The universality of symmetric power L-functions and their Rankin-Selberg L-functions

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1 The universality of symmetric power L-functions and their Rankin-Selberg L-functions Hongze Li, Jie Wu To cite this version: Hongze Li, Jie Wu. The universality of symmetric power L-functions and their Rankin-Selberg L- functions. Journal of the Mathematical Society of Japan, Maruzen Company Ltd, 2007, 59 2), pp <hal > HAL Id: hal Submitted on 7 Sep 2006 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 J. Math. Soc. Japan, to appear The universality of symmetric power L-functions and their Rankin-Selberg L-functions H.-Z. Li ) Shanghai) & J. Wu Nancy) Abstract. We establish the universality theorem for the first four symmetric power L-functions of automorphic forms and their associated Rankin-Selberg L-functions. This generalizes some results of Laurin cikas & Matsumoto and Matsumoto respectively.. Introduction ccsd , version - 7 Sep 2006 The automorphic L-function is a powerful tool to study arithmetic, algebraic and geometric objects. Many results will follow from the known or conjectured analytic properties of automorphic L-functions. It is therefore important to explore an L-function in various analytic aspects. Here, we are concerned with the universality property. Roughly speaking, a function f has the universality property if every non-vanishing analytic function can be approximated uniformly on compact subsets in the half critical strip D 2 ) by translations of this function f, where D 2 ) denotes.) Dσ 0 ) := s C : σ 0 < Re s < } for any σ 0 <. According to Linnik-Ibragimov, it was conjectured that the universality property is intrinsic to all Dirichlet series which can be analytically continued to left of their abscissa of absolute convergence. The universality of the Riemann zeta-function ζs) was first discovered by Voronin [23]. More precisely he proved the following: Let K be a closed disc of radius r < 4 centered at s = 3 4, and ϕs) a non-vanishing analytic function in the interior of K and continuous on K. Then for any ε > 0, there is a real number t such that.2) sup ζs + it) ϕs) < ε. In 98, Bagchi [] developed a new method to deduce the universality property of ζs) and obtained a result stronger than.2), as follows. Let K be a compact subset of D 2 ) with connected complement and ϕs) a non-vanishing analytic function in the interior of K and continuous on K. Then for any ε > 0, we have.3) lim inf T T meas } t [0, T] : sup ζs + it) ϕs) < ε > 0, ) Project supported by NSFC , the Scientific Research Foundation for the Returned Overseas Chinese Scholars and Shuguang Plan of Shanghai 2000 Mathematics Subject Classification: F66 Key words and phrases: Automorphic L-function, Universality

3 2 H.-Z. Li & J. Wu where meas ) is the Lebesgue measure. This result was generalized by different authors to many other L-functions such as Dirichlet L-functions, Dedekind L-functions, Hurwitz L-functions, Lerch L-functions, etc. A detailed historical account can be found in [5]. In this paper we are interested in the universality of automorphic L-functions. For a positive even integer k such that k = 2 or k 6 2), we denote by H k the set of all Hecke primitive eigencuspforms of weight k for the full modular group SL2, Z). The Fourier series expansion of f H k at the cusp is fz) = λ f n)n k )/2 e 2πinz Im z > 0), n= where λ f n) is the nth normalized) Fourier coefficient of f, verifying.4) λ f m)λ f n) = ) mn λ f d 2 d m,n) for any integers m and n. In particular it is a multiplicative function of n. According to Deligne, for any prime number p there is α f p) such that.5) λ f p ν ) = α f p) ν + α f p) ν α f p) ν ν ) and.6) α f p) =. In particular λ f ) = and λ f n) is real. For m N, the mth symmetric power L-function attached to f H k and its Rankin-Selberg L-function are defined as αf p) m 2j p s).7) Ls, sym m f) := p and 0 j m αf p) 2m i j) p s).8) Ls, sym m f sym m f) := p 0 i, j m for σ >, respectively. The products over primes in.7) and.8) admit Dirichlet series representation.9) Ls, F) = λ F n)n s n= for σ >, where F = sym m f or sym m f sym m f, and λ F n) is a multiplicative function. Following from.6), we have for n, dm+ n) if F = sym m f,.0) λ F n) d m+) 2n) if F = sym m f sym m f, 2) For k 2, 4, 6, 8, 0, 4}, there is no cusp forms of weight k for the full modular group SL2, Z) see [2])

