On the universality for functions in the Selberg class
Jorn¨ Steuding∗
August 2002
Abstract. We prove a universality theorem for functions in the Selberg class.
Keywords: universality, L-functions, Selberg class. AMS subject classification numbers: 11M06, 11M41.
1 Introduction and statement of the main result
Let s = σ + it be a complex variable. The Riemann zeta-function is given by
∞ 1 1 −1 ζ(s)= s = 1 − s for σ>1, n=1 n p p ! X Y and by analytic continuation elsewhere, except for a simple pole at s =1;hereand in the sequel p denotes always a prime number, and the product above is taken over all primes. The Euler product gives a first glance on the close connection between ζ(s) and the distribution of prime numbers. However, the Riemann zeta-function has interesting function-theoretical properties beside. Voronin [32] proved a remarkable universality theorem for ζ(s), namely that any non-vanishing continuous function g(s) 1 on the disc {s ∈ C : |s|≤r} with 0
1 here and in the sequel meas A stands for the Lebesgue measure of the set A.The theorem states that the set of translates on which ζ(s) approximates a given function g(s) has a positive lower density! Meanwhile, it is known that there exists a rich zoo of Dirichlet series with this universality property; we mention only some significant examples: Voronin [32] proved joint universality for Dirichlet L-functions to pairwise non-equivalent characters; Reich [26] obtained universality for Dedekind zeta-functions; Bagchi [1] for certain Hurwitz zeta-functions; Laurinˇcikas [10], [11], [12] for certain Dirichlet series with multiplicative coefficients, Lerch zeta-functions and Matsumoto zeta-functions; Matsumoto [15] for Rankin-Selberg L-functions; Laurinˇcikas and Matsumoto [13] for L-functions attached to modular forms and, jointly with the author [14], for L-functions associated to new- forms (resp. elliptic curves); Mishou [17] for Hecke L-functions. It is expected that all functions given by a Dirichlet series and analytically continuable to the left of the half plane of absolute convergence, which satisfy some natural growth conditions, are universal. For a nice survey on this topic see [16]. The aim of this paper is to prove a universality theorem for functions in the Selberg class; the proof makes use of the positive density method, introduced by Laurinˇcikas and Matsumoto [13]. Selberg [27] defined a general class of Dirichlet series having an Euler product, analytic continuation and a functional equation, and formulated some fundamental conjectures concerning them. This class of functions is of special interest in the context of the generalized Riemann hypothesis; it is expected that for every function in the Selberg class the analogue of the Riemann hypothesis holds, i.e. that the nontrivial zeros lie on the critical line. For surveys on the Selberg class we refer to [8] and [20]. The Selberg class S consists of Dirichlet series ∞ a(n) F (s):= s n=1 n X satisfying the hypotheses:
• Ramanujan hypothesis: a(n) n for every >0;
• Analytic continuation: there exists a non-negative integer m such that (s − 1)mF (s) is an entire function of finite order;
• Functional equation: there are numbers Q>0,λj > 0,µj with Re µj ≥ 0, and some complex number ω with |ω| =1suchthat
r s ΛF (s):=Q Γ(λj s + µj )F (s)=ωΛF (1 − s); j=1 Y
2 • Euler product: F (s)satisfies
∞ b(pk) F (s)= Fp(s)withFp(s)=exp ks , p p ! Y kX=1
k kθ 1 where b(p ) p for some θ<2. By the latter axiom it is easily seen that the coefficients a(n) are multiplicative, and that for each prime p ∞ a(pk) (1) F (s)= , p pks kX=0 which converges absolutely for σ>0; this is proved in [4]. ThedegreeofF ∈Sis defined by
r dF =2 λj . jX=1 Despite of plenty of identities for the Gamma-function this quantity is well-defined; if
NF (T ) counts the number of zeros of F ∈Sin the region 0 ≤ σ ≤ 1, |t|≤T (according dF to multiplicities), then one can show by contour integration that NF (T ) ∼ π T log T . Denote by Sd all elements F ∈Swith degree dF equal to d. It is conjectured that all F ∈Shave integral degree. All known examples of Dirichlet series in the Selberg class are automorphic L-functions, and for all of them it turns out that the related −s Euler factors Fp are the inverse of polynomials in p (of bounded degree). Examples for functions in the Selberg class are the Riemann zeta-function and shifts of Dirichlet L-functions L(s + iθ, χ) attached to primitive characters χ and with θ ∈ R (degree 1); normalized L-functions associated to holomorphic cuspforms and (conjecturally) nor- malized L-functions attached to Maass waveforms (degree 2); Dedekind zeta-functions to number fields K (degree [K : Q]). The notion of a primitive function is very fruitful for studying the structure of the
Selberg class. A function 1 6≡ F ∈Sis called primitive if the equation F = F1F2 with
Fj ∈S,j =1, 2, implies F = F1 or F = F2. The central claim concerning primitive functions is
Conjecture [Selberg [27]] Denote by aF (n) the coefficients of the Dirichlet series representation of F ∈S.
