : Hn,0 → Hn defined by the coordinatewise restriction is continuous, hence, applying a property of the weak convergence of probability measures [3], Theorem 5.1, we obtain the lemma.

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The joint universality of zeta-functions attached to certain cusp forms

3. Supports of certain random elements Let G1 , . . . , Gn be simply connected regions on C, and H(G1 , . . . , Gn ) = H(G1 ) × . . . × H(Gn ). In this section we will consider supports of H(G1 , . . . , Gn )valued random elements. Lemma 2. Let X and Y be two independent H(G1 , . . . , Gn )-valued random elements with distributions P and Q, respectively. Then the distribution of the sum X + Y is the convolution P ∗ Q of P and Q. Proof. Suppose that X and Y are defined on (Ω0 , F, P), and let A ∈ B(H(G1 , . . . , Gn )). From the independence of X and Y we have that the distribution P(X + Y ∈ A) of X + Y is equal to the product ¡ ¢ (P × Q) (x, y) : x + y ∈ A , x, y ∈ H(G1 , . . . , Gn ). However, denoting by IA the indicator function of the set A, by the Fubini theorem we find Z ¡ ¢ (P × Q) (x, y) : x + y ∈ A = IA (x + y)d(P × Q) Z

µ

Z

H(G1 ,...Gn )×H(G1 ,...,Gn )

= H(G1 ,...Gn )

Z

=

¶ IA (x + y)P (dx) Q( dy)

H(G1 ,...Gn )

P (A − y)Q(dy) = (P ∗ Q)(A). H(G1 ,...Gn )

Denote by SX the support of the random element X. Lemma 3. Let X and Y be two independent H(G1 , . . . , Gn )-valued random elements. Then the support SX+Y of the sum is the closure of the set © ª S = f ∈ H(G1 , . . . , Gn ) : f = x + y with x ∈ SX , y ∈ SY .

Proof. Let {Kjm } be a sequence of compact subsets of Gj such that Gj =

∞ [

Kjm ,

m=1

Kjm ⊂ Kj,m+1 , and if Kj is a compact and Kj ⊂ Gj , then Kj ⊆ Kjm for some m, j = 1, . . . , n. For fj , gj ∈ H(Gj ) we put ρj (fj , gj ) =

∞ X ∞ X m=1 m=1

2−m

ρjm (fj , gj ) , 1 + ρjm (fj , gj )

A. Laurinˇ cikas, Kohji Matsumoto

63

where ρjm (fj , gj ) = sup |fj (s) − gj (s)|,

j = 1, . . . , n.

s∈Kjm

Then ρj is a metric on H(Gj ) which induces its topology, j = 1, . . . , n. For f = (f1 , . . . , fn ), g = (g1 , . . . , gn ) ∈ H(G1 , . . . , Gn ) we put ρ(f , g) = max ρj (fj , gj ). 16j6n

Then ρ is a metric on H(G1 , . . . , Gn ) which induces its topology. Now we begin the proof of the lemma. Let x ∈ SX , y ∈ SY , and f = x + y. We take an arbitrary positive number δ, and put © ª A = g ∈ H(G1 , . . . , Gn ) : ρ(f , g) < δ . Moreover, let P and Q be the distribution of random elements X and Y , respectively. Then we have Z (P ∗ Q)(A) = P (A − g)Q(dg) H(G1 ,...,Gn )

Z

>

P (A − g)Q(dg) {g: ρ(y,g)< δ2 }

¡ δ ¢ > P {g : ρ(x, g) < } 2 =P

µ ©

Z Q(dg)

{g: ρ(y,g)< δ2 }

¶ µ ¶ © δª δª g : ρ(x, g) < Q g : ρ(y, g) < > 0, 2 2

since by the definition of the support each multiplier is positive. This and Lemma 2 show that S ⊆ SX+Y . But the set SX+Y is closed, hence S ⊆ SX+Y . It remains to show that S ⊇ SX+Y . Suppose that there exists a point f such that f ∈ SX+Y but f 6∈ S. Since f ∈ SX+Y , for any δ > 0 we have Z P (A − g)Q(dg) > 0.

