THE JOINT UNIVERSALITY FOR GENERAL DIRICHLET SERIES

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1 Annales Univ. Sci. Budapest., Sect. omp. 22 (2003) THE JOINT UNIVERSALITY FOR GENERAL DIRIHLET SERIES A. Laurinčikas (Vilnius, Lithuania) In honour of Professor Karl-Heinz Indlekofer on the occasion of his 60th birthday 1. Introduction In 1975 S.M. Voronin [14] discovered one more remarkable property of the Riemann zeta-function ζ(s), s = σ + it. Roughly speaking he proved that any analytic function can be approximated uniformly on some sets by translations ζ(s + iτ). The latter property of ζ(s) is called the universality. To state the last version of Voronin s theorem we need some notations. Let meas{a} denote the Lebesgue measure of the set A, and let, for T > 0, ν T (...) = 1 meas{τ [0, T ] :...}, T where in place of the dots a condition satisfied by τ is to be written. As usual, stands for the complex plane. Voronin s theorem ([5]). Let K be a compact subset of the strip {s 1 : 2 < σ < 1} with connected complement, and let g(s) be a non-vanishing continuous function on K which is analytic in the interior K. Then, for every ε > 0, ( lim inf ν T T s K ζ(s + iτ) g(s) ) < ε > 0. Later the universality of zeta-functions was studied by many mathematicians, among them by S.M. Gonek, A. Reich, B. Bagchi, K. Matsumoto, H. Mishou, W. Schwarz, J. Steuding, by the author, and by others. Partially ported by the Lithuanian Science and Studies Foundation.

2 236 A. Laurinčikas The paper [11] is devoted to the universality of general Dirichlet series (1) a m e λms, where a m and {λ m } is an increasing sequence, λ m +. Denote by σ a the abscissa of absolute convergence of series (1), and let f(s), for σ > σ a, be its sum. The universality for the function f(s) requires several additional conditions. Let the system of exponents {λ m } of series (1) be linearly independent over the field of rational numbers. We pose that f(s) cannot be represented by an Euler product over primes in the half-plane σ > σ a. Moreover, we pose that f(s) is meromorphically continuable to the half-plane σ > σ 1 with some σ 1 < σ a, and that it is analytic in the strip D = { s : σ 1 < σ < σ a }. For conditions of the continuation see, for example, [13]. assume that, for σ > σ 1, the estimates Furthermore we f(s) = B t α, t t 0, α > 0, and T f(σ + it) 2 dt = BT, T, T are satisfied. Here and what follows B denotes a quantity bounded by some constant. Moreover, we require some conditions on the sequences {a m } and {λ m }. Let, for x > 0, r(x) = 1, λ m x and c m = a m e λmσa. Suppose that there exists a real θ > 0 with the property as x, and that λ m x c m 2 = θr(x)(1 + o(1)) c m d for some d > 0. Finally, we pose that (2) r(x) = 1 x κ + B

3 The joint universality for general Dirichlet series 237 with κ 1 and positive 1, and B 2. Then in [11] the following statement was obtained. Theorem A. Suppose that the function f(s) satisfies all the conditions stated. Let K be a compact subset of the strip {s : σ 1 < σ < σ a } with connected complement, and let g(s) be a continuous function on K which is analytic in the interior of K. Then, for every ε > 0, ( lim inf ν T T s K f(s + iτ) g(s) ) < ε > 0. The aim of this note is to obtain a joint universality theorem for general Dirichlet series, i.e. to prove that a collection of analytic functions can be simultaneously approximated by translations of general Dirichlet series. We recall that the joint universality for Dirichlet L-functions L(s, χ) was proved independently by S.M. Voronin [15-16], S. M. Gonek [4] and B. Bagchi [1-2]. For example, in [15] we find the following statement. Theorem B. Suppose that 0 < r < 1/4. Let χ 1,..., χ n be pairwise nonequivalent Dirichlet characters, and let f 1 (s),..., f n (s) be continuous and non-vanishing on s r functions which are analytic on s < r. Then, for every ε > 0, ( lim inf max max T (s L + 3 ) ) 4 + iτ, χ j f j (s) < ε > 0. s r The paper [6] contains a joint conditional universality theorem for Matsumoto zeta-functions, while in [7] the joint universality for Lerch zeta-functions is obtained. The paper [8] is devoted to the joint universality for zetafunctions attached to certain cusp forms. In [9] the joint universality of twisted automorphic L-functions is proved. During the conference Theory of the Riemann zeta and allied functions at Oberwolfach in 2001 Professor E. Bombieri in discussion with the author noted that joint universality theorems for zeta-functions are an interesting and important problem of analytic number theory. It seems to be that the joint universality for general Dirichlet series is rather complicated problem. Therefore, we limit ourselves by the investigation of a collection of general Dirichlet series with the same sequence of exponents {λ m }. Let, for σ > σ aj, the series f j (s) = a mj e λ ms

