Siberian Mathematical Journal, Vol. 49, No. 4, pp. 612–627, 2008 c 2008 Garbaliauskien˙e V., Genys J., and Laurinˇ Original Russian Text Copyright  cikas A.

DISCRETE UNIVERSALITY OF THE L-FUNCTIONS OF ELLIPTIC CURVES V. Garbaliauskiene, ˙ J. Genys, and A. Laurinˇ cikas

UDC 511

Abstract: A discrete universality theorem is obtained in the Voronin sense for the L-functions of elliptic   is rational for curves. We use the difference of an arithmetical progression h > 0 such that exp 2πk h some k = 0. A limit theorem in the space of analytic functions plays a crucial role in the proof. Keywords: elliptic curve, L-function, limit theorem, probability measure, random element, space of analytic functions, universality, weak convergence

1. Introduction Denote by N, Z, Q, R, and C the sets of all positive integers, integers, rational, real, and complex numbers, respectively. Let E be an elliptic curve given by the Weierstrass equation y 2 = x3 + ax + b,

a, b ∈ Z.

Suppose that the discriminant Δ = −16(4a3 + 27b2 ) of E is nonzero. Then it is well known that E is a nonsingular curve. Given a prime p, denote by ν(p) the number of solutions of the congruence y 2 ≡ x3 + ax + b (mod p), and put λ(p) = p − ν(p). The classical result of Hasse asserts that √ |λ(p)| ≤ 2 p.

(1)

Let s = σ + it be a complex variable. Then the L-function LE (s) of the elliptic curve E is defined, for σ > 32 , by   −1  1 λ(p) −1  λ(p) . 1− s 1 − s + 2s−1 LE (s) = p p p p|Δ

pΔ

In view of (1) the product defining LE (s) converges uniformly on the compact subsets of the halfplane {s ∈ C : σ > 32 } and defines there an analytic function. Moreover, the Shimura–Taniyama–Weil conjecture on the analytic continuation and functional equation for LE (s) has been proven recently [1]. Therefore, now it is known that the function LE (s) continues analytically to an entire function and satisfies the functional equation  √ s  √ 2−s N N Γ(s)LE (s) = ± Γ(2 − s)LE (2 − s), 2π 2π where N is the conductor of the curve E, and Γ(s), as usual, denotes the gamma-function. For these and other facts from the theory of elliptic curves, see [2] for example. The universality of LkE (s), k ∈ N, was obtained in [3]. Let meas{A} stand for the Lebesgue measure of a measurable set A ⊂ R. The third author was partially supported by the Lithuanian Foundation of Studies and Science. ˇ Siauliai; Vilnius. Translated from Sibirski˘ı Matematicheski˘ı Zhurnal, Vol. 49, No. 4, pp. 768–785, July–August, 2008. Original article submitted February 13, 2007. 612

c 2008 Springer Science+Business Media, Inc. 0037-4466/08/4904–0612 

Theorem 1 [3]. Let K be a compact subset of the strip D = {s ∈ C : 1 < σ < 32 } with connected complement, and let f (s) be a continuous nonvanishing function on K analytic in the interior of K. Then, for every ε > 0,     1 lim inf meas τ ∈ [0, T ] : sup LkE (s + iτ ) − f (s) < ε > 0. T →∞ T s∈K For k = 1, this theorem also is a consequence of the main result of [4]. If the analog of the Riemann hypothesis for LE (s) is true, i.e., all nontrivial zeros of LE (s) lie on the critical line σ = 1, then [3] the function L−k E (s), k ∈ N, is also universal in the sense of Theorem 1. We recall that S. M. Voronin proved [5] the universality of the Riemann zeta-function ζ(s). Later, many mathematicians, among them A. Reich, S. M. Gonek, B. Bagchi, K. Matsumoto, J. Steuding, ˇ zeviˇcien˙e, R. Kaˇcinskait˙e, the second author, W. Schwarz, Y. Mishou, H. Bauer, R. Garunkˇstis, R. Sleˇ et al. improved and generalized Voronin’s theorem for some classical zeta-functions and classes of Dirichlet series. The Linnik–Ibragimov conjecture says that all functions in some half-plane given by Dirichlet series are universal in the Voronin sense provided that they analytically continuable to the left of the absolute convergence half-plane and satisfying some natural growth conditions. In [6], the discrete universality was obtained of LE (s). Let, for N ∈ N, μN (. . . ) =

1 {0 ≤ m ≤ N : . . . } , N +1

where in place of dots a condition satisfied by m is to be written. Moreover, let h > 0 be a fixed number.   Theorem 2 [6]. Suppose that exp 2πk is an irrational number for all k ∈ Z \ {0}. Let K and f (s) h be the same as in Theorem 1. Then, for every ε > 0, lim inf μN (sup |LE (s + imh) − f (s)| < ε) > 0. N →∞

s∈K

Since by the Hermite–Lindemann theorem ek , k ∈ Z \ {0}, is irrational, we can take in Theorem 2, for example, h = 2π.   In this paper we consider the general case of h where exp 2πk can be rational for some values of h k ∈ Z \ {0}.   is rational. Let K and Theorem 3. Suppose that there exist integers k = 0 such that exp 2πk h f (s) be the same as in Theorem 2. Then, for every ε > 0, lim inf μN (sup |LE (s + imh) − f (s)| < ε) > 0. N →∞