(P × Q)(A) = H(G1 ,...,Gn )

The later inequality is possible only if there exists a point u ∈ SY such that P (A − u) > 0. Therefore there exists a point v ∈ SX in the sphere {g : ρ(f − u, g) < δ}. Then ρ(f , u + v) < δ and f 0 = u + v ∈ S. Moreover, if δ → 0, then f 0 → f . Thus f ∈ S, and this contradiction proves that S ⊇ SX+Y . Now let {Am } be a sequence of sets on H(G1 , . . . , Gn ). By LimAm denote a set of such f ∈ {H(G1 , . . . , Gn )} that every neighbourhood of f contains at least one point belonging to almost all sets Am .

The joint universality of zeta-functions attached to certain cusp forms

64

Lemma 4. Let Pn and P be probability measures on (H(G1 , . . . , Gn ), B(H(G1 , . . . , Gn ))) and let Pn converge weakly to P as n → ∞. Then SP ⊆ LimSPn . Proof. Let f ∈ SP , and, for ε > 0, Aε = {g : ρ(f , g) < ε}. For a fixed f the boundaries of the spheres ρ(f , g) < ε do not intersect for different ε. Therefore we can choose ε that the Aε should be the continuity set of P . Then the properties of the weak convergence yield lim Pn (Aε ) = P (Aε ) > 0.

n→∞

So, we have Pn (Aε ) > 0 for n > n0 (f , ε). For these n > n0 the distance of f from SPn does not exceed ε. Hence, since ε is an arbitrary number, we find that f ∈ LimSPn . Therefore SP ⊆ LimSPn . Lemma 5. Let {Xn } be a sequence of independent H(G1 , . . . , Gn )-valued random elements such that the series ∞ X Xm m=1

converges almost surely. Then the support of the sum of the later series is the closure of the set of all f ∈ H(G1 , . . . , Gn ) which may be written as a convergent series ∞ X f= f m , f m ∈ SXm . m=1

Proof. Let {Xn } be given on (Ω0 , F, P), and ∞ X

X=

X m = Ln + R n ,

m=1

where Ln =

n X

Xm ,

m=1

∞ X

Rn =

Xm .

m=n+1

Since the series of the lemma converges almost surely, for any ε > 0 ¡ ¢ P ω ∈ Ω0 : ρ(Rn , 0) > ε → 0, n → ∞.

(2)

Let Pn (A) = P(Ln ∈ A),

P (A) = P(X ∈ A),

A ∈ B(H(G1 , . . . , Gn )).

Then the above relations imply the weak convergence of Pn to P as n → ∞. Therefore in view of Lemma 4 SX ⊆ LimSLn .

(3)

A. Laurinˇ cikas, Kohji Matsumoto

65

Now let f 0 ∈ LimSLn , and, for any δ > 0, © ª Aδ = f : ρ(f , f 0 ) < δ . Then there exists n1 such that for n > n1 P(Ln ∈ Aε ) = Pn (Aε ) > 0.

(4)

Define Bδ = {f : ρ(f , 0) < δ}. Then by (2) for n > n2 P(Rn ∈ Bδ ) > 0.

(5)

Let Qn (A) = P(Rn ∈ A), A ∈ B(H(G1 , . . . , Gn )). Then we have that P = Pn ∗ Qn . Hence and from (4), (5) and Lemma 2 we obtain Z P (A2δ ) =

Pn (A2δ − g)Qn (dg) H(G1 ,...,Gn )

Z

Z

Pn (A2δ − g)Qn (dg) > Pn (Aδ )

> Bδ

Qn (dg)

Bδ

= Pn (Aδ )Qn (Bδ ) > 0 for n > n3 = max(n1 , n2 ). This means that f0 ∈ SX . Therefore SX ⊇ LimSLn . This and (3) imply SX = LimSLn . (6) Since X1 , . . . , Xn are independent, by Lemma 3 we have that SLn is the closure of the set of all f ∈ H(G1 , . . . , Gn ) which can be written as a sum f=

n X m=1

f m,

f m ∈ SXm .