A: For all F ∈S exists a positive integer nF such that |a (p)|2 F = n log log x + O(1); p F pX≤x
3 B: for any primitive function F ,
a (p)a (p) log log x + O(1) if F = G, F G = p x p O(1) if F 6= G. X≤ In a sense, primitive functions are expected to form an orthonormal system. Note that the prime number theorem (see [31], §3.14) implies 1 (2) =loglogx + C + O exp −c log x , p pX≤x q where c is some positive constant. However, for our purpose we have to introduce a subclass. Denote by S˜ the subset of the Selberg class consisting of F ∈Ssatisfying the following axioms:
• Euler product:foreachprimep there exist complex numbers αj(p), 1 ≤ j ≤ m, such that
m −1 αj (p) F (s)= 1 − s ; p j=1 p ! Y Y
• Ramanujan-Petersson conjecture: for all but finitely many p we have
|αj(p)| =1, 1 ≤ j ≤ m;
• Mean-square: there exist a positive constant κ such that 1 lim |a(p)|2 = κ, x→∞ π(x) p x X≤ where π(x) counts the prime numbers p ≤ x.
Note that Bombieri and Hejhal [2], resp. Bombieri and Perelli [3], defined similar subclasses for their investigations on the value distribution of L-functions. In Chapter 5 we shall give a motivation for introducing the first two axioms in the context of the Langlands program; the axiom on the mean square is closely related to Selberg’s conjectures. If we assume additionally κ ∈ N in the axiom on the mean-square, we may deduce Selberg’s Conjecture A. On the other side, a stronger version of Selberg’s Conjecture A,
2 |aF (p)| 1 = nF log log x + cF + o , p x p log x! X≤ where cF is some constant depending on F , would imply the asymptotic formula on the mean-square with κ = nF ; this is easily seen by partial integration. Now we are in the position to state the main theorem:
4 Theorem 1 Suppose that F ∈ S˜.LetK be a compact subset of the strip 1 1 D := s ∈ C :max , 1 − <σ<1 2 dF with connected complement, and let g(s) be a non-vanishing continuous function on K which is analytic in the interior of K. Then, for any >0, 1 lim inf meas τ ∈ [0,T]:max|F (s + iτ) − g(s)| < > 0. T s K →∞ T ∈ Obvious examples for functions satisfying the conditions of the theorem above are the Riemann zeta-function and Dirichlet L-functions L(s, χ) to primitive characters χ. Further examples are Dedekind zeta-functions, Hecke L-functions, Rankin-Selberg L-functions, Artin L-functions and L-functions associated to newforms; in Chapter 5 we shall give a further example. Before we start with the proof we note some consequences on the value distribution and functional independence. As in the case of the Riemann zeta-function (see [10]) one can show that i) the set
(n) {(F (s + iτ),F0(s + iτ),...,F (s + iτ)) : τ ∈ R},
n+1 lies for fixed s ∈ D everywhere dense in C , and ii) if F1(z),...,FN (z) are continuous n+1 functions on C , not all identically zero, then, for some s ∈ C,
N k 0 (n) s Fk(F (s),F (s),...,F (s)) 6=0. kX=1 2 Mean-square formulae and a limit theorem
First of all we shall prove a mean-square estimate for the coefficients of the Dirichlet series of F .
Lemma 2 As x →∞,
2 |a(n)|2 x(log x)m −1. n x X≤ Proof. By the identity
∞ m −1 m ∞ k a(n) αj(p) αj(p) s = 1 − s = 1+ ks , n=1 n p j=1 p ! p j=1 p ! X Y Y Y Y kX=1 valid for σ>1, and the unique prime factorization of the integers, we deduce
m kj a(n)= αj(p) , ν pYkn k1+...X+km=ν jY=1 5 ν ν ν+1 where p kn means that p |n but p 6|n. Takingintoaccountthat|αj(p)|≤1, we find
|a(n)|≤ 1=dm(n), ν pYkn k1+...X+km=ν say. Thus it is sufficient to find a mean-square estimate for the function dm(n); note that dm(n) is a multiplicative arithmetic function (appearing as coefficients in the Dirichlet series expansion of ζ(s)m). Consequently, we may write
2 dm(n) = g(d) Xd|n with some multiplicative function g.Since
ν m (m + ν − 1)! dm(p )=]{(k1,...,km) ∈ N0 : k1 + ...+ km = ν} = , ν!(m − 1)! 2 2 2 2 we find g(1) = dm(1) =1,g(p)=dm(p) − dm(1) = m − 1, and by induction m2ν g(pν )=d (pν )2 − d (pν−1)2 ∼ . m m ν! Hence we obtain ∞ ν 2 g(d) g(p ) dm(n) ≤ x ≤ x 1+ ν n x d p x ν=1 p ! X≤ dX≤x Y≤ X m2 − 1 ∞ m2ν m2 − 1 = x 1+ + ν = x 1+ + O(x). p x p ν=2 ν!p ! p x p ! Y≤ X Y≤ Now a well-known formula due to Mertens gives the estimate of the lemma.