s∈K

Remark. A similar theorem by the same method can be proved for the L-functions of new forms. 2. A Limit Theorem The proof of Theorem 3 is based on a limit theorem in the space of analytic functions for the function LE (s). However, a discrete limit theorem with h satisfying the hypotheses of Theorem 3 in the mentioned space for LE (s)is not known. If exp 2πk is rational for some integers k = 0, it suffices to consider only positive k’s with the h property. Let  k0 be the smallest of them. Then in [7] it was noted that the others k are the multiples 0 0 =m of k0 . Put exp 2πk h n0 with m0 , n0 ∈ N, (m0 , n0 ) = 1.  We take the unit circle γ ⊂ C and define the infinite-dimensional torus Ω = p γp , where γp = γ for all primes p. With the product topology and pointwise multiplication Ω is a compact topological abelian group. Let ω(p) denote the projection of ω ∈ Ω onto the coordinate space γp , and, for m ∈ N, we put  ω(m) = ω α (p), pα ||m

613

where pα ||m means that pα | m but pα+1  m. Thus, ω(m) is a completely multiplicative function and |ω(m)| = 1. Denote the class of Borel sets of a space S by B(S). Define Ωh = {ω ∈ Ω : ω(m0 ) = ω(n0 )}. Then Ωh is a closed subgroup of Ω; therefore, it is also a compact topological group, and the probability Haar measure mhH can be defined on (Ωh , B(Ωh )). This gives the probability space (Ωh , B(Ωh ), mhH ). Let D0 = {s ∈ C : σ > 1}, and, for s ∈ D0 and ωh ∈ Ωh ,   −1  λ(p)ωh (p) −1  λ(p)ωh (p) ωh2 (p) + 2s−1 . 1− 1− LE (s, ωh ) = ps ps p pΔ

p|Δ

Given a domain G in the complex plane, denote by H(G) the space of analytic functions on G equipped with the topology of uniform convergence on compacta. Proposition 1. LE (s, ωh ) is an H(D0 )-valued random element defined on (Ωh , B(Ωh ), mhH ). Proof. Denote by mH the probability Haar measure on (Ω, B(Ω)), and let pr1 , . . . , prk be an arbitrary finite collection of prime numbers, and Ar1 , . . . , Ark ∈ B(γ). Moreover, let g : Ω → Ωh be the same measurable function as in [7, p. 342]. We also recall that if P is a probability measure on (S, B(S)) then every measurable function u : S → S1 induces the unique probability measure P u−1 defined by P u−1 (A) = P (u−1 A), A ∈ B(S1 ). Denote by gp its restriction to the coordinate space γp . Since {ω(p)} is a sequence of independent random variables on (Ω, B(Ω), mH ) and mhH = mH g −1 , we obtain mhH (ωh ∈ Ωh : ωh (pr1 ) ∈ Ar1 , . . . , ωh (prk ) ∈ Ark ) = mH g −1 (ωh ∈ Ωh : ωh (pr1 ) ∈ Ar1 , . . . , ωh (prk ) ∈ Ark )

= mH ω ∈ Ω : ω(pr1 ) ∈ gp−1 Ar1 , . . . , ω(prk ) ∈ gp−1 Ark r1 rk

= mH ω ∈ Ω : ω(pr1 ) ∈ gp−1 Ar1 ) · . . . · mH (ω ∈ Ω : ω(prk ) ∈ gp−1 Ark r r 1

k

= mhH (ωh ∈ Ωh : ωh (pr1 ) ∈ Ar1 ) · . . . · mhH (ωh ∈ Ωh : ωh (prk ) ∈ Ark ). Thus, {ωh (p)} is a sequence of independent random variables on (Ωh , B(Ωh ), mhH ). To prove the proposition it suffices to show that the product −1  λ(p)ωh (p) ωh2 (p) + 2s−1 1− ps p pΔ

converges uniformly on the compact subsets of D0 almost surely. To this end, it suffices clearly to obtain the almost sure convergence on the compact subsets of D0 of the series  xp (s, ωh ), (2) pΔ h (p) where xp (s, ωh ) = λ(p)ω . Denote the expectation of the random element ξ by Eξ. Clearly, Eωh (p) = 0. ps Therefore, Exp (s, ωh ) = 0 for each prime p. Moreover,

E|xp (s, ωh )|2 ≤ and in view of (1)



λ2 (p) , p2σ

E|xp (s, ωh )|2 < ∞

pΔ

for s ∈ D0 . Hence by a well-known theorem (see [9, Theorem 1.2.11]) the series (2) converges almost surely for each fixed s ∈ D0 . Therefore, by Corollary 2.1.3 of [9] the series (2) converges uniformly on the compact subsets of D0 for almost all ωh ∈ Ωh with respect to mhH . The proposition is proved. Now we are ready to state a limit theorem in the space of analytic functions for LE (s). 614

Theorem 4. Suppose that h is the same as in Theorem 3. Then the probability measure def