4 The universality of symmetric power L-functions and their Rankin-Selberg L-functions 3 where d z n) is the nth coefficient of the Dirichlet series ζs) z. The case F = sym f in.0) is commonly known as Deligne s inequality. According to [3, Section 3.2.] and [0, Proposition 2.], the gamma factors of Ls, sym m f) and Ls, sym m f sym m f) are, respectively, n Γ C s + ν + 2 )k )) if m = 2n +.) L s, sym m ν=0 f) := n ) Γ R s + δ 2 n ) Γ C s + νk ) if m = 2n ν= and m Γ C s) n+ ) m ν+ Γ C s + νk ) if m = 2n +.2) L s, sym m f sym m ν= f) = m Γ R s)γ C s) n ) m ν+ Γ C s + νk ) if m = 2n, ν= where Γ R s) := π s/2 Γs/2), Γ C s) := 22π) s Γs) and if 2 n, δ 2 n = 0 otherwise. For F = sym m f or F = sym m f sym m f where f H k and m =, 2, 3, 4, it is known that the function Λs, F) := L s, F)Ls, F) is entire on C and satisfies the functional equation.3) Λs, F) = ε F Λ s, F) with ε F = ± see [5, 7, 8, 9] for F = sym m f and [0, 20] for F = sym m f sym m f). For the universality property of Ls, F), we have the following result. Theorem. Let m 4, 2 k such that k = 2 or k 6, f H k and F = symm f or F = sym m f sym m f. Define.4) σ F := m + ) if F = sym m f, m + ) 2 if F = sym m f sym m f. Let K be a compact subset of Dσ F ) with connected complement and ϕs) a non-vanishing analytic function in the interior of K and continuous on K. Then for any ε > 0, we have lim inf T T meas } t [0, T] : sup Ls + it, F) ϕs) < ε > 0. Remark. i) The particular case F = sym f = f of Theorem was first investigated by Ka cėnas & Laurin cikas [6] and established completely by Laurin cikas & Matsumoto [3]. Another particular case F = sym f sym f = f f was considered by Matsumoto [6] recently. ii) Theorem is established only for m 4 due to the lack of knowledge about the high symmetric powers.

5 4 H.-Z. Li & J. Wu iii) The reason why Theorem holds only for Dσ F ) instead of D 2 ) is that the estimate T Ls, F) 2 dτ f T T ) is only achieved for σ > σ F see 5.3) below), where σ F is defined as in Theorem. It seems interesting to improve this estimate further so that Theorem can hold for D 2 ). iv) It is possible to generalize without too much difficulty) Theorem to the case of the congruence subgroup Γ 0 N) with square-free N, as what Laurin cikas, Matsumoto & Steuding [5] did for Ls, f). Like [3] and [6], we shall use Bagchi s method to prove Theorem. Interested readers are referred to [] for an excellent paradigm on Bagchi s method.) One of their main tools is Rankin s asymptotic formula λ f p) 2 x log x p x for x see [8], theorem 2). However, such a prime number theorem for the symmetric mth power L-function with m 2 is not available. In Section 2, we shall establish this result based on [20] and [0], which is clearly of independent interest and may have many other applications. As in [4], we can deduce the following as simple consequences of Theorem. Corollary 2. Let m, k, f, F and σ F be as in Theorem. For σ F < σ < and any positive integer J, define a mapping ψ : R C J by Then ψr) is dense in C J. ψτ) := Lσ + iτ, F), L σ + iτ, F),...,L J ) σ + iτ, F) ). Corollary 3. Let m, k, f, F and σ F be as in Theorem, and J be a non negative integer. If the continuous functions g j : C J C 0 j J) satisfy J s j g j Ls, F), L s, F),..., L J ) s, F) ) 0 j=0 for all s C, then g j 0 0 j J). Acknowledgement. We began working on this paper in April 2004 during the visit of the second author to Shanghai Jiaotong University, and finished in February 2005 when the first author visited l Université Henri Poincaré Nancy ). We are indebted to both institutions for invitations and support. The second author would like to thank the GRD de Théorie des Nombres au CNRS for support. We would express our sincere gratitude to K. Matsumoto for his kind help in our study of his joint paper with Laurin cikas [3]. Finally the authors wish to thank the referee for pointing out a mistake in the earlier version. 2. The prime number theorem for symmetric power L-functions Let m N, 2 k such that k = 2 or k 6 and f H k. From.6), the product.8) is absolutely convergent for σ >. Thus we can define Λ symm f sym m fn) by the relation 2.) L L s, symm f sym m f) = n= Λ symm f sym m fn) n s for σ >. The aim of this section is to prove the following result.