Furthermore, we need a mean-square estimate for the integral over F . Therefore we apply Lemma 3 (Potter [23]) Suppose that the functions ∞ ∞ an bn A(s)= s and B(s)= s n=1 n n=1 n X X have a half plane of convergence, are of finite order, and that all singularities lie in a subset of the complex plane of finite area. Further, assume the estimates
2 β+ 2 β+ |an| x and |bn| x , nX≤x nX≤x as x →∞, and that A(s) and B(s)
c( a −σ) A(s)=h(s)B(1 − s), where h(s) |t| 2 as |t|→∞,andc is some positive constant. Then T ∞ 2 1 |an| lim |A(σ + it)|2 dt = T →∞ 2σ 2T −T n=1 n Z X a 1 1 for σ>max{ 2, 2 (β +1)− c }.
6 In view of the functional equation
r Γ(λj (1 − s)+µj) F (s)=ωQ1−2σ F (1 − s), j=1 Γ(λj s + µj ) Y and by Stirling’s formula, we obtain for fixed σ
( 1 −σ) d F (σ + it) |t| 2 F |F (1 − σ + it)| as |t|→∞.
Hence, we deduce by the Phragmen-Lindel¨of principle that F (s) is an entire function of finite order, and satisfies the estimate
d (1−σ) F (3) F (σ + it) |t| 2 for 0 ≤ σ ≤ 1, as |t|→∞.
With regard to Lemma 2 application of Lemma 3 yields
Corollary 4
1 T ∞ |a(n)|2 1 1 lim |F (σ + it)|2 dt = for σ>max , 1 − . T →∞ 2σ T 0 n=1 n 2 dF Z X In order to prove Theorem 1 we need a limit theorem in the space of analytic functions. Therefore, denote by H(D) the space of analytic functions on D equipped with the topology of uniform convergence on compacta, and by B(S)theclassof
Borel sets of a topological space S.Letγ = {s ∈ C : |s| =1} and Ω = p γp, where γp = γ for each prime p. With product topology and pointwise multiplicationQ this infinite-dimensional torus Ω is a compact topological abelian group. Therefore the probability Haar measure mH on (Ω, B(Ω)) exists. This gives a probability space
(Ω, B(Ω), mH ). Let ω(p) stand for the projection of ω ∈ Ω to the coordinate space γp. Then {ω(p):p ∈ P} is a sequence of independent random variables defined on the probability space (Ω, B(Ω), mH). Define for ω ∈ Ω
m −1 αj(p)ω(p)) (4) F (s, ω)= 1 − s . p p ! Y jY=1 In [11] it was proved that the product converges for almost all ω ∈ Ω uniformly on compact subsets of D. Further, it was shown that L(s, ω)isanH(D)-valued random element on the probability space (Ω, B(Ω), mH ) (functions in S˜ form a subclass of Matsumoto zeta-functions considered in [11]). Let P denote the distribution of the random element L(s, ω), i.e.
P(A)=mH (ω ∈ Ω:F (s, ω) ∈ A)forA ∈B(H(D)).
Then, by the axioms of S˜ and Corollary 4, we obtain as a simple consequence of the limit theorem in [11]
7 Theorem 5 The probability measure PT , defined by 1 PT (A) = lim meas{τ ∈ [0,T]:F (s + iτ) ∈ A} for A ∈B(H(D)), T →∞ T converges weakly to P,asT →∞.
For M>0 define 1 1 DM = s ∈ C :max , 1 − <σ<1, |t| Since DM ⊂ D we obtain, by the induced topology, that F (s, ω)isfors ∈ DM also an H(DM )-valued random element on the probability space (Ω, B(Ω), mH ). If Q denotes the distribution of F (s, ω)on(H(DM ), B(H(DM ))), we deduce from Theorem 5 Corollary 6 The probability measure QT , defined by 1 QT (A) = lim meas{τ ∈ [0,T]:F (s + iτ) ∈ A} for A ∈B(H(DM )), T →∞ T converges weakly to Q,asT →∞. 3 A denseness result In view of the Euler product of F (s) and (4) we define for b(p) ∈ γ, s ∈ DM ,andeach prime p m αj(p)b(p) (5) gp = gp(s)=gp(s, b(p)) = − log 1 − s . j=1 p ! X The key to prove the universality result is the following Theorem 7 The set of all convergent series p gp(s) is dense in H(DM ). P For the proof we will need the following Lemma 8 Let {yp} be a sequence in H(DM ) which satisfies o 1 ) if µ is a complex measure on (C, B(C)) with compact support contained in DM such that yp dµ < ∞, p ZC X then r s dµ(s)=0 for any r ∈ N ∪{0}; ZC 8 o 2 ) the series p yp converges on H(DM ); o P 3 ) for every compact K ⊂ DM 2 sup |yp(s)| < ∞. p s∈K X Then the set of all convergent series b(p)yp with b(p) ∈ γ, p X is dense in H(DM ). This lemma is a particular case of Theorem 6.3.10 of [10]. Further we recall some statements on functions of exponential type. A function k(s) analytic in the closed angular region | arg s|≤ϕ0 where 0 <ϕ0 ≤ π, is said to be of