PN (A) = μN (LE (s + imh) ∈ A) ,

A ∈ B(H(D0 )),

converges weakly to the distribution of the random element LE (s, ωh ) as N → ∞. We begin the proof of Theorem 4 with limit theorems for Dirichlet polynomials. Let pn (s) =

n 

am m−s ,

am ∈ C,

m=1

n be a Dirichlet rpolynomial. Denote by p1 , . . . , pr the distinct prime numbers that divide m=1 m, and define Ωr = j=1 γpj , where γpj = γ for j = 1, . . . , r. Suppose that n > max(m0 , n0 ). By analogy with the case of Ω we can define ω(m) for ω ∈ Ωr . Moreover, we put Ωhr = {ω ∈ Ωr : ω(m0 ) = ω(n0 )}, and define the probability measure

imh QhN (A) = μN (pimh 1 , . . . , pr ) ∈ A on (Ωr , B(Ωr )). Lemma 1. The probability measure QhN converges weakly to the Haar measure mhr on the probability space (Ωhr , B(Ωhr )) as N → ∞. The proof of the lemma is given in [7, Lemma 3]. Let g(m), m ∈ N, be a completely multiplicative function, |g(m)| = 1, g(m0 ) = g(n0 ), and n 

pn (s, g) =

am g(m)m−s .

m=1

On (H(D0 ), B(H(D0 )) define two probability measures



PN,pn (A) = μN pn (s + imh) ∈ A , PN,pn ,g (A) = μN pn (s + imh, g) ∈ A . Theorem 5. The probability measures PN,pn and PN,pn ,g both converge weakly to the same probability measure on (H(D0 ), B(H(D0 ))) as N → ∞. Proof. Consider the function u : Ωr → H(D0 ) defined as u(x1 , . . . , xr ) =

n 



am m−s

m=1

αj p m j 1≤j≤r

α

xj j

−1

,

(x1 , . . . , xr ) ∈ Ωr .

Obviously, u is continuous on Ωr ; moreover, imh pn (s + imh) = u(pimh 1 , . . . , pr ).

This, the continuity of u, Lemma 1, and the properties of the weak convergence (see [8, Theorem 5.1]) show that PN,pn = QhN u−1 converges weakly to mhr u−1 as N → ∞. Similarly, taking v(x1 , . . . , xr ) =

n  m=1

am m−s

 αj p m j 1≤j≤r

α

−1

xj j g −αj (pj )

,

(x1 , . . . , xr ) ∈ Ωr ,

615

we find that PN,pn ,g converges weakly to mhr v −1 as N → ∞. However, v(x1 , . . . , xr ) = u1 (x1 , . . . , xr ), (x1 , . . . , xr ) ∈ Ωr , where

u1 (x1 , . . . , xr ) = x1 g −1 (p1 ), . . . , xr g −1 (pr ) . Therefore, the invariance of the Haar measure mhr with respect to the translation by points in Ωhr yields −1 = mhr u−1 . mhr v −1 = mhr (u(u1 ))−1 = (mhr u−1 1 )u

This proves the theorem. The next step in the proof of Theorem 4 is a limit theorem in the space of analytic functions for absolutely convergent Dirichlet series. Given σ1 > 12 and n ∈ N, put LE,n (s) =

∞  m σ 1   λ(m) exp − ms n

m=1

and LE,n (s, ωh ) =

∞  m σ 1   λ(m)ωh (m) , exp − ms n

ωh ∈ Ωh .

m=1

Here the coefficients λ(m) are defined, for σ > 32 , by ∞  λ(m) . LE (s) = ms m=1

1

The estimate (1) implies the estimate λ(m)  m 2 d(m), where d(m) is the divisor function. Therefore, it follows from [7, pp. 347–348] that the series for LE,h (s) and LE,h (s, ωh ) both converge absolutely for σ > 1. On (H(D0 ), B(H(D0 ))) define the two probability measures PN,n (A) = μN (LE,n (s + imh) ∈ A),

N,n (A) = μN (LE,n (s + imh, ωh ) ∈ A). P

N,n both converge weakly to the same probability Theorem 6. The probability measures PN,n and P measure Pn on (H(D0 ), B(H(D0 ))) as N → ∞. Proof. The proof runs in the same way as that of Theorem 6 in [7], where a similar assertion was obtained for probability measures on the complex plane. However, the spaces C and H(D0 ) differ one from another by topology. Therefore, we must make some modifications. It is well known (for example, see [9, Lemma 1.1.7] or [10]) that there exists a sequence {Kn } of compact subsets of D0 such that ∞  Kn , D0 = n=1

and Kn can be chosen to satisfy Kn ⊂ Kn+1 , and if K is a compact and K ⊂ D0 then K ⊆ Kn for some n. Given f, g ∈ H(D0 ), put ρ(f, g) =

∞ 

2−n

n=1

ρn (f, g) , 1 + ρn (f, g)

where ρn (f, g) = sup |f (s) − g(s)|. s∈Kn

Then ρ is a metric on H(D0 ) inducing the topology of uniform convergence on compacta. 616