4 238 A. Laurinčikas converges absolutely, j = 1,..., n. Suppose that f j (s) is meromorphically continuable to the half-plane σ > σ 1j with some σ 1j < σ aj, all poles being included in a compact set, that it is analytic in the strip D j = {s : σ 1j < < σ < σ aj }, and that f j (s) cannot be represented by an Euler product over primes in the region σ > σ aj, j = 1,..., n. We also require that, for σ > σ 1j, the estimates (3) f j (σ + it) = B t α j, t t 0, α j > 0, and (4) T T f j (σ + it) 2 dt = BT, T, hold, j = 1,..., n. Moreover, we need some conditions on the sequences {a mj } and {λ m }. We pose that the system {log 2} {λ m } is linearly independent over the field of rational numbers. Let c mj = a mj e λ mσ aj, j = 1,..., n. Suppose that there exist r n sets N k, N k1 N k2 = for k 1 k 2, N = r k=1 N k, such that c mj = b kj for m N k, k = 1,..., r, j = 1,..., n. We set B = b b 1n b r1... b rn We also pose that the sequence of exponents {λ m } satisfies the relation (2), and that (5) 1 = κ k r(x)(1 + o(1)), x, λm x with positive κ k, k = 1,..., r. Theorem. Suppose that conditions (2) (4) are satisfied, and that rank (B) = n. Let K j be a compact set of the strip D j with connected complement, and let g j (s) be a continuous function on K j which is analytic in the interior of K j, j = 1,..., n. Then, for any ε > 0, ( lim inf ν T T. fj (s + iτ) g j (s) ) < ε > 0.

5 The joint universality for general Dirichlet series A limit theorem for the functions f j (s) For any region G on the complex plain, by H(G) denote the space of analytic on G functions equipped with the topology of uniform convergence on compacta. Let N > 0, and let D j,n = { s : σ 1j < σ < σ aj, t < N }, j = 1,..., n, H n,n = H n (D 1,N,..., D n,n ) = H(D 1,N )... H(D n,n ). Denote by B(S) the class of Borel sets of the space S, and define the probability measure ( (f1 P T (A) = ν T (s 1 + iτ),..., f n (s n + iτ) ) ) A, A B(H n,n ). To prove the theorem we need a limit theorem for the measure P T as T in the sense of the weak convergence of probability measures. Moreover, the limit measure in such theorem must be explicitly given. For this, define the following topological structure. Let γ denote the unit circle on, and Ω = γ m, where γ m = γ for all m N. The infinitedimensional torus Ω is a compact topological Abelian group, therefore on (Ω, B(Ω)) the probability Haar measure m H exists, and we obtain a probability space (Ω, B(Ω), m H ). Let ω(m) stand for the projection of ω Ω onto the coordinate space γ m. Now on the probability space (Ω, B(Ω), m H ) we define the H n,n -valued random element f(s 1,..., s n ; ω) by the formula f(s 1,..., s n ; ω) = ( f 1 (s 1, ω),..., f n (s n, ω) ), where f j (s j, ω) = a mj ω(m)e λmsj, s j D j,n, j = 1,..., n,