6 The universality of symmetric power L-functions and their Rankin-Selberg L-functions 5 Proposition 2.. Let m 4, 2 k such that k = 2 or k 6 and f H k. Then for x, we have 2.2) 2.3) 2.4) Λ symm f sym m fn) x, n x λf p m ) 2 log p x, p x λf p m ) 2 x log x. p x This proposition will be referred as the prime number theorem for the coefficients of symmetric power L-functions associated with newforms. It plays a key role in our proof of Theorems & 2, and is of independent interest. The case m = was considered by Rankin [8]. We shall prove this proposition with the non-vanishing property on σ = in standard way. To this end, we firstly prove two preliminary lemmas. Let m N, 2 k such that k = 2 or k 6 and f H k. We define αf p) 2m l) p s) α f p) 2m l) p s)} l+) 2.5) Ψ f,m s) := p 0 l<m for σ > and 2.6) Λ f,m n) = 2 m j= m + j)cos[2jθ fp)ν] log p if n = p ν, 0 otherwise, where α f p) is determined by.5).6) and θ f p) [0, π] is chosen such that α f p) = e iθ fp). Lemma 2.. Let m N, 2 k such that k = 2 or k 6 and f H k. Then for σ >, we have 2.7) 2.8) Ψ f,m Ψ f,m s) = log Ψ f,m s) = n= n=2 Λ f,m n) n s, Λ f,m n) n s log n. Proof. By the Deligne inequality, the Euler product Ψ f,m s) converges absolutely for σ >. Taking logarithmic derivative on both sides of 2.5), we have, for σ >, Ψ f,m Ψ f,m s) = p = p αf p) 2m l) p s log p l + ) α f p) 2m l) p + α fp) 2m l) p s ) log p s α f p) 2m l) p s 0 l<m ν 0 l<m l + ) [α fp) 2m l)ν + α f p) 2m l)ν ] log p p sν, which is equivalent to 2.7). Integrating 2.7) on the half-line s + t : t 0}, we obtain 2.8).

7 6 H.-Z. Li & J. Wu Lemma 2.2. Let m 4, 2 k such that k = 2 or k 6 and f H k. Then for σ and s, we have Ls, sym m f sym m f) 0. Proof. Noticing that 0 i, j m i+j=l l + if 0 l m = 2m l + if m < l 2m and using.8), we can write, for σ >, 2.9) Ls, sym m f sym m f) = ζs) m+ Ψ f,m s). As usual we denote by Λn) von Mangoldt s function. Following from 2.6), 2.7), 2.9) and the classical relations ζ ζ s) = n= Λn) n s, log ζs) = n=2 Λn) n s log n σ > ), we infer that 2.0) Λ symm f sym m fn) = m + )Λn) + Λ f,m n) n ) and 2.) log Ls, sym m f sym m f) = n=2 Λ symm f sym m fn) n s log n σ > ). get Next we calculate Λ symm f sym m fp ν ). Write ϑ ν = θ f p)ν for notational convenience, we Λ f,m p ν )log p) = 2 cosl2ϑ ν ) l m j m l+ = 2 j m l m j+ cosl2ϑ ν ), by 2.6). On the other hand, we have l m j+ cosl2ϑ ν ) = Re l m j+ e il2ϑν ) e im j+2)2ϑ ν ) e i2ϑν = Re e i2ϑν ) e = Re e im j+2)ϑν eim j+)ϑν im j+)ϑν e iϑν e iϑν = cos[m j + 2)ϑ ν] sin[m j + )ϑ ν ] sinϑ ν. Inserting it into the preceding formula and applying the identity 2 cosαsin β = sinα + β) sinα β),