Let, for M ∈ N, LE,n,M (s) =

M  m σ 1   λ(m) exp − ms n

m=1

and

M  m σ 1   λ(m)ωh (m) exp − , ms n

LE,n,M (s, ωh ) =

ωh ∈ Ωh ,

m=1

and define the two probability measures PN,n,M (A) = μN (LE,n,M (s + imh) ∈ A),

N,n,M (A) = μN (LE,n,M (s + imh, ωh ) ∈ A) P

N,n,M converge weakly to the on (H(D0 ), B(H(D0 ))). Then by Theorem 5 both measures PN,n,M and P same measure, say, Pn,M as N → ∞. First of all we will check that the family of probability measures {Pn,M } is tight for n fixed.  B(Ω,  P)) with values mh, Let θN be a random variable defined on a certain probability space (Ω, m = 0, 1, . . . , N , and 1 P(θN = mh) = , m = 0, 1, . . . , N. N +1 Then, putting XN,n,M (s) = LE,n,M (s + iθN ), we have by Theorem 5 that D

XN,n,M (s) −→ Xn,M (s),

(3)

N →∞

D

where Xn,M (s) is an H(D0 )-valued random element with distribution Pn,M and −→ stands for convergence in distribution. By the Chebyshev inequality, for Ml > 0, P( sup |XN,n,M (s)| > Ml ) ≤ s∈Kl

N  1 sup |LE,n,M (s + imh)|, (N + 1)Ml s∈Kl

(4)

m=0

where {Kl } is the above-defined sequence of compact sets. Since the series for LE,n (s) converges absolutely on D0 , we have N 1  sup lim sup sup |LE,n,M (s + imh)| ≤ Rl < ∞. (5) M ≥1 N →∞ N + 1 s∈Kl m=0

We now take Ml =

2l

Rl ε

, where ε is an arbitrary positive number. Then (4) and(5) yield, for l ∈ N, lim sup P( sup |XN,n,M (s)| > Ml ) ≤ N →∞

s∈Kl

ε . 2l

(6)

The function u : H(D0 ) → R, given by the formula u(f ) = sups∈Kl |f (s)|, f ∈ H(D0 ), is continuous. Therefore, in view of (3) D sup |XN,n,M (s)| −→ sup |Xn,M (s)|. s∈Kl

This and (6) show that

N →∞ s∈K l

ε P sup |Xn,M (s)| > Ml ≤ l . 2 s∈Kl

(7)

Let Hε = {f ∈ H(D0 ) : sups∈Kl |f (s)| ≤ Ml , l ∈ N}. Then by the compactness principle (the Montel theorem) [9, Theorem 5.5.2] Hε is compact, and by (7) P (Xn,M (s) ∈ Hε ) ≥ 1 − ε 617

for all M ∈ N or

Pn,M (Hε ) ≥ 1 − ε

for all M ∈ N. This shows that the family of probability measures {Pn,M } is tight. Hence by the Prokhorov theorem (see [8]) it is relatively compact. By the definition of LE,n,M (s) and LE,n (s), for σ > 1, lim LE,n,M (s) = LE,n (s),

M →∞

and, since the series for LE,n (s) converges absolutely, the convergence is uniform on the compact subsets of D0 . Therefore, for every ε > 0, lim lim sup μN (ρ(LE,n,M (s + imh), LE,n (s + imh)) ≥ ε)

M →∞ N →∞

≤ lim lim sup M →∞ N →∞

n  1 ρ(LE,n,M (s + imh), LE,n (s + imh)) = 0. (N + 1)ε m=0

From this, taking XN,n (s) = LE,n (s + iθN ), we obtain lim lim sup P(ρ(XN,n,M (s), XN,n (s)) ≥ ε) = 0.

M →∞ N →∞

(8)

Let {Pn,M1 } be a subsequence of {Pn,M } which converges weakly to Pn , say, as M1 → ∞. Then D

Xn,M1 −→ Pn .

(9)

M1 →∞

The space H(D0 ) is separable. Therefore, (3), (8), (9), and Theorem 4.2 from [8] show that D

XN,n −→ Pn ,

(10)

N →∞

i.e. the measure PN,n converges weakly to Pn as N → ∞. Moreover, (10) shows that Pn is independent of the choice of the subsequence {Pn,M1 }. Thus, D

Xn,M −→ Pn .

(11)

M →∞

Now, reasoning similarly for the random elements  N,n,M (s, ωh ) = LE,n,M (s + iθN , ωh ), X

 N,n (s, ωh ) = LE,n (s + iθN , ωh ), X

N,n also converges weakly to Pn as N → ∞. we obtain in view of (11) that the probability measure P Theorem 6 is proved. To prove Theorem 4 it remains to pass from LE,n (s) to LE (s). For this we need an approximation in the mean of LE (s) and LE (s, ωh ) by LE,n (s) and LE,n (s, ωh ), respectively. Lemma 2. Let K be a compact subset of D0 . Then N  1 lim lim sup sup |LE (s + imh) − LE,n (s + imh)) = 0. n→∞ N →∞ (N + 1) s∈K m=0

Proof. It follows from [1] and [4, p. 79] that the function LE (s), for σ > 1, is of finite order, i.e., LE (σ + it)  |t|α , 618

|t| ≥ t0 > 0,

α > 0,

(12)

and the mean square of LE (s), for σ > 1, is bounded: 1 T

T |LE (σ + it)|2 dt  1 as T → ∞.

(13)

0

Using the Cauchy formula, we derive from (13) that, for σ > 1, 1 T

T

|LE (σ + it)|2 dt  1 as T → ∞.