6 240 A. Laurinčikas and let P f be the distribution of the random element f(s 1,..., s n ; ω), i.e. P f (A) = m H ( ω Ω : f(s1,..., s n ; ω) A ), A B(H n,n ). Lemma 1. The probability measure P T P f as T. converges weakly to the measure Proof. Let D j = {s : σ > σ 1j }, M n = M( D 1 )... M( D n ), and ( (f1 P T (A) = ν T (s 1 + iτ),..., f n (s n + iτ) ) ) A, A B(M n ), where M(G) denotes the space of meromorphic on G functions equipped with the topology of uniform convergence on compacta. We put Ĥ n = Ĥn( D 1,..., D n ) = H( D 1 )... H( D n ), and define on (Ω, B(Ω), m H ) an Ĥn-valued random element f(s 1,..., s n ; ω) by the formula f(s 1,..., s n ; ω) = ( f(s1, ω),..., f(s n, ω) ), where f j (s j, ω) = a mj ω(m)e λmsj, s j D j, j = 1,..., n. Then in [12] it was proved that the probability measure P T converges weakly to the distribution of the random element f(s 1,..., s n ; ω) as T. Note that in [12] the condition λ mj c j (log m) θj, with positive c j and θ j, j = 1,..., n, is covered by (2) (4). Let M n,n = M(D 1,N )... M(D n,n ). Since the function h : Mn M n,n, defined by the coordinatewise restriction, is continuous, hence in view of Theorem 5.1 from [3], we obtain the assertion of the lemma.

7 The joint universality for general Dirichlet series Functions of exponential type We recall that an entire function g(s) is of exponential type if lim r log g(re iθ ) r < uniformly in θ, θ π. Now we state some lemmas on the functions of exponential type. Lemma 2. Let g(s) be an entire function of exponential type, and let {ξ m } be a sequence of complex numbers. Moreover, let α 1, α 2 and α 3 be real positive numbers satisfying 1 0 lim x log g(±ix) x 2 0 ξ m ξ n α 2 m n ; 3 0 ξ m lim m m = α 3; 4 0 α 1 α 2 < π. Then lim m α 1 ; log g(ξ m ) ξ m = lim r log g(r). r The lemma is a version of Bernstein s theorem, for the proof see Theorem of [5]. Lemma 3. Let µ be a complex-valued measure on (, B()) having compact port contained in the half-plane σ > σ 0. Define and pose that g(s) 0. Then g(s) = e sz dµ(z) lim r log g(r) r > σ 0. Proof of the lemma can be found in [5], Theorem

8 242 A. Laurinčikas 4. The port of the random element f This section is devoted to the port of the measure P f. We recall that the minimal closed set S Pf H n,n such that P f (S Pf ) = 1 is called the port of P f. The set S Pf consists of all g = (g 1 (s 1 ),..., g n (s n )) H n,n such that for every neighbourhood G of g the inequality P f (G) > 0 is satisfied. The port of distribution of the random element X is called the port of X and is denoted by S X. We begin with some auxiliary lemmas. Lemma 4. Let {X m } be a sequence of independent H n,n -valued random elements such that the series converges almost surely. Then the port of the sum of the latter series is the closure of the set of all g H n,n which may be written as convergent series g = X m g m, g m S Xm. Proof of the lemma is given in [8]. Lemma 5. Let {g m } = {(g 1m,..., g nm )} be a sequence in H n,n which satisfies: 1 0 If µ 1,..., µ n are complex measures on (, B()) with compact ports contained in D 1,N,..., D n,n, respectively, such that n g jm dµ j <, j=1 then for j = 1,..., n, r = 0, 1, 2,...; 2 0 The series s r dµ j (s) = 0 g m