Now if f ∈ SX , then there exists a sequence {g n : g n ∈ SLn } and lim g n = f . This together with (6) yield the assertion of the lemma.

n→∞

4. The denseness of one set of series In this section we will consider convergent series in H(G1 , . . . , Gn ). Lemma 6. Let {f m } = {(f1m , . . . , fnm )} be a sequence in H(G1 , . . . , Gn ) which satisfies:

66

The joint universality of zeta-functions attached to certain cusp forms

a) If µ1 , . . . , µn are complex measures on (C, B(C)) with compact supports contained in G1 , . . . , Gn , respectively, such that n Z ∞ X ¯ ¯X ¯ fjm dµj ¯ < ∞, m=1 j=1

then

C

Z sr dµj (s) = 0 C

for j = 1, . . . , n, r = 0, 1, 2, . . .; b) The series

∞ X m=1

fm

converges in H(G1 , . . . , Gn ); c) For any compacts K1 ⊆ G1 , . . . , Kn ⊆ Gn ∞ X n X

sup |fjm (s)|2 < ∞.

m=1 j=1 s∈Kj

Then the set of all convergent series ∞ X m=1

am f m

with am ∈ γ is dense in H(G1 , . . . , Gn ). Proof. The lemma for G1 = . . . = Gn was proved in [1], Lemma 5.2.9, and in the case n = 1 it can be found in [15], Theorem 6.3.10. Let Kj be a compact subset of Gj , j = 1, . . . , n. We choose a simply connected region Uj such that Kj ⊆ Uj , U j is a compact subset of Gj and the boundary ∂Uj of Uj , j = 1, . . . , n, is an analytic simple closed curve. Consider the Hardy space H 2 (Uj ), see for definition [15], Section 6.3, which is a Hilbert space, j = 1, . . . , n. Now let H 2 (U1 , . . . , Un ) = H 2 (U1 ) × . . . × H 2 (Un ). Define for f = (f1 , . . . , fn ), g = (g1 , . . . , gn ) ∈ H 2 (U1 , . . . , Un ) the inner product by (f , g) =

n X

(fj , gj ),

j=1

where (fj , gj ) is the inner product on H 2 (Uj ), j = 1, . . . , n. Thus we have that H 2 (U1 , . . . , Un ) is a Hilbert space again.

A. Laurinˇ cikas, Kohji Matsumoto

67

From the proof of Theorem 6.3.10 from [15] we have kf m k = (f m , f m ) =

n X

(fj , fj ) =

j=1

=B

n X

n X

kfj k2

j=1

sup |fj (s)|2 .

j=1 s∈U j

Therefore in view of the condition c) and of choice of Uj ∞ X m=1

kf m k2 < ∞.

2

Now let g ∈ H (U1 , . . . , Un ) be such that ∞ X m=1

|f m , g| < ∞.

(7)

Using the formula for the inner product in H 2 (Uj ), see [15], Section 6.3, we obtain that there exists a complex measure µj on (C, B(C)) with support contained in ∂Uj , j = 1, . . . , n, such that n Z X fjm dµj . (f , g) = j=1

C

Hence by (7)

¯ Z ∞ ¯X X ¯ n ¯ ¯ fjm dµj ¯¯ < ∞, ¯ m=1 j=1 C and therefore by the condition a) of the lemma Z sr dµj = 0 C for j = 1, . . . , n, r = 0, 1, 2, . . .. This means that gj is orthogonal to all the polynomials, j = 1, . . . , n. Since the polynomials are dense in the topology of H 2 (Uj ), Theorem 6.3.9 of [15], hence we obtain that gj = 0, j = 1, . . . , n, and g = 0. Consequently, ∞ X |f m , g| = ∞ m=1

2

for 0 6= g ∈ H (U1 , . . . , Un ). Therefore by Theorem 6.1.16 form [15] we obtain that the set of all convergent series in H 2 (U1 , . . . , Un ) ∞ X m=1