0

This, (13), and Gallagher’s lemma (see [11, Lemma 1.4]) show now that, for σ > 1, N 1  |LE (σ + imh + iτ )|2  1 + |τ | as N → ∞. N +1

(14)

m=0

It is not difficult to see that, for σ > 1, 1 LE,n (s) = 2πi where

σ1+i∞

LE (s + z)ln (z) σ1 −i∞

s ln (s) = Γ σ1



s σ1



dz , z

ns .

Now we change the contour in the last integral. Let σ2 > 1 and σ2 < σ. Since the integrand has a simple pole at z = 1, the residue theorem together with (12) yields 1 LE,n (s) = 2πi

σ2 −σ+i∞ 

σ2 −σ−i∞

z LE (s + z)ln (z) . + LE (s). z

(15)

Let L be a closed contour in D0 enclosing K, and let δ denote the distance from K to L. Then by the Cauchy integral formula  1 |LE (z + imh) − LE,n (z + imh)||dz|. sup |LE (s + imh) − LE,n (s + imh)| ≤ 2πδ s∈K L

Therefore, for sufficiently large N , N 1  sup |LE (s + imh) − LE,n (s + imh)| N +1 m=0 s∈K  N 1  1 |LE (z + imh) − LE,n (z + imh)||dz|  N +1 2πδ m=0



|L| sup N + 1 s∈L

L 2N 

|LE (σ + imh) − LE,n (σ + imh)|,

(16)

m=0

where |L| is the length of L. By (15) ∞ LE (σ + imh) − LE,n (σ + imh) 

|LE (σ2 + imh + iτ )||ln (σ2 − σ + iτ )| dτ. −∞

619

Therefore, we find in view of (14) that 2N 1  |LE (σ + imh) − LE,n (σ + imh)| N +1 m=0

∞ 

 |ln (σ2 − σ + iτ )|

 2N 2 1  2 |LE (σ2 + imh + iτ )| dτ N +1 1

m=0

−∞

∞ |ln (σ2 − σ + iτ )|(1 + |τ |) dτ.



(17)

−∞

We can choose δ and σ2 such that the inequality σ2 − σ ≤ −c < 0 be satisfied. Then ∞ lim sup

n→∞ σ≤−c −∞

|ln (σ + iτ )|(1 + |τ |) dτ = 0.

This, (16), and (17) prove the lemma. Lemma 3. Let K be a compact subset of D0 . Then N 1  lim lim sup sup |LE (s + imh, ωh ) − LE,n (s + imh, ωh )| = 0 n→∞ N →∞ N + 1 s∈K m=0

for almost all ωh ∈ Ωh . Proof. Since {ωh (m)} is a sequence of pairwise orthogonal random variables (see [7, p. 342]); using Rademacher’s theorem on the series of pairwise orthogonal random variables (for example, see [12, p. 458]) 1 and the estimate λ(m)  m 2 d(m), we infer that the series ∞  λ(m)ωh (m) ms

m=1

for almost all ωh ∈ Ω converges uniformly on the compact subsets of the half-plane D0 . By the proof of Proposition 1 the product   −1  λ(p)ωh (p) −1  λ(p)ωh (p) ωh2 (p) + 1− 1 − ps ps p2s−1 pΔ

p|Δ

also converges uniformly on the compact subsets of D0 for almost all ωh ∈ Ωh . Since, for σ > 32 ,    −1 ∞  λ(m)ωh (m)  λ(p)ωh (p) −1  λ(p)ωh (p) ωh2 (p) = + 2s−1 ; 1− 1− ms ps ps p m=1

p|Δ

pΔ

therefore, this equality remains true by analytic continuation for almost all ωh ∈ Ωh in the half-plane σ > 1. Hence it follows from Lemma 9 of [7] that, for σ > 1, N 

|LE (σ + imh, ωh )|2  N

as N → ∞

(18)

m=0

for almost all ωh ∈ Ωh . Note that Lemma 9 of [7] deals with a wide class of Dirichlet series including LE (s). Now in view of (18) the proof of the lemma coincides with that of Lemma 2. 620

Proof of Theorem 4. We note first that the probability measures PN and N (A) = μN (LE (s + imh, ωh ) ∈ A), P

A ∈ B(H(D0 )),

both converge weakly to the same probability measure P on (H(D), B(H(D))) as N → ∞. Taking into account Theorem 6 and Lemmas 2 and 3, we can derive this by using the same arguments as in the proof of Theorem 6. In this case we consider the H(D0 )-valued random element XN,n (s) = LE,n (s + iθN ) instead of XN,n,M (s), and the random element YN (s) = LE (s + iθN ) instead of XN,n . Similar changes  N,n (s, ωh ).  N,n,M (s, ωh ) and X are also used for the random elements X The identification of the limit measure runs by a standard method. We take a continuity set A of the measure P . Then by the first part of the proof lim μN (LE (s + imh, ωh ) ∈ A) = P (A).

N →∞

(19)

Now we fix the set A, and let θ be a random variable on (Ωh , B(Ωh ), mhH ) given by  1 if LE (s, ωh ) ∈ A, θ(ωh ) = 0 if LE (s, ωh ) ∈ A. 