9 The joint universality for general Dirichlet series 243 converges in H n,n ; 3 0 For any compacts K 1 D 1,N,..., K n D n,n, j=1 Then the set of all convergent series with a m γ is dense in H n,n. n g jm (s) 2 <. a m g m Proof of the lemma can be found in [8]. Let ν be a complex measure on (, B()) with compact port contained in { } s : min (σ 1j σ aj ) < σ < 0, t < N. Define w(z) = e sz dν(s), z. Lemma 6. Suppose that, for some k, w(λ m ) <. Then s l dν(s) = 0, l = 0, 1, 2,.... Proof. Let g(s) = w(s) in Lemma 3. Since, for x > 0, w(±x) e Nx dν(s), condition 1 0 of Lemma 2 is satisfied with α 1 = N. Now we fix an α 3 satisfying 0 < α 3 < π N,

10 244 A. Laurinčikas and a real number ξ with (6) 1 α 3 ξ > 2, where the constants 1 and 2 are given by formula (2). Define A = {l N : r ( ] (l ξ)α 3, (l + ξ)α 3 with w(r) 1 r κ }. We have w(λ m ) w(λm ) l A m l A m 1 λ κ m, where m denotes the sum over all m N k satisfying the inequalities (l ξ)α 3 < λ m (l + ξ)α 3. This and the hypothesis of the lemma yield (7) l A a<λm b 1 λ κ m < with a = (l ξ)α 3, b = (l + ξ)α 3. Summing by parts and applying conditions (2) and (5), we find 1 = = 1 b κ a<λm b b 1 + κ a ( a<λm u a<λm b λ κ m ) du 1 ( 1κ k ((l b κ (1 + o(1)) ) κ ( ) ) κ + ξ)α3 (l ξ)α3 = 1κ k α3l κ κ ( ( b κ (1 + o(1)) 1 + ξ l = 2 1κ k α κ 3l κ κξ b κ l u κ+1 ( r(b) r(a) ) κ k (1 + o(1)) bκ 2κ k b κ 2(1 + o(1)) = ) κ ( 1 ξ ) ) κ 2κ k 2 l b κ (1 + o(1)) = (1 + o(1)) + B l 2 2κ k 2 b κ (1 + o(1)) as l. Therefore, (6) and (7) yield (8) l A 1 l <.

11 The joint universality for general Dirichlet series 245 Let Then from (8) we find A = {a l : l N}, a 1 < a 2 <.... a l (9) lim l l = 1. By the definition of the set A there exists a sequence ξ l such that (a l ξ)α 3 < ξ l (a l + ξ)α 3 and This and (9) show that and Moreover, in view of (9) lim l w(ξ l ) 1 ξ κ l ξ l lim l l. = α 3, log w(ξ l ) ξ l 0. ξ m ξ n > a m a n α 3 α 2 m n with some positive constant α 2. Therefore, applying Lemma 2, we find that (10) lim r log w(r) r 0. On the other hand, by Lemma 3, if w(s) 0, then lim r log w(r) r > 0, and this contradicts (10). onsequently, w(s) 0, and the lemma follows by differentiation. Lemma 7. whole of H n,n. The port of the random element f(s 1,..., s n ; ω) is the Proof. Let, for m = 1, 2,..., f m (s 1,..., s n ; ω(m)) =

12 246 A. Laurinčikas = ( f m (s 1, ω),..., f m (s n, ω) ) = ( a m1 ω(m)e λ ms 1,..., a mn ω(n)e λ ms n ). It follows from the definition of Ω that {ω(m)} is a sequence of independent random variables with respect to the measure m H. Hence {f m (s 1,..., s n ; ω(m)} is a sequence of independent H n,n -valued random elements defined on the probability space (Ω, B(Ω), m H ). The port of each random variable ω(m) is the unit circle γ. Therefore, the set {f m (s 1,..., s n ; a) : a γ} is the port of the random element f m (s 1,..., s n ; ω(m)). Hence in virtue of Lemma 4 the closure of the set of all convergent series f m (s 1,..., s n ; a m ), a m γ, is the port of the random element f(s 1,..., s n ; ω). To prove the lemma it remains to check that the latter set is dense in H n,n. For this we will apply Lemma 5. Let µ 1,..., µ n be complex measures on (, B()) with compact ports contained in D 1,N,..., D n,n, respectively, such that (11) n j=1 a mj e λms dµ j (s) <. Now let Then we have that h j (s) = s σ aj, j = 1,..., n. µ j h 1 j (A) = µ j (h 1 j A), A B(), is a complex measure of (, B()), with compact port contained in D j,n = = {s : σ 1j σ aj < σ < 0, t < N}, j = 1,..., n. Now (11) can be rewritten in the form n c mj j=1 e λ ms dµ j h 1 j This together with hypotheses on c mj leads to (s) <. (12) n b kj j=1 e λms dµ j h 1 j (s) <, k = 1,..., r.