exponential type if log |k(r exp(iϕ))| lim sup < ∞ for |ϕ|≤ϕ0, r→∞ r uniformly in ϕ. Later we shall use Lemma 9 Let µ be a complex Borel measure on (C, B(C)) with compact support con- tained in the half plane σ>σ0. Moreover, let k(s)= exp(sz) dµ(z) for s ∈ C, ZC and k(s) 6≡ 0. Then log |k(r)| lim sup >σ0. r→∞ r This is Lemma 6.4.10 of [10]. Lemma 10 (Bernstein) Let k(s) be an entire function of exponential type, and let {ξm : m ∈ N} be a sequence of complex numbers. Moreover, let λ, η and ω be real positive numbers such that o log |k(±iy)| 1 ) lim supy→∞ y ≤ λ, o 2 ) |ξm − ξn|≥ω|m − n|, o ξm 3 ) limm→∞ m = η, 4o) λη < π. 9 Then log |k(ξ )| log |k(r)| lim sup m = lim sup . m→∞ |ξm| r→∞ r This lemma is a version of Bernstein’s theorem; for the proof see [10], Theorem 6.4.12. Now we are in the position to give the Proof of Theorem 7. Defineg ˜p =˜gp(s)=gp(s, 1). First we prove that the set of all convergent series (6) b(p)˜gp(s)withb(p) ∈ γ p>NX is dense in H(DM ). Let {˜b(p):˜b(p) ∈ γ} be a sequence such that the series g˜p if p>N, (7) ˜b(p)ˆgp withg ˆp =ˆgp(s)= p 0ifp ≤ N, X converges in H(DM ). We show that such a sequence {˜b(p)} exists. By the Taylor expansion of the logarithm, a(p) g˜ (s)= + r (s)withr (s) p−2σ. p ps p p The series p rp(s) converges uniformly on compact subsets of DM . Moreover we see, as in the proofP that L(s, ω, f) is a random element, that the series ω(p)a(p) s p p X converges uniformly for almost all ω ∈ Ω on compact subsets of DM .Consequently, there exists a sequence {˜b(p):˜b(p) ∈ γ} such that the series ˜b(p)a(p) s p p X converges in H(DM ). This proves, together with the convergence of p rp(s), that (7) converges in H(DM ). P Now let fp = fp(s)=˜b(p)ˆgp(s). To prove the denseness of the set of all convergent series (6) it is sufficient to show that the set of all convergent series (8) b(p)fp with b(p) ∈ γ p X is dense in H(DM ). For this we will verify the hypotheses of Lemma 8. Obviously, hypotheses 2o and 3o are fulfilled. To prove hypothesis 1o let µ be a complex Borel measure with compact support contained in DM such that (9) fp(s)dµ < ∞. p ZC X 10 Define ˜b(p)a(p) h (s)= , p ps then |fp(s) − hp(s)| < ∞ p X uniformly on compact subsets of DM . By the Ramanujan-Petersson conjecture, we π may define angles θp ∈ [0, 2 ]by m (10) |a(p)| = α (p) = m cos φ for prime p. j p j=1 X In view of (9), (11) cos φp|ρ(log p)| < ∞, p X where ρ(z)= exp(−sz)dµ(s). ZC Now we apply Lemma 10 with k(s)=ρ(s). By the definition of ρ(s)wehave |ρ(±iy)|≤exp(My) | dµ(s)| ZC for y>0. Therefore, log |ρ(±iy)| lim sup ≤ M, y→∞ y o π and the condition 1 of Lemma 10 is valid with α = M.Fixanumberη with 0 <η< M , and define 1 1 A = n ∈ N : ∃ r ∈ n − η, n + η with |ρ(s)|≤exp(−r) . 4 4 √ κ Further, fix a number φ with 0 <φ (12) |ρ(log p)| < ∞. pX∈Pφ Now 1 |ρ(log p)|≥ 0 |ρ(log p)|≥ 0 , p p p pX∈Pφ nX6∈A X nX6∈A X 11 0 where n denotes the sum over all primes p ∈ Pφ satisfying the inequalities P 1 1 n − η 1 1 where α =exp n − 4 η ,β =exp n + 4 η .Letπφ(x)=]{p ≤ x : p ∈ Pφ}, then we obtain for α ≤ u ≤ β 2 2 cos φp ≤ 1+φ 1 α By partial summation, the axiom on the mean square of the coefficients of F ∈ S˜ yields 1 κ (14) cos2 φ = |a(p)|2 ∼ π(x), p m2 m2 pX≤x pX≤x as x →∞. Hence, κ 2 m2 − φ πφ(u) − πφ(α) ≥ + o(1) (π(u) − π(α)) 1 − φ2 ! for u ≥ α(1 + δ), as n →∞. Thus, we obtain by partial sumation β κ 2 β 1 dπφ(u) 2 − φ dπ(u) = ≥ m + o(1) p α u 1 − φ2 ! α u pX∈Pφ Z Z α κ 2 1 2 − φ 1 log(1 + δ) 1 1 ≥ m − + o(1) + O , p 1 − φ2 2 η ! n n2 pX∈Pφ α ξk log |ρ(ξk)| lim = η and lim sup ≤−1. k→∞ k k→∞ ξk Applying Lemma 10, we obtain log |ρ(r)| (18) lim sup ≤−1. r→∞ r However, by Lemma 9, if ρ(z) 6≡ 0, then log |ρ(r)| lim sup > 0, r→∞ r contradicting (18). Therefore ρ(z) ≡ 0, and by differentiation sr dµ(s)=0 for r =0, 1, 2,.... ZC Thus also hypothesis 1o of Lemma 8 is satisfied. Therefore, we obtain by Lemma 8 the denseness of all convergent series (8), and hence of all convergent series (6). Let y(s) ∈ H(DM ), K be a compact subset of DM and >0. Fix N such that ∞ 1 (19) sup < . νσ s∈K p>N ν=2 νp 4m X X By the denseness of all convergent series (6) in H(DM ) we see that there exists a sequence {˜b(p):˜b(p) ∈ γ} such that (20) sup y(s) − g˜ (s) − ˜b(p)˜g (s) < . p p s∈K p≤N p>N 2 X X Setting 1ifp ≤ N, b(p)= ˜b(p)ifp>N, then (19) and (20) imply sup y(s) − gp(s) =sup y(s) − g˜p(s) − gp(s) s∈K s∈K p p≤N p>N X X X ≤ sup y(s) − g˜ (s) − ˜b(p)˜g (s) +sup ˜b( p)˜g (s) − g (s) p p p p s∈K p≤N p>N s∈K p>N p>N X X X X ∞ 1 ≤ +2 m sup <. νσ 2 s∈K p>N ν=2 νp X X Since y(s),K and are arbitrary, the theorem is proved. 