8 The universality of symmetric power L-functions and their Rankin-Selberg L-functions 7 it follows that Λ f,m p ν )log p) = sin ϑ ν = sin ϑ ν j m l m sin[2m j + ) + ]ϑν } sinϑ ν ) sin[2l + )ϑ ν ] m. Similarly, we have l m sin[2l + )ϑ ν ] = Im l m e i2l+)ϑν ) ) e = Im e iϑν eim+)2ϑν i2ϑν e i2ϑν ) = Im e im+2)ϑν eimϑν e imϑν e iϑν e iϑν = sin[m + 2)ϑ ν] sinmϑ ν ) sin ϑ ν = sin2 [m + )ϑ ν ] sin 2 ϑ ν sinϑ ν. Combining this with the previous relation, we deduce that ) 2 sin[m + Λ f,m p ν )log p) )ϑν ] = m, sinϑ ν which implies, via 2.0), ) 2 sin[m + 2.2) Λ symm f sym m fp ν )ϑν ] ) = log p. sin ϑ ν In particular, we obtain with.5) that 2.3) ) 2 sin[m + )θf p)] Λ symm f sym m fp) = log p sin θ f p) 2 = e fp)) im 2j)θ log p 0 j m = λf p m ) 2 log p. Now we are ready to prove Lemma 2.2. Suppose Ls, sym m f sym m f) has a zero at +iτ 0 of order l, where τ 0 0. Consider the function gs) := Ls, sym m f sym m f) 3 Ls + iτ 0, sym m f sym m f) 4 Ls + i2τ 0, sym m f sym m f) 2. Since Ls, sym m f sym m f) is holomorphic except for a simple pole at s =, gs) is holomorphic for σ and the zero at s = is of order 4l 3. But from 2.), we have for σ >, log gs) = n 2 Λ symm f sym m fn) 3 + 4n iτ 0 n s + 2n i2τ0). log n

9 8 H.-Z. Li & J. Wu Together with 2.6), 2.0) and 2.2), we deduce, for σ >, log gσ) = n 2 = n 2 0. Λ symm f sym m fn) cosτ0 n σ log n) + 2 cos2τ 0 log n) ) log n Λ symm f sym m fn) + 2 cosτ0 n σ log n) ) 2 log n Thus gσ) for σ >, and g cannot have a zero of order 4l 3 ) at σ =. This contradiction completes our proof. Next we shall apply Theorem II.7. of [22] to prove Proposition 2.. Define Gs) := L L s +, symm f sym m f) s + s. Since Λs, sym m f sym m f) is holomorphic except for simple poles at s = 0,, the function Gs) is analytically continued to a meromorphic function on C. By Lemma 2.2, we have L + iτ, sym m f sym m f) 0. Thus Gs) is holomorphic in an open set containing the half-plane σ 0. In particular we have G2σ + iτ) Gσ + iτ) σ sup 0 θ, τ T G θ + iτ) for T > 0, 0 σ 2 and τ T. From this we deduce T T G2σ + iτ) Gσ + iτ) dτ = o) σ 0+) for each fixed T > 0. Now Theorem II.7. of [22] is applied with F = L /L, a = c = and w = 0 to yield the asymptotic formula 2.2). From 2.3), we can write where we have, via 2.6) and 2.0), n xλ symm f sym m fn) = p x R := p ν x, ν 2 p x /2 ν log x/log p λ f p m ) 2 log p + R, Λ symm f sym m fp ν ) m + ) 2 m x /2. p x /2 log x m + ) 2 log p Thus 2.2) implies 2.3). Finally 2.4) follows from 2.3) by integration by parts.