αm f m

The joint universality of zeta-functions attached to certain cusp forms

68

with αm ∈ γ is dense in H 2 (U1 , . . . , Un ). Let f = (f1 , . . . , fn ) ∈ H(G1 , . . . , Gn ) and ε > 0. Since the convergence in the H 2 (Uj ) topology implies the uniform convergence on compact subsets of Uj , j = 1, . . . , n, we deduce that there exists a sequence {αm , αm ∈ γ} such that the series ∞ X

αm fjm

m=1

converges uniformly on Kj for all j = 1, . . . , n, and ¯ ¯X n X ¯ ∞ ¯ ε sup ¯¯ αm fjm (s) − fj (s)¯¯ < . 4 j=1 s∈Kj m=1 Hence there exists a natural number M such that n X

M ¯X ¯ ε sup ¯ αm fjm (s) − fj (s)¯ < , 2 j=1 s∈Kj m=1

(8)

and in view of the condition b) n X

∞ X

¯ sup ¯

j=1 s∈Kj

Let

½ am =

m=M +1

¯ ε fjm (s)¯ < . 2

αm , 1 6 m 6 M , 1, m > M.

Then (8) and (9) yield n X

∞ ¯X ¯ sup ¯ am fjm (s) − fj (s)¯ < ε,

j=1 s∈Kj

m=1

and the lemma is proved. 5. The support of the random element ϕ Let, for |z| < 1, log(1 + z) = z − and define

z2 z3 + − ..., 2 3

¡ ¢ f p (s1 , . . . , sn , ap ) = f1p (s1 , ap ), . . . , fnp (sn , ap ) µ µ ¶ µ ¶ α1 (p)ap β1 (p)ap = − log 1 − − log 1 − ,..., ps1 ps1 µ ¶ ¶¶ µ βn (p)ap αn (p)ap − log 1 − , − log 1 − s n p psn sj ∈ DjN ,

j = 1, . . . , n, ap ∈ γ.

(9)

A. Laurinˇ cikas, Kohji Matsumoto

69

Lemma 7. The set of all convergent series X f p (s1 , . . . , sn , ap ) p

is dense in Hn . Proof. Let µ µ ¶ µ ¶ α1 (p) β1 (p) fep = fep (s1 , . . . , sn ) = − log 1 − s1 − log 1 − s1 , . . . , p p µ ¶ µ ¶¶ αn (p) βn (p) − log 1 − sn − log 1 − sn . p p We take p0 > 0 and define ½ fbp = fbp (s1 , . . . , sn ) =

fep (s1 , . . . , sn ) if p > p0 , 0 if p 6 p0 .

The products for random elements ϕ(sj , ω, Fj ) converge uniformly on compact subsets of Dj,N , j = 1, . . . , n, for almost all ω ∈ Ω. Therefore there exists a sequence {b ap , b ap ∈ γ} such that the series X b ap fb p

p

converges in Hn , the details can be found in [23], p. 349. We will prove that the set of all convergent series X (10) ap fbp , ap ∈ γ, p

is dense in Hn . Denoting g p = (g1p , . . . , gnp ) = b ap fbp , we see that for this it suffices to show the denseness of the set of all convergent series X (11) ap g(p), ap ∈ γ. p

By choise of b ap the series

X p

gp

converges in Hn . Since |αj (p)| 6 p(κj −1)/ 2 , we have that

|βj (p)| 6 p(κj −1)/ 2 ,

j = 1, . . . , n,

¶ αn (p) + βn (p) α1 (p) + β1 (p) + r1p (s1 ), . . . , + rnp (sn ) ps1 psn µ ¶ c1 (p) cn (p) = + r1p (s1 ), . . . , sn + rnp (sn ) s 1 p p

fep (s1 , . . . , sn ) =

µ

(12)

70

The joint universality of zeta-functions attached to certain cusp forms

with

rjp (sj ) = Bpκj −2σj −1 ,

j = 1, . . . , n.