Then

θdmhH = mhH (ωh ∈ Ωh : LE (s, ωh ) ∈ A) = PLE (A),

Eθ =

(20)

Ωh

where PLE (A) is the distribution of the random element LE (s, ωh ). Let ah = {p−ih : p is prime}, and define a transformation fh of Ωh taking fh (ωh ) = ah ωh for ωh ∈ Ωh . Then fh is a measurable measure preserving transformation on (Ωh , B(Ωh ), mhH ). It was proved in [7, Lemma 7] that the transformation fh is ergodic. This and the classical Birkhoff theorem yield N 1  θ (fhm (ωh )) = Eθ lim N →∞ N + 1

(21)

m=0

for almost all ωh ∈ Ωh . However, the definitions of θ and fh show that the left-hand side of (21) is μN (LE (s + imh, ωh ) ∈ A). Therefore, in view of (20) and (21) lim μN (LE (s + imh, ωh ) ∈ A) = PLE (A)

N →∞

for almost all ωh . This and (19) prove that P (A) = PLE (A) for every continuity set of P . Hence, P (A) = PLE (A) for all A ∈ B(H(D0 )). Let M > 0 be an arbitrary number, and   3 DM = s ∈ C : 1 < σ < , |t| < M . 2 Clearly, DM ⊂ D0 . Therefore, LE (s, ωh ) is also an H(DM )-valued random element defined on (Ωh , B(Ωh ), mhH ). Denote its distribution by PLE ,M . Corollary 1. The probability measure μN (LE (s + imh) ∈ A), A ∈ B(H(DM )), converges weakly to PLE ,M as N → ∞.  Proof. Since the function f : H(D0 ) → H(DM ) given by f (g) = g(s)s∈D , g ∈ H(D0 ), is M continuous, the corollary is a simple consequence of Theorem 4 and Theorem 5.1 of [8]. 621

3. Proof of Theorem 3 In order to prove Theorem 3 we must show that the support of PLE ,M is SM = {g ∈ H(DM ) : g(s) = 0 or g(s) ≡ 0 on DM }. We start with one denseness result. Given ap ∈ γ, put  λ(p)a

if p|Δ, − log 1 − ps p fp (s, ap ) = 2

ap λ(p)a if p  Δ. − log 1 − ps p + p2s−1  Lemma 4. The set of all convergent series p fp (s, ap ) is dense in H(DM ). Proof. We will prove that, for every fixed p0 , the set of all convergent series  fp (s, 1)ap , ap ∈ γ,

(22)

(23)

p>p0

is dense in H(DM ). For brevity, put  fˆp (s, 1) =

fp (s, 1) if p > p0 , 0 if p ≤ p0 .

We have by (1) that, for p > p0 ,

λ(p) fˆp (s, 1) = s + rp (s) p  with rp (s)  p1−2σ . Therefore, the series p rp (s) converges uniformly on the compact subsets of DM .  h (p) Also, in the proof of Proposition 1 it was obtained that the series p λ(p)ω converges uniformly on ps the compact subsets of DM for almost all ωh ∈ Ωh with respect to mhH . These remarks show that there exists a sequence {ˆ ap : a ˆp ∈ γ} such that the series  ap (24) fˆp (s, 1)ˆ p

converges in H(DM ). Now let gp (s) = fˆp (s, 1)ˆ ap . Clearly, to prove the denseness of the set of all convergent series (23), it suffices to show that the set of all convergent series  gp (s)ap , ap ∈ γ, (25) p

is dense in H(DM ). To this end we will apply Theorem 6.3.10 of [9]. Let μ be a complex Borel measure on (C, B(C)) with compact support in DM such that     gp (s) dμ < ∞.   p

(26)

C

 λ(p)ˆ a Put hp (s) = ps p . Then, obviously, the series p |gp (s) − hp (s)| converges uniformly on the compact subsets of DM . Therefore, (26) implies       hp (s) dμ < ∞. p

622

C

Hence,



    |λ(p)| p−s dμ < ∞.

p

(27)

C

   M = s ∈ C : 1 < σ < 1 , u(s) = s − 1 , and μu−1 (A) = We now transform the domain DM . Let D 2 2  M , and in view μ(u−1 A), A ∈ B(C). Then μu−1 is a complex Borel measure with compact support in D of (27)  |λp ||k(log p)| < ∞, (28) p

where λp = λ(p)p

− 12

and

 k(z) =

e−sz dμu−1 (s),

z ∈ C.

C

We now apply a version of Bernstein’s theorem (see [9, Theorem 6.4.12]) to k(z). We observe first that, for y positive,  |k(±iy)| ≤ eM y | dμu−1 (s)|. C

Hence, lim sup y→∞

log |k(±y)| ≤ M. y

Therefore, condition a) of the above-mentioned theorem is satisfied with α = M . We take a positive π number β, β < M , and define the set A = {m ∈ N : ∃r ∈ ((m − 1/4)β, (m + 1/4)β], k(r) ≤ e−r }. Moreover, let θ, 0 < θ < 1, be fixed, and let Pθ denote the set of prime numbers satisfying |λp | > θ. Then (28) yields  |k(log p)| < ∞. (29) p∈Pθ

Moreover,

where



p



|k(log p)| ≥

p∈Pθ

  m∈A /

p

|k(log p)| ≥

  1 , pp

(30)

m∈A /

denotes the sum over all primes p ∈ Pθ such that

(m − 1/4)β < log p ≤ (m + 1/4)β. 