10 The universality of symmetric power L-functions and their Rankin-Selberg L-functions 9 3. Bagchi s method and proof of Theorem In this section, we present Bagchi s method in our case and first formulate it as three propositions. At the end of this section, we shall apply Propositions 3.2 and 3.3 to prove Theorem. The proof of these three propositions will be given in sections 4, 5 and 6, respectively. Let m 4, 2 k such that k = 2 or k 6, f H k, F = symm f or sym m f sym m f. Let σ F and Dσ F ) be defined as in.4) and.), respectively. Denote by H F the space of analytic functions on Dσ F ) equipped with the topology of uniform convergence on compact subsets of Dσ F ). Let γ := s C : s = } be the unit torus and Ω := p where γ p = γ for all prime numbers p. With the product topology and componentwise multiplication, Ω is a compact abelian topological group. Hence there is a unique probability Haar measure µ h on Ω, BΩ)) 3) and we have µ h = p µ h,p, where µ h,p is the Haar measure on γ p, Bγ p )) see [9], Theorem 5.4). For every ω = ω p } Ω, we extend it to a completely multiplicative function, by defining ω n := ωp ν. p ν n In view of.0), we can prove, similar to Lemma 5..6 and Theorem 5..7 of [], that there is a subset Ω Ω with µ h Ω) = such that for any ω Ω the series ω n λ F n)n s and the product n γ p, ω pλ ν F p ν )p νs p ν 0 are uniformly convergent on compact subsets of the half-plane σ > 2, and the equality 3.) Ls, F; ω) := ω n λ F n)n s = ω pλ ν F p ν )p νs n p holds. Clearly for σ > 2 and ω Ω, we have Ls, F; ω) = + ωp λ F p)p s + Op 2σ ) ). p ν 0 Therefore for any ω Ω, the series 3.2) L s, F; ω) := p log ω p λ F p)p s) 3.3) L p 0 s, F; ω) := p>p 0 ω p log λ F p)p s) are uniformly convergent on compact subsets of the half-plane σ > 2, where p 0 3 is an arbitrarily fixed constant. Moreover, we introduce two subsets of H F : 3.4) L F := L s, F; ω) : ω Ω } and L F,p 0 := L p 0 s, F; ω) : ω Ω }. The first auxiliary result of Bagchi s method is the denseness of L F, which is important in the proof of Proposition 3.3 below. 3) For any space X, we denote by BX) the class of all Borel subsets of X.

11 0 H.-Z. Li & J. Wu Proposition 3.. Let m 4, 2 k such that k = 2 or k 6, f H k and F = symm f or sym m f sym m f. i) For any fixed p 0 3, the set L F,p 0 is dense in H F. ii) The set L F is dense in H F. Define three probability measures on H F, BH F ) ) 3.5) P F,T A) := T meas t [0, T] : Ls + it, F) A }, 3.6) QFA) := µh P F A) := µ h ω Ω : Ls, F; ω) A} ), 3.7) ω Ω : log Ls, F; ω) A} ), for A BH F ). The next limit theorem is one of the keys of Bagchi s method. Proposition 3.2. Let m 4, 2 k such that k = 2 or k 6, f H k and F = symm f or sym m f sym m f. Then the probability measure P F,T converges weakly to P F as T. The third key step of Bagchi s method is to determine the support of the probability measure P F on H F, BH F )). By definition, a point s S is said to be in the support of a probability measure P on S, BS)) iff every open neighborhood of s has strictly positive measure. The set of all such points is called the support of P, denoted by SP). Clearly SP) is the smallest closed subset of S such that P SP) ) = see [2], Chapter ). The support of a S-valued random variable Y on the probability space X, BX), µ) is the support of the probability measure P Y on S, BS)) where P Y A) = µy A) A BS)), called the distribution of Y. Proposition 3.3. Let m 4, 2 k such that k = 2 or k 6, f H k and F = symm f or sym m f sym m f. With the previous notation, we have the following results: i) The support of the probability measure Q F on H F, BH F )) is the whole space H F. ii) The support of the probability measure P F on H F, BH F )) is S 0 := ϕs) H F : ϕs) 0 for any s Dσ F ) or ϕs) 0 }. Now we apply Propositions 3., 3.2 and 3.3 to prove Theorem. Let K be a compact subset of D σ F ) with connected complement. Let ϕs) be a nonvanishing continuous functions on K which is analytic in the interior of K. By Lemma of [3], for any ε > 0 we can find a polynomial ps) such that ps) 0 on K and 3.8) sup ϕs) ps) < 4 ε. Since ps) has only finitely many zeros, we can find a region G such that K G and ps) 0 on G. We choose log ps) to be analytic in the interior of G. Applying Lemma of [3] to log ps) again, we find another polynomial qs) such that 3.9) sup ps) e qs) < 4 ε. From 3.8) and 3.9), we deduce, for any T > 0, 3.0) t [0, T] : sup Ls+it, F) e qs) ε } < t [0, T] : sup Ls+it, F) ϕs) < ε }. 2