13 4 The support of the measure QT Now we identify the measure QT , defined in Corollary 6. Lemma 11 The support of the measure QT is the set SM = {ϕ ∈ H(DM ):ϕ(s) 6=0 for s ∈ DM , or ϕ(s) ≡ 0}. In order to prove this lemma we make use of the following two lemmas. Lemma 12 (Hurwitz) Let {fn(s)} be a sequence of functions analytic on DM such that fn(s) → f(s) uniformly on DM ,asn →∞. Suppose that f(s) 6≡ 0, then an interior point s0 of DM isazerooff(s) if, and only if, there exists a sequence {sn} in DM such that sn → s0,asn →∞,andfn(sn)=0for all n large enough. A proof of Hurwitz’ theorem can be found in [31], section 3.4.5. Lemma 13 Let {Xn} be a sequence of independent H(G)-valued random elements, ∞ where G is a region in C, and suppose that the series n=1 Xn converges almost everywhere. Then the support of the sum of this series isP the closure of the set of all ϕ ∈ H(G) which may be written as a convergent series ∞ ϕ = ϕn with ϕn ∈ SXn , n=1 X where SXn is the support of the random element Xn. This is Theorem 1.7.10 of [10]. Now we can give the Proof of Lemma 11. The sequence {ω(p)} is a sequence of independent random variables on the probability space (Ω, B(Ω), mH). Define xp(s)=gp(s, ω(p)), then {xp(s)} is a sequence of independent H(DM )-valued random elements. The support of each ω(p) is the unit circle γ, and therefore the support of the random elements xp(s) is the set {ϕ ∈ H(DM ):ϕ(s)=yp(s, b)withb ∈ γ}, wherey ˜p(s, b)=gp(s, b). Consequently, by Lemma 13, the support of the H(DM )- valued random element log L(s, ω, f)= xp(s) p X 14 is the closure of the set of all convergent series p fp(s) in the notation of section 3. By Theorem 7 the set of these series is dense in PH(DM ). The map h : H(DM ) → H(DM ),f7→ exp(f) is a continuous function sending log L(s, ω, f)toL(s, ω, f)andH(DM )toSM \{0}. Therefore, the support SL of L(s, ω, f)containsSM \{0}. On the other hand, the support of L(s, ω, f) is closed. By Lemma 12 it follows that SM \{0} = SM .Thus, SM ⊂ SL. Since the Ramanujan-Petersson hypothesis is satisfied, the functions m αj(p)ω(p) exp(gp(s, ω(p))) = 1 − s j=1 p ! Y are non-zero for s ∈ DM ,ω ∈ Ω. Hence, L(s, ω, f)isanalmostsurelyconvergent product of non-vanishing factors. If we apply Lemma 12 again, we conclude that L(s, ω, f) ∈ SM almost surely. Therefore SL ⊂ SM . The lemma is proved. Now we are in the position to give the Proof of Theorem 1. Since K is a compact subset of D, there exists a number M such that K ⊂ DM . First we suppose that g(s) has a non-vanishing analytic continuation to H(DM ). Denote by Φ the set of functions ϕ ∈ H(DM ) such that sup |ϕ(s) − g(s)| <. s∈K By Lemma 11 the function g(s) is contained in the support SL of the random element F (s, ω). Since by Corollary 6 the measure QT converges weakly to Q,asT →∞,and the set Φ is open, it follows from the properties of weak convergence and support that (21) lim inf νT sup |F (s + iτ) − g(s)| < ≥ Q(Φ) > 0. T →∞ s∈K ! Now let g(s) be as in the statement of the theorem. Here we have to apply a well-known approximation result for polynomials (a proof can be found in [33]): Lemma 14 (Mergelyan) Let K be a compact subset of C with connected comple- ment. Then any continuous function g(s) on K which is analytic in the interior of K is approximable uniformly on K by polynomials in s. Thus, there exists a sequence {pn(s)} of polynomials such that pn(s) → g(s)asn →∞ uniformly on K.Sinceg(s) is non-vanishing on K,wehavepm(s) 6=0onK for sufficiently large m,and (22) sup |g(s) − pm(s)| < . s∈K 4 15 Since the polynomial