Let Kj be an arbitrary compact subset of Dj,N , j = 1, . . . , n. Then (12) implies n XX

sup |gjp (s)|2 < ∞.

j=1 s∈Kj

p

To apply Lemma 6 for the set of the series (11) it remains to check the condition a) of the latter lemma. Let µj be a complex measure on (C, B(C)) with compact support contained in Dj,N , j = 1, . . . , n, such that n Z X¯X ¯ ¯ gjp dµj ¯ < ∞. p

j=1

(13)

C

We put hjp (sj ) = b ap cj (p)p−sj , j = 1, . . . , n. Then in view of (12) n XX

sup |gjp (s) − hjp (s)| < ∞.

j=1 s∈Kj

p

Combining this with (13), we obtain n Z X¯X ¯ ¯ hjp (s)dµj (s)¯ < ∞, p

j=1

C

and, by the definition of hjp , Z n X¯X ¯ ¯ cj (p) p−s dµj (s)¯ < ∞. p

Now let DN = {s ∈ C :

1 2

j=1

C

< σ < 1, |t| < N }, and let

hj (s) = s − Then we have that

(14)

κj − 1 , 2

j = 1, . . . , n.

−1 µj h−1 j (A) = µj (hj A),

A ∈ B(C),

is a complex measure on (C, B(C)) with compact support contained in DN , j = 1, . . . , n. Now (14) can be rewritten in the form Z n X¯X ¯ ¯ ¯ cjp p−s dµj h−1 j (s) < ∞. p

j=1

C

(15)

A. Laurinˇ cikas, Kohji Matsumoto

71

Using the condition (B) on cjp , we deduce from (15) X

¯X ¯ Z ¯ n ¯ ¯ < ∞, |ρk (p)|¯¯ λjk p−s dµj h−1 (s) j ¯

p

j=1

Taking µ ek (A) =

k = 1, . . . , n.

(16)

C

n X

λjk µj h−1 j (A),

A ∈ B(C),

j=1

Z

e−sz de µk (z),

ηk (z) =

z ∈ C,

k = 1, . . . , n,

C we write (16) as X

¯ ¯ |ρk (p)|¯ηk (log p)¯ < ∞,

k = 1, . . . , n.

p

Since ρk (p), k = 1, . . . , n, satisfies the condition (A), similarly as in [23], Lemma 6, we derive that ηk (z) ≡ 0, k = 1, . . . , n. This, the definition of µ ek and the condition (C) show that Z e−sz dµj h−1 j = 1, . . . , n. j (s) ≡ 0, C Hence by the differentiation we find that Z sr dµj h−1 j = 1, . . . , n, j (s) = 0,

r = 0, 1, 2, . . . ,

C and together with definition of hj this implies Z sr dµj (s) = 0,

j = 1, . . . , n,

r = 0, 1, 2, . . . .

C Thus all conditions of Lemma 6 are satisfied, and we have the denseness of all convergent series (11) and therefore that of series (10). It remains the final part of the proof. Let x0 (s1 , . . . , sn ) = (x01 (s1 ), . . . , x0n (sn )) ∈ Hn , Kj be a compact subset of Dj,N , j = 1, . . . , n, and ε > 0. We choose p0 to satisfy n ∞ X XX |αj (p)|l + |βj (p)|l ε sup (17) < . lσj lp 4 j=1 sj ∈Kj p>p 0

l=2

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The joint universality of zeta-functions attached to certain cusp forms

The denseness of the set of all convergent series (10) shows that there exists a sequence {e ap , e ap ∈ γ} such that n X

X X ¯ ¯ ε sup ¯x0j (sj ) − fejp (sj ) − e ap fejp (sj )¯ < . 2 p>p j=1 sj ∈Kj p6p0

We take

½ ap =

0

1 e ap

if p 6 p0 , if p > p0 .