12 The universality of symmetric power L-functions and their Rankin-Selberg L-functions On the other hand, the set G := g H F : sup } gs) e qs) < 2 ε belongs to G BH F ) and is open in H F, thus we have 3.) P F,T G) = T meas t [0, T] : Ls + it, F) G } = t T meas [0, T] : sup Ls + it, F) e qs) < 2 }. ε By Proposition 3., the measure P F,T G) converges weakly to P F G) as T. With 3.0) and 3.), Theorem..8 of [] leads to lim inf T liminf T P F G). T meas T meas t [0, T] : sup } Ls + it, F) ϕs) < ε t [0, T] : sup } Ls + it, F) e qs) < 2 ε Obviously e qs) S 0 = SP F ) and G is a neighbourhood of e qs). Therefore P F G) > 0. This completes the proof of Theorem. 4. Proof of Proposition 3. In order to prove Proposition 3., we first apply our result in Section 2 to establish a preliminary lemma, which is a generalization of the key lemma in Laurinčikas & Matsumoto [3]. Lemma 4.. Let m 4, 2 k such that k = 2 or k 6 and f H k. For every δ [0, ), define P δ = P δ sym m f) := p : p is prime such that λ f p m ) δ }. Let η > 0 and c > + η be two fixed constants. For any a 2 and + η)a < b ca, we have p P δ a<p b p δ 2 m + ) 2 δ 2 + o c,δ,η) where o c,δ,η ) is a quantity tending towards 0 as a. } a+η)<p b p, Proof. Define π δ x) := # p x : p P δ }. In particular we have P 0 = P the set of all prime numbers) and π 0 x) = πx) := # [, x] P ). Clearly it is sufficient to prove that for any + η)a < u b, δ 2 } πu) 4.) π δ u) π δ a) m + ) 2 δ 2 + o ) c,δ,η) πa), since the desired inequality follows from 4.) via a simple integration by parts.

13 2 H.-Z. Li & J. Wu 4.2) For a u b, the Deligne inequality λ f p m ) m + allows us to write a<p u λf p m ) 2 m + ) 2 a<p u, p P δ + δ 2 According to 2.4) of Proposition 2., we have a<p u a<p u, p/ P δ [ m + ) 2 δ 2][ π δ u) π δ a) ] + δ 2[ πu) πa) ]. λf p m ) 2 = πu) + o)} πa) + o)} = πu) πa) + o πu) ). Since + η)a < u ca, a simple calculation shows, via the prime number theorem, that Combining these two estimates yields 4.3) πu) πa) = u + o)} a + o)} log u log a ηa log a + o c,η)} a<p u η 2c πu). λf p m ) 2 = [ πu) πa) ] + o c,η )}. Now the desired inequality 4.) follows from 4.2) and 4.3). Now we are ready to prove Proposition 3.. Proof of Proposition 3.. Fix a ω 0 = ω 0 p } Ω, then the series 4.4) L p 0 s, F; ω 0 ) := p>p 0 ω 0 p log λ F p)p s) converges in H F. To prove assertion i), we shall apply Lemma 4 of [3] to this series. In fact, it suffices to verify the condition a) there, since conditions b) and c) are plainly satisfied. Let µ be a complex measure on C, BC)) with compact support in Dσ F ) such that 4.5) p C log λ F p)p s) dµs) <. Since σ F > 2, we see easily 4.6) λ F p) ρlog p) < p with ρz) := e sz dµs). We shall prove that 4.6) leads to 4.7) ρz) 0, C

14 The universality of symmetric power L-functions and their Rankin-Selberg L-functions 3 which implies the validity of condition a) in Lemma 4 of [3] since for any non-negative integer r, C sr dµs) = 0 by differentiating 4.7) r-times with respect to z and taking z = 0. Noticing that L p 0 s, F; ω) = L p 0 s, F; ω/ ω 0 ) ω 0) and ω/ ω 0 = ω p / ω 0 p} Ω, Lemma 4 of [3] shows that L F,p 0 is dense in H F. It remains to prove 4.7). Firstly we write ρz) = e zs dµ s), C where the measure µ is defined by µ A) = µ A) for A BC) with A := a : a A}. Clearly µ supports in s C : < σ < 2 }. Thus ρz) verifies all conditions of Lemma 5 of [3]. If ρz) 0, then this lemma implies 4.8) lim sup r log ρr) r >. Next we shall apply Lemma 7 of [3] to deduce an opposite inequality. This follows a contradiction, and hence 4.7) holds true. Since the support of µ is compact and is contained in Dσ F ), we have ρ±iy) e My dµs) y > 0), C where M = M µ is a positive constant such that µ supports in σ F, ) [ M, M]. Thus ρz) satisfies condition a) of Lemma 7 of [3] with α = M. Fix a positive number β < π/m, which assures condition d) of Lemma 7 of [3]. A similar calculation to 2.3) allows us to obtain λf p m ) if F = sym m f, 4.9) λ F p) = λ f p m ) 2 if F = sym m f sym m f. Define L := l N : r l 4 )β, l + 4 )β] such that ρr) e r}. By using 4.6) and 4.9), we can deduce, for any fixed δ [0, ), > p Now we apply Lemma 4. with δ 2 l/ L δ 2 l/ L λ F p) ρlog p) δ 2 p P δ l 4 )β<log p l+ 4 )β p P δ l 4 )β<log p l+ 4 )β p P δ ρlog p) ρlog p) p. a := expl 4 )β}, b := expl + 4 )β}, c := exp 2 β}