 

 Take a = exp m − 14 β and b = exp m + 14 β . Then (29) and (30) show that   1 < ∞. p p∈P m∈A /

(31)

θ a
Putting πθ (x) =



1,

p≤x p∈Pθ

we find by the definition of λp and (1) that, for a < u ≤ b,    λ2p ≤ 4 1 + θ2 1 = (4 − θ2 )(πθ (u) − πθ (a)) + θ2 (π(u) − π(a)), a
a
(32)

a
623

where, as usual, π(x) =



1.

p≤x

By the Shimura–Taniyama–Weil conjecture proved partially in [13] and completely in [1] the function LE (s) coincides with the L-function of some new form of weight 2 for some Hecke subgroup. Therefore, from [14, Theorem 8.4] it follows that 1  2 λp = 1. x→∞ π(x) lim

p≤x

Let δ be a small positive constant and u ≥ a(1 + δ). Then this and (32) imply, as m → ∞,   1 − θ2 + o(1) (π(u) − π(a)). (πθ (u) − πθ (a)) ≥ 4 − θ2 Summation by parts yields, as m → ∞, b   b   1  dπθ (u)  1 − θ2 1 − θ2 dπ(u) = ≥ ≥ + o(1) + o(1) p u 4 − θ2 u 4 − θ2 p∈P

θ a
a

a

 a(1+δ)
1 . p

(33)

From the classical formula 1 p≤x

p

= log log x + c1 + O(e−c2

√

x

) as x → ∞

we obtain as m → ∞ that  a(1+δ)
1 1 log(1 + δ) 1 = − +O p 2 β m



1 m2

 .

This and (33), as m → ∞, lead to the inequality      1 1 1 − θ2 1 log(1 + δ) 1 ≥ − +O . p 4 − θ2 2 β m m2 p∈P θ a
Thus, in view of (31), if δ is sufficiently small then  1 < ∞. m

(34)

m ∈A

Write A = {am } where a1 < a2 < . . . . Then (34) shows that lim

m→∞

am = 1. m

(35)

By the definition of A, there exists a sequence {ξm } such that (am − 1/4)β < ξm ≤ (am + 1/4) β and

624

|k(ξm )| ≤ e−ξm .

(36)

Hence by (35) ξm =β m→∞ m

(37)

lim

and

log |k(ξm )| ≤ −1. (38) ξm m→∞ The relation (37) is condition c) of Theorem 6.4.12 from [9]. Moreover, by (36) β |ξm − ξn | ≥ |am − an |β − ≥ c3 |m − n|. 2 Therefore, condition b) of Theorem 6.4.12 of [9] holds, too. Consequently, the latter theorem and (38) imply that log |k(r)| ≤ −1. (39) lim sup r r→∞ However, by Lemma 6.4.10 from [9], if k(z) ≡ 0 then log |k(r)| > −1, lim sup r r→∞ and this contradicts (39). Hence k(z) ≡ 0, and by differentiation we find that  sr dμu−1 (s) = 0, r = 0, 1, 2, . . . . lim sup

C

This and the definition of μu−1 show  also that sr dμ(s) = 0,

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

C

So we have  proved that condition a) of Theorem 6.3.10 of [7] holds. Moreover, by the definition of gp (s) the series p gp (s) converges in H(DM ), and, for every compact subset K of DM ,  sup |gp (s)|2 < ∞. p s∈K

Therefore, the sequence {gp (s)} satisfies all hypotheses of Theorem 6.3.10 from [9]. Hence the set of all convergent series (25) is dense in H(DM ). It is easy now to complete the proof of Lemma 4. Really, let K be a compact subset of DM , x(s) ∈ H(DM ), and let ε be an arbitrary positive number. We fix p0 such that    ∞ |λ(p)|k ε sup < . (40) kσ kp 4 s∈K p>p 0

k=2

ap ∈ γ} such that In virtue of the denseness of all convergent series (23) there exists a sequence { ap :      ε  sup x(s) − fp (s, 1) − fp (s, 1) ap  < . (41) 2 s∈K p>p p≤p0

0



We take ap =

1  ap

if p ≤ p0 , if p > p0 .

Then by (40) and (41)            sup x(s) − fp (s, ap ) ≤ sup x(s) − fp (s, 1) − fp (s, 1) ap  s∈K

s∈K

p

p≤p0

p>p0

   ∞   |λ(p)|k ε  fp (s, 1) ap − fp (s, ap ) << + 2 sup < ε, + sup  2 kpkσ s∈K p>p s∈K p>p p>p 0

0

0

k=2

and the lemma is proved. We are now ready to consider the support of PLE ,M . 625