pm(s) has only finitely many zeros, there exists a region G1 whose complement is connected such that K ⊂ G1 and pm(s) 6=0onG1. Hence there exists a continuous branch log pm(s)onG1,andlogpm(s) is analytic in the interior of G1. Thus, by Lemma 14, there exists a sequence {qn(s)} of polynomials such that qn(s) → log pn(s)asn →∞uniformly on K. Hence, for sufficiently large k sup |pm(s) − exp(qk(s))| < . s∈K 4 Thus and from (22) we obtain (23) sup |g(s) − exp(qk(s))| < . s∈K 2 From (21) we deduce lim inf νT sup |F (s + iτ) − exp(qk(s))| < > 0. T →∞ s∈K 2! This proves in connection with (23) the theorem. 5 Langlands program and power L-functions In this final chapter we shall give a motivation for introducing the subclass S˜ and give a further application of our universality theorem. The Langlands program tries to unify number theory and representation theory. These two disciplines are linked by L-functions associated to automorphic representa- tions and the relations between the analytic properties and the underlying algebraic structures; for an introduction to the Langlands program see [6] and [18]. For the sake of simplicity we now deal only with Q (and not with an arbitrary number field K). Denote by A the adele ring of Q. Further, let π be an automorphic cuspidal representation of GLm(Q), i.e. an irreducible unitary representation of GLm(A)which appears in its regular representation on GLm(Q)\GLm(A). Then π can be factored into a direct product π = ⊗pπp with each πp being an irreducible unitary representation of GLm(Qp)ifp<∞,whereQp is the field of p-adic numbers, and of GLm(R)ifp = ∞. For all but a finite number of places p the representation πp is unramified. We define the L-function associated to π by m −1 αj(p) (24) L(s, π)= 1 − s , p j=1 p ! Y Y and the completed L-function by m s − αj (∞) Λ(s, π)=L(s, π∞)L(s, π)withL(s, π∞)= Γ ; j=1 2 ! Y 16 here the numbers αj (p), 1 ≤ j ≤ m are determined from the local representations πp,p≤∞. Jacquet [7] showed that any Λ(s, π) satisfies after a suitable normalization the functional equation 1 s− 2 Λ(s, π)=πNπ Λ(1 − s, π˜), whereπ ˜ is the congradient representation of π, Nπ ∈ N is the conductor of π and π is of modulus 1, and is the sign of the functional equation. For m = 1 one obtains simply the Riemann zeta-function and the Dirichlet L- functions whereas for m =2onegetsL-functions associated to modular forms. It is expected that all zeta-functions arising in number theory are but special realizations of L-functions to automorphic representations constructed above. On the other side it is expected that all functions in the Selberg class are automorphic L-functions. M.R. Murty [19] proved that, assuming Selberg’s conjecture, i) if π is any irreducible cus- pidal automorphic representation of GLm(A) which satisfies the Ramanujan-Petersson conjecture, then L(s, π) is primitive (in the sense of the Selberg class), and ii) if K is a Galois extension of Q with solvable group G,andχ is an irreducible character of G of degree m, then there exists an irreducible cuspidal automorphic representation π of GLm(A) such that L(s, χ)=L(s, π). This motivates the special form of the Euler product in the definition of the subclass S˜; the axiom on the mean square was already discussed in the context of the Selberg conjectures. It only remains to consider the Ramanujan-Petersson conjecture. The Ramanujan τ-function is defined by the power series ∞ ∞ ∆(z):= τ(n)qn = q (1 − qn)24 with q =exp(2πiz), Re z>0; n=1 n=1 X Y ∆(z) is a normalized cuspform of weight 12 to the full modular group SL2(Z). Ramanu- 11 jan [24] conjectured that τ(n) is multiplicative and satisfies the estimate |τ(p)|≤2p 2 . This was proved by Mordell and Deligne [5], respectively. Petersson [22] extended this to the coefficients of modular forms of level N. It is expected that this holds for all L- functions of arithmetical nature; after a suitable normalization of the coefficients (such 1 that the L-function satisfies a functional equation with point symmetry at s = 2 ): Conjecture [Ramanujan-Petersson] With the notation from above, if πp is unram- ified for p<∞, then |αj (p)| =1 for 1≤ j ≤ m, and if π∞ is unramified, then Re αj(∞)=0for 1 ≤ j ≤ m. With view to all these widely believed conjectures it might be possible that the subclass S˜ coincides with the Selberg class. 