15 4 H.-Z. Li & J. Wu and η > 0 such that + η < c. It follows that δ 2 δ 2 } ) > m + ) 2 δ 2 + o c,δ,η) δ 2 δ 2 } ) m + ) 2 δ 2 + o c,δ,η) l/ L +η)a<p b l/ L p log + η) 2 β ) )} l + O l 2, which implies 4.0) l/ L l <. If we write L = a, a 2,... } with a < a 2 <..., it is easy to see that a n 4.) lim n n =. In fact we have [x] = L [, x] + N L) [, x]. But 4.0) implies N L [, x] x + x + x x<l x, l/ L l> x, l/ L Thus L [, x] x, which is equivalent to 4.). By the definition of L, there exists a sequence r n } such that Then /l = ox). a n 4 )β < r n a n + 4 )β and ρr n) e rn. r n lim n n = β and limsup n This shows that condition c) of Lemma 7 of [3] is satisfied. For any integers m and n such that m > n, we have log ρr n ) r n. r m r n a m a n 2 )β 2 a m a n )β 2 m n)β. Thus the condition b) of Lemma 7 of [3] is also satisfied. Now we can apply Lemma 7 of [3] to write lim sup r log ρr) r = limsup n log ρr n ) r n. This contradicts to 4.8), and the proof of assertion i) completes. Next we shall use the result in assertion i) to prove ii). Let K be a compact subset of Dσ F ) and ϕ H F. For any ε > 0, we take p 0 3 such that 4.2) sup p>p 0 ν 2 λ F p) ν νp νσ < ε 2. Since ϕs) + p p 0 log λ F p)p s) H F,

16 The universality of symmetric power L-functions and their Rankin-Selberg L-functions 5 assertion i) shows that there is ω = ω p } Ω such that 4.3) sup ϕs) + log λ F p)p s) L p 0 s, F; ω) < ε 2. p p 0 Taking ω p := if p p0 ω p if p > p 0 and ω = ω p}, the inequalities 4.2) and 4.3) imply sup ϕs) L s, F; ω ) sup ϕs) + log λ F p)p s) L p 0 s, F; ω) p p 0 + sup log ωp λ F p)p s ) ω p log λ F p)p s ) p>p 0 < ε 2 + sup λ F p) ν νp νσ < ε. p>p 0 ν 2 This completes the proof. 5. Proof of Proposition 3.2 Obviously Proposition 3.2 is a particular case of Theorem 2 of [2]. Thus it suffices to verify all assumptions there, that is, to show that there is a positive constant c for which 5.) Ls, F) f τ c σ > σ F, τ ), and 5.2) T Ls, F) 2 dτ f T σ > σ F, T ). By using.),.2) and.3), a standard Phragmén-Lindelöf argument allows us to obtain the convex bound for Ls, F), i.e. 5.) with c = m + + δ 2 m )/4 if F = sym m f and c = m + ) 2 /4 if F = sym m f sym m f. A detailed proof can be found in [0]. In order to verify 5.2), we can apply theorem 4 of Perelli [7], where an estimate of this type was established for a general class of L-functions. In view of.) and.2), it is easy to see that Ls, F) lies in the class considered in Perelli [7] with evident choice of parameters. Therefore Theorem 4 of Perelli [7] gives 5.3) T Ls, F) 2 σ)/ σf )+ε dτ f,ε T uniformly for 2 σ < and T, which implies 5.2). This completes the proof. 6. Proof of Proposition 3.3 By the definition, ω p } is a sequence of independent random variables defined on the probability space Ω, BΩ), µ h ), and the support of each ω p is the unit circle γ. Hence )} log ωp ν λ Fp ν )p νs ν 0