Then this and (17) yield n X

X ¯ ¯ sup ¯x0j (sj ) − fjp (sj , ap )¯

j=1 sj ∈Kj

n X 6

p

X X ¯ ¯ sup ¯x0j (sj ) − fejp (sj ) − ap fejp (sj )¯

j=1 sj ∈Kj

p6p0

p>p0

n X

X ¯X ¯ ε + sup ¯ e ap fejp (sj ) − fjp (sj , e ap )¯ < 2 p>p0 j=1 sj ∈Kj p>p0 µXX ¶ n ∞ X |α( p)|l + |βj (p)|l +2 sup < ε. lplσj j=1 sj ∈Kj p>p0 l=2

This shows that X ¢ ¡ f p (s1 , . . . , sn , ap ) < ε, ρ x0 (s1 , . . . , sn ), p

and the lemma is proved. Let © ª Sj,N = f ∈ H(Dj,N ) : f (s) 6= 0 for any s ∈ Dj,N or f (s) ≡ 0 ,

j = 1, . . . , n,

and Sn = S1,N × . . . × Sn,N . Lemma 8. The support of the random element ϕ(s1 , . . . , sn , ω; F1 , . . . , Fn ) is the set Sn . Proof. By the definition {ω(p)} is a sequence of independent random variables on the probability space (Ω, B(Ω), mH ), and the support of each ω(p) is the unit circle γ. Therefore ½ µ ¶−1 µ ¶−1 α1 (p)ω(p) β1 (p)ω(p) log 1 − + log 1 − ,..., ps1 ps1 µ ¶−1 ¶−1 ¾ µ αn (p)ω(p) βn (p)ω(p) log 1 − + log 1 − psn psn

A. Laurinˇ cikas, Kohji Matsumoto

73

is a sequence of independent Hn -valued random elements, and the set ½ ¡ ¢ f p = f1p (s1 ), . . . , fnp (sn ) ∈ Hn : fjp (sj ) = fjp (sj , a) µ ¶ µ ¶ ¾ αj (p)a βj (p)a = − log 1 − − log 1 − , j = 1, . . . , n, a ∈ γ psj psj is the support of each element. By Lemma 5 the support of Hn -valued random element X ª ©X fnp (sn , ω(p)) (18) f1p (s1 , ω(p)), . . . , p

p

is the closure of the set of all convergent series X f p (s1 , . . . , sn ; ap ),

ap ∈ γ.

p

By Lemma 7 the latter set is dense in Hn . Hence the support of the random element (18) is Hn . Using the Hurwitz theorem hence we deduce as in [23] that the support of the random element ϕ(s1 , . . . , sn , ω; F1 , . . . , Fn ) is the set Sn . 6. Proof of the theorem The proof of the theorem runs in the same way as in the case n = 1 [23]. First we suppose that the functions fj (s) have non-vanishing analytic continuation to H(Dj,N ), j = 1, . . . , n. Denote by G the set of all (g1 , . . . , gn ) ∈ Hn such that sup sup |gj (s) − fj (s)|
0. 4

Now let the functions fj (s), j = 1, . . . , n, be the same as in the statement of the theorem. By the Mergelyan theorem [29] there exists polynomials pj (s), pj (s) 6= 0 on Kj , such that ε sup sup |fj (s) − pj (s)| < . (19) 4 16j6n s∈Kj Moreover, since pj (s) 6= 0 on Kj ,by the Mergelyan theorem again there exists polynomials gj (s), j = 1, . . . , n, such that ¯ ¯ ε sup sup ¯pj (s) − egj (s) ¯ < . 4 16j6n s∈Kj

74

The joint universality of zeta-functions attached to certain cusp forms

This and (19) imply

¯ ¯ ε sup sup ¯fj (s) − egj (s) ¯ < . 2 16j6n s∈Kj

(20)

However, egj (s) 6= 0 for all s. Therefore by the first case µ lim inf νT T →∞

¯ ¯ ε sup sup ¯ϕ(s + iτ, Fj ) − egj (s) ¯ < 2 16j6n s∈Kj

¶ > 0.

Hence and from (20) we have the assertion of the theorem.

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Kai kuriu‘ paraboliniu‘ formu‘ dzeta funkciju‘ rinkinio universalumas A. Laurinˇcikas, Kohji Matsumoto Straipsnyje i‘rodytas normuotu‘ eigenformu‘ dzeta funkciju‘ rinkinio universalumas. I‘ rodymui naudojamos ortogonalumo sa‘lygos (A)–(C).

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