Lemma 5. The support of PLE ,M is SM . Proof. We have seen in the proof of Proposition 1 that {ωh (p)} is a sequence of independent random variables on (Ωh , B(Ωh ), mhH ). Hence, keeping the notation of (22), we have that {fp (s, ωh (p))} is a sequence of independent H(DM )-valued random elements on (Ωh , B(Ωh ), mhH ). The support of the random element fp (s, ωh (p)) is {g ∈ H(DM ) : g(s) = fp (s, a), a ∈ γ}. Consequently, by Theorem 1.7.10 of [9] the support of the H(DM )-valued random element  log LE (s, ωh ) = fp (s, ωh (p)) p

is the closure of the set of all convergent series  fp (s, ap ), ap ∈ γ. p

By Lemma 4 the latter set is dense in H(DM ). The map u : H(DM ) → H(DM ), given by the formula u(g) = eg , g ∈ H(DM ), is a continuous function sending log LE (s, ωh ) onto LE (s, ωh ) and H(DM ) onto SM \{0}. Thus, the support of LE (s, ωh ) includes SM \{0}. However, the support of LE (s, ωh ) is a closed set. Therefore, by the Hurwitz theorem (see [9, Lemma 6.5.5]) SM \ {0} = SM , and thus the support of LE (s, ωh ) contains SM . On the other hand, LE (s, ωh ) is an almost sure convergent product of nonzero factors, hence by the Hurwitz theorem again we find that the support of LE (s, ωh ) lies in SM almost surely. This together with the above remark proves the lemma. Proof of Theorem 3. Clearly, there exists a number M > 0 such that K ⊂ DM . We suppose first that the function f (s) has nonvanishing analytic continuation to DM . Let G denote the set of functions g ∈ H(DM ) such that sup |g(s) − f (s)| < ε. s∈K

The set G is open, and by Lemma 5 the function f (s) belongs to the support of the random element LE (s, ωh ). Therefore, Corollary 1, the properties of weak convergence of probability measures, and Theorem 1.1.8 of [9] yield lim inf μN (sup |LE (s + imh) − f (s)| < ε) ≥ PLE ,M (G) > 0. N →∞

s∈K

(42)

Now let f (s) satisfy the hypotheses of Theorem 3. Then by the Mergelyan theorem (see [15] for example) there exists a polynomial pn (s), pn (s) = 0 on K, such that sup |f (s) − pn (s)| < ε/4.

s∈K

(43)

Since the polynomial pn (s) has finitely many zeros, there exists a domain G1 with connected complement, K ⊂ G1 , and pn (s) = 0 on G1 . Hence there is a continuous branch of log pm (s) on G1 analytic in the interior of G1 . By the Mergelyan theorem again there exists a polynomial qm (s) such that sup |pn (s) − eqm (s) | < ε/4.

s∈K

This and (43) show that

sup |f (s) − eqm (s) | < ε/2.

s∈K

However, eqm (s) = 0. Therefore, (42) implies lim inf μN (sup |LE (s + imh) − eqm (s) | < ε/2) > 0. N →∞

This and (44) prove the theorem. 626

s∈K

(44)

References 1. Breuil C., Conrad B., Diamond F., and Taylor R., “On the modularity of elliptic curves over Q: wild 3-adic exercises,” J. Amer. Math. Soc., 14, No. 4, 843–939 (2001). 2. Washington L. C., Elliptic Curves. Number Theory and Cryptography, Chapman and Hall/CRC, London, New York (2003). 3. Garbaliauskien˙e V. and Laurinˇcikas A., “Some analytic properties for L-functions of elliptic curves,” Proc. Inst. Math., 13, No. 1, 1–8 (2005). 4. Laurinˇcikas A., Matsumoto K., and Steuding J., “The universality of L-functions associated to new forms,” Izv. Math, 67, No. 1, 77–90 (2003). 5. Voronin S. M., “A theorem on the ‘universality’ of the Riemann zeta-function,” Math. USSR-Izv., 9, 443–453 (1975). 6. Garbaliauskien˙e V. and Laurinˇcikas A., “Discrete value distribution of L-functions of elliptic curves,” Publ. Inst. Math., 76, No. 90, 65–71 (2004). 7. Kaˇcinskait˙e R. and Laurinˇcikas A., “On the value distribution of the Matsumoto zeta-function,” Acta Math. Hungar., 105, No. 4, 339–359 (2004). 8. Billingsley P., Convergence of Probability Measures, John Wiley & Sons, New York (1968). 9. Laurinˇcikas A., Limit Theorems for the Riemann Zeta-Function, Kluwer Acad. Publ., Dordrecht; Boston; London (1996). 10. Conway J. B., Functions of One Complex Variable, Springer-Verlag, New York (1973). 11. Montgomery L. M., Topics in Multiplicative Number Theory, Springer-Verlag, Berlin (1971). 12. Lo`eve M., Probability Theory, Van Nostrand, Toronto (1955). 13. Wiles A., “Modular elliptic curves and Fermat’s last theorem,” Ann. Math., 144, 443–551 (1995). 14. Murty M. R. and Murty V. K., Non-Vanishing of L-Functions and Applications, Birkh¨ auser, Basel etc. (1997). 15. Walsh J. L., Interpolation and Approximation by Rational Functions in the Complex Domain, Amer. Math. Soc., Providence, RI (1960) (Amer. Math. Soc. Collog. Publ.; 20). ˙ V. Garbaliauskiene; J. Genys; ˇ ˇ Siauliai University, Siauliai, Lithuania E-mail address: [email protected] ˇikas A. Laurinc Vilnius University, Vilnius, Lithuania E-mail address: [email protected]

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