13 The joint universality for general Dirichlet series 247 Taking µ k (A) = we write (12) in the form n j=1 b kj µ j h 1 j (A), A B(), v k (z) = e sz d µ k (s), z, k = 1,..., r, v k (λ m ) <, k = 1,..., r. learly, µ k, k = 1,..., r, is a complex measure on (, B()) with compact port contained in { s : min (σ 1j σ aj ) < σ < 0, Now Lemma 3 shows that v k (z) 0, and thus } t < N. s l d µ k (s) = 0, l = 0, 1, 2,..., k = 1,..., r. Hence, using the definition of µ k and the properties of the matrix B, we obtain that s l dµ j h 1 j (s) = 0, l = 0, 1, 2,..., j = 1,..., n, and this together with definition of the function h j implies the relations (13) In the proof that s l dµ j (s) = 0, l = 0, 1, 2,..., j = 1,..., n. f j (s j, ω) = a mj ω(m)e λ ms j is an H(D j )-valued random element, D j = {s : σ > σ 1j }, it is proved in [10] that, for almost all ω Ω, the series for f j (s j, ω) converges uniformly

14 248 A. Laurinčikas on compact subsets of D j, j = 1,..., n. {b m : b m γ} such that Therefore, there exists a sequence f m (s 1,..., s n ; b m ) converges in H n,n. Moreover, in [10] it was obtained that, for σ > σ 1j, a mj 2 e 2λmσ <, j = 1,..., n. Hence, by the well-known property of Dirichlet series, see orollary of [5], we have that for any compacts K j D j,n, j = 1,..., n, j=1 n fmj (s, b m ) 2 <. Since b m = 1, condition (13) is valid also for f(s 1,..., s n ; b m ). Thus we have that all conditions of Lemma 3 for f m (s 1,..., s n ; b m ) are satisfied, and therefore the set of all convergent series a mf m (s 1,..., s n ; b m ) with a m γ is dense in H n,n. Hence the set of all convergent series f m (s 1,..., s n ; a m ), a m γ, is dense in H n,n, and the closure of this set is the whole of H n,n. The lemma is proved. 5. Proof of the Theorem The proof is similar to that given in [8]. First we pose that the functions g j (s) can be continued analytically to the whole of D j,n, respectively, j = = 1,..., n. Denote by G the set of all (y 1,..., y n ) H n,n such that yj (s) g j (s) ε < 4.

15 The joint universality for general Dirichlet series 249 The set G is open. Therefore, Lemma 1, properties of the weak convergence and Lemma 5 show that ( lim inf f j (s + iτ) g j (s) < ε ) P f (G) > 0. T 4 Now let the functions g j (s), j = 1,..., n, be the same as in the statement of the theorem. By the Mergelyan theorem, see, for example, [17], there exist polynomials p j (s), j = 1,..., n, such that (14) By the first part of the proof we have ( (15) lim inf ν T T p j (s) g j (s) < ε 2. f j (s + iτ) p j (s) < ε ) > 0. 2 In view of (14) { τ : { τ : f j (s + iτ) p j (s) < ε } 2 fj (s + iτ) g j (s) } < ε. Therefore this and (15) yield that ( lim inf ν T T fj (s + iτ) g j (s) ) < ε > 0, and the theorem is proved. Now we give an example. Let a mj be a periodic sequence with period r n, j = 1,..., n, and λ m = (m + α) β with some transcendental α > 0 and β (0, 1). Then f j (s) = a mj e (m+α)β s converges absolutely for σ > σ aj = 0, j = 1,..., n, and c mj = a mj is constant on the set N k = {m N : m k (mod r)}, j = 1,..., n. Moreover, r(x) = (m+α) β x 1 = x 1/β + B.