17 We conclude with an application of our universality theorem to some typical L- functions in the Langlands setting. Symmetric power L-functions became important by Serre’s reformulation of the Sato-Tate conjecture [28]. However, before we can give a definition of symmetric power L-functions we have to recall some facts from the theory of modular forms. Let f(z) be a normalized cusp form of weight k and level N.In particular, f(z) has a Fourier expansion ∞ k−1 f(z)= a(n)n 2 exp(2πinz). n=1 X By the theory of Hecke operators, the coefficients turn out to be multiplicative, and we may attach to f its L-function ∞ a(n) a(p) −1 a(p) 1 −1 L(s, f)= s = 1 − s 1 − s + 2s ; n=1 n p ! p p ! X pY|N pY6|N both series and product converge absolutely for σ>1. Hecke proved that L(s, f)has an analytic continuation to an entire function and satisfies a functional equation of Riemann type. With regard to Deligne’s celebrated proof of the Ramanujan-Petersson conjecture [5] we may define an angle θp ∈ [0,π] by setting a(p)=2cosθp for each prime p; this should be compared with the angles defined by (10). Now let k be an even positive integer. For any non-negative integer m the symmetric m-th power L-function attached to f is given by m −1 exp(iθp(m − 2j)) Lm(s, f):= 1 − s j=0 p ! pY6|N Y for σ>1. It can be shown that ζ(2s) L (s, f)=ζ(s),L(s, f)=L(s, f)andL (s, f)= L(s, f ⊗ f), 0 1 2 ζ(s) where L(s, f ⊗f) is the Rankin-Selberg convolution L-function. Shimura [30] obtained the analytic continuation and functional equation in case of m =2;form>2this is an open problem. Serre [28] conjectured that if p ranges over the set of prime numbers, then the angles θp are uniformly distributed with respect to the Sato-Tate 2 2 measure π sin θ dθ (in analogy to a similar conjecture on elliptic curves due to Sato and Tate). Furthermore, Serre proved that the non-vanishing of Lm(s, f) on the abscissa of convergence σ =1forallm ∈ N would imply the Sato-Tate conjecture for newforms, namely β 1 2 2 lim ]{p ≤ x : α<θp <β} = sin θ dθ. x→∞ π(x) π Zα 18 However, in the case of L-functions associated to modular forms it would be sufficient to have an analytic continuation to σ ≥ 1 for proving the Sato-Tate conjecture; see [21]. If this is known to hold for all Lr(s, f),r ≤ 2(m + 1), then one can deduce asymptotic formulae for the 2mth-power moments of cos θp. Taking deep results of Hecke, Ogg, Shahidi and Shimura into account, M.R. Murty and V.K. Murty [20] proved that if 1 Lr(s, f) has an analytic continuation up to σ ≥ 2 for all r ≤ 2(m +1), then 1 2r 1 2r lim (2 cos θp) = for r ≤ m +1, x→∞ r+1 r π(x) p x X≤ 1 2r+1 lim (2 cos θp) =0 forr ≤ m. x→∞ π(x) p x X≤ This implies Lm(s, f) ∈ S˜ for m =0, 1 unconditionally, and for m ≥ 2 conditionally (depending on the analytic continuation). Thus, Theorem 1 yields the universality of Lm(s, f)form =0, 1 unconditionally, and for m ≥ 2ifallLm(s, f)haveanalyticcon- tinuation throughout C. By the powerful methods of the Langlands program Shahidi [29] obtained analytic continuation to σ ≥ 1form ≤ 4 (in particular cases more is known, see [9]). Acknowledgements. The author is very grateful to professors R. Garunkˇstis, A. Laurinˇcikas, K. Matsumoto for several discussions on universality and to Prof. M.R. Murty for submitting material on Artin L-functions and the Selberg class. References [1] B. Bagchi, The statistical behaviour and universality properties of the Riemann zeta-function and other allied Dirichlet series, Ph.D.Thesis, Calcutta, Indian Sta- tistical Institute, 1981 [2] E. Bombieri, D.A. Hejhal, On the distribution of zeros of linear combinations of Euler products, Duke Math. J. 80 (1995), 821-862 [3] E. Bombieri, A. Perelli, Distinct zeros of L-functions, Acta Arith. 