17 6 H.-Z. Li & J. Wu is a sequence of independent H F -valued random elements, and the set } ϕ H F : ϕs) = log a ν λ F p ν )p ), νs a γ ν 0 is the support of H F -valued random element log ν 0 ων pλ F p ν )p νs). Consequently, by theorem.7.0 of [] see also [3], lemma 0), the support of the H F -valued random element log Ls, F; ω) = p ) log ωp ν λ F p ν )p νs is the closure of L F, i.e. the whole space H F by Proposition 3.ii). This proves the first assertion. Now we consider any element ϕ = ϕs) of S 0 0} and its neighbourhood G in S 0 0}. Since the map exp : H F S 0 0} is onto and continuous, we see that exp ϕ) H F exists, and exp G) is a neighbourhood of exp ϕ) in H F. According to i), H F is the support of log Ls, F; ω), so Q F exp G)) > 0, where Q F is the distribution of log Ls, F; ω), defined by 3.7). But Q F exp G)) = P F G), where P F is the distribution of Ls, F; ω) given by 3.6). Hence P F G) > 0. This implies that any ϕ S 0 0} is an element of the support of Ls, F; ω). Thus ν 0 S 0 0} SP F ). By Lemma 9 of [3], we have S 0 0} = S 0. Since SP F ) is closed, we deduce 6.) S 0 SP F ). Let Ω Ω be described as in Section 3. Then for any ω Ω, we have ωp α f p) m 2j p s) if F = sym m f, p 0 j m 6.2) Ls, F; ω) = ωp α f p) 2m i j) p s) if F = sym m f sym m f. p 0 i, j m Since every factor on the right-hand side of 6.) is non-zero, the function Ls, F; ω) is also non-vanishing. Thus Ls, F; ω) : ω Ω} S0 0} and P F S 0 0}) = µ h ω Ω : Ls, F; ω) S0 0} ) µ h Ω) = P F S 0 ) =. Since SP F ) is the smallest closed subset of H F such that P F SPF ) ) = and S 0 is closed, we must have 6.3) SP F ) S 0. Now the required result follows from 6.) and 6.3).

18 The universality of symmetric power L-functions and their Rankin-Selberg L-functions 7 7. Proofs of Corollaries 2 and 3 The proofs of Corollaries 2 and 3 will follow closely those of Theorems 2 and 3 of [4], but we reproduce here the details for the convenience of readers. Let s 0,..., s J be complex numbers such that s 0 0. Inductively on J, we easily see that there is a polynomial ps) = J j=0 b js j such that e ps) ) j) s=0 = s j 0 j J ). Let σ F < σ <, and K be a compact subset of Dσ F ) with connected complement such that σ is contained in the interior of K. We denote by δ the distance of σ from the boundary of K. Then for any ε > 0, Theorem assures that we find a real τ for which supls + iτ, F) e ps σ) < εδj 2 J J! holds. Then, using Cauchy s integral formula we have L j) σ + iτ, F) s j = j! Ls + iτ, F) e ps σ) 2π s σ ) j+ ds < ε s σ =δ/2 for 0 j J, which implies Corollary 2. Next we prove Corollary 3. Without loss of generality, we suppose g J 0. Then there exists a bounded region G C J and a constant B 0 > 0 such that g J B 0 in G. Let σ Dσ F ). According to Corollary 2, we can find a sequence of real numbers τ n such that X n = Lσ + iτ n, F), L σ + iτ n, F),..., L J ) σ + iτ n, F) ) G. By the assumption of Corollary 3, we have J s j g j Ls, F), L s, F),..., L J ) s, F) ) = s J g J Ls, F), L s, F),..., L J ) s, F) ) j=0 for all s C. Letting s = σ + iτ n and dividing both sides by σ + iτ n ) J, we obtain J σ + iτ n ) j J g j X n ) = g J X n ). j=0 Since G is bounded, g j X n ) is bounded 0 j J ). Hence the left-hand side of above tends to zero as n. On the other hand, g J X n ) B 0 > 0. This contradiction finishes the proof