16 250 A. Laurinčikas learly, elements b kj can be chosen so that rank (B) = n. Therefore, assuming that f j (s) is analytically continuable to a half-plane σ > σ 1j with σ 1j < σ aj, and the estimates (3) and (4) are satisfied, we obtain the joint universality of functions f 1 (s),..., f n (s). References [1] Bagchi B., The statistical behaviour and universality properties of the Riemann zeta-function and other allied Dirichlet series, Ph.D. Thesis, Indian Statistical Institute, alcutta, [2] Bagchi B., Joint universality theorem for Dirichlet L-functions, Math. Z., 181 (1982), [3] Billingsley P., onvergence of probability measures, John Wiley & Sons, New York, [4] Gonek S.M., Analytic properties of zeta and L-functions, Ph.D. Thesis, University of Michigan, [5] Laurinčikas A., Limit theorems for the Riemann zeta-function, Kluwer, [6] Laurinqikas A., O nul h line inyh kombinaci i dzeta-funkcii Matsumoto, Lietuvos Mat. Rinkinys, 38 (2) (1998), (On the zeros of linear combinations of the Matsumoto zeta-functions, Lith. Math. J., 38 (2) (1998), ) [7] Laurinčikas A. and Matsumoto K., The joint universality and the functional independence for Lerch zeta-functions, Nagoya Math. J., 157 (2000), [8] Laurinčikas A. and Matsumoto K., The joint universality of zetafunctions attached to certain cusp forms, Proc. Sci. Seminar Faculty of Phys. and Math., Šiauliai University, 5 (2002), [9] Laurinčikas A. and Matsumoto K., The joint universality of twisted automorphic L-functions, J. Math. Soc. Japan (to appear). [10] Laurinčikas A., Schwarz W. and Steuding J., Value distribution of general Dirichlet series III., Analytic and probabilistic methods in number theory, Proc. of the Third Intern. onf. in Honour of J. Kubilius, Palanga, Lithuania, September 2001, (eds. A. Dubickas, A. Laurinčikas and E. Manstavičius), TEV, Vilnius, 2002, [11] Laurinčikas A., Schwarz W. and Steuding J., The universality of general Dirichlet series, Analysis (to appear).

17 The joint universality for general Dirichlet series 251 [12] Laurinqikas A. i Steuding I., Mnogomerna predelьna teorema dl obwih r dov Dirihle, Lietuvos Mat. Rinkinys, 42 (4) (2002), (Laurinčikas A. and Steuding J., A joint limit theorem for general Dirichlet series, Lith. Math. J., 42 (4) (2002), ) [13] Mandelbrojt S., Séries de Dirichlet, Gauthier-Villars, Paris, [14] Voronin S.M., Teorema ob universalьnosti dzeta-funkcii Rimana, Izv. AN SSSR, Ser. matem., 39 (1975), (Voronin S.M., Theorem on the universality of the Riemann zeta-function, Math. USSR Izv., 9 (1975), ) [15] Voronin S.M., O funkcionalьno i nezavisimosti L-funkci i Dirihle, Acta Arith., 27 (1975), (Voronin S.M., On the functional independence of Dirichlet L-functions) [16] Voronin S.M., Analitiqeskie svo istva proizvod wih funkci i Dirihle arifmetiqeskih obъektov, Dis. doktora fiz.-mat. nauk, Moskva, (Voronin S.M., Analytic properties of generating Dirichlet functions of arithmetical objects, DSc Thesis, Moscow, 1977.) [17] Walsh J.L., Interpolation and approximation by rational functions in the complex domain, Amer. Math. Soc. oll. Publ. 20, A. Laurinčikas Dept. of Probability Theory and Number Theory Vilnius University Naugarduko 24 LT-2600 Vilnius, Lithuania