83 (1998), 271-281 [4] J.B. Conrey, A. Ghosh, On the Selberg class of Dirichlet series: small degrees, Duke Math. J. 72 (1993), 673-693 [5] P. Deligne, La Conjecture de Weil I, II, Publ. I.H.E.S. 43 (1974), 273-307; 52 (1981), 313-428 [6] S. Gelbart, An elementary introduction to the Langlands program, Bull. Amer. Math. Soc. 10 (1984), 177-219 19 [7] H. Jacquet, Principal L-functions of the linear group, Proc. Symp. Pure Math. Vol 33, AMS part II (1979), 63-86 [8] J. Kaczorowski, A. Perelli, The Selberg class: a survey, Number theory in progress. Proceedings of the international conference in honor of the 60th birth- day of Andrej Schinzel, Zakopane 1997. Vol. 2: Elementary and analytic number theory. De Gruyter 1999, 953-992 [9] H.H. Kim, F. Shahidi, Symmetric cube L-functions for GL2 are entire, Ann. Math. 150 (1999), 645-662 [10] A. Laurincikasˇ , Limit theorems for the Riemann zeta-function,KluwerAcad- emic Publishers, Dordrecht 1996 [11] A. Laurincikasˇ , On the limit distribution of the Matsumoto zeta-function I,II, Acta Arith. 79 (1997), 31-39; Liet. Matem. Rink. 36 (1996), 464-485 (in Russian) [12] A. Laurincikasˇ , The universality of the Lerch zeta-functions, Liet. Matem. Rink. 37 (1997), 367-375 (in Russian); Lith. Math. J. 37 (1997), 275-280 [13] A. Laurincikas,ˇ K. Matsumoto, The universality of zeta-functions attached to certain cusp forms, Acta Arith. 98 (2001), 345-359 [14] A. Laurincikas,ˇ K. Matsumoto, J. Steuding, The universality of L- functions associated to newforms, (to appear) [15] K. Matsumoto, The mean values and universality of Rankin-Selberg L-functions, in: Number Theory, Proceedings of the Turku Symposium on Number theory in memory of Kustaa Inkeri (1999), M. Jutila, T. Mets¨ankyl¨a (Eds.), Walter de Gruyter, Berlin, New York 2001, 201-221 [16] K. Matsumoto, Probabilistic value-distribution theory of zeta-functions, Sugaku 53 (2001), 279-296 (in Japanese) [17] H. Mishou, The universality theorem for L-functions associated with ideal class characters, Acta Arith. 98 (2001), 395-410 [18] M.R. Murty, A motivated introduction to the Langlands program, Advances in Number theory, F. Gouvea and N. Yui, eds., Clarendon Press Oxford 1993, 37-66 [19] M.R. Murty, Selberg’s conjectures and Artin L-functions, Bull. Amer. Math. Soc. 31 (1994), 1-14 [20] M.R. Murty, V.K. Murty, Non-vanishing of L-functions and applications, Birkh¨auser 1997 20 [21] V.K. Murty, On the Sato-Tate conjecture, Number theory related to Fermat’s Last Theorem, (ed. N. Koblitz), Birkh¨auser 1982, 195-205 [22] H. Petersson, Konstruktion der s¨amtlichen L¨osungen eienr Riemannschen Funktionalgleichung durch Dirichletreihen mit Eulerscher Produktentwicklung II, Math. Ann. 117 (1940/41), 39-64 [23] H.S.A. Potter, The mean values of certain Dirichlet series I, Proc. London Math. Soc. 46 (1940), 467-468 [24] S. Ramanujan, On certain arithmetical functions, Trans. Camb. Phil. Soc. 22 (1916), 159-184 [25] A. Reich, Universelle Wertverteilung von Eulerprodukten, Nach. Akad. Wiss. G¨ottingen, Math.-Phys. Kl. (1977), 1-17 [26] A. Reich, Wertverteilung von Zetafunktionen, Arch. Math. 34 (1980), 440-451 [27] A. Selberg, Old and new conjectures and results about a class of Dirichlet series, Proceedings of the Amalfi Conference on Analytic Number Theory, Amalfi 1989, Salerno (1992), 367-385 [28] J.-P. Serre, Abelian l-adic representations and elliptic curves, Research Notes in Mathematics, AK Peters 1968 [29] F. Shahidi, Symmetric power L-functions for GL(2), in: Elliptic curves and related objects, ed. H. Kisilevsky and M.R. Murty, CRM Proceedings and Lecture Notes, Vol. 4 (1994), 159-182 [30] G. Shimura, On modular forms of half integral weight, Ann. Math. 97 (1973), 440-481 [31] E.C. Titchmarsh, The theory of functions, Oxford 1932 [32] S.M. Voronin, Theorem on the ’universality’ of the Riemann zeta-function,Izv. Akad. Nauk SSSR, Ser. Matem., 39 (1975 (in Russian); Math. USSR Izv. 9 (1975), 443-445 [33] J.L. Walsh, Interpolation and approximation by rational functions in the complex domain, Amer. Math. Soc. Coll. Publ. vol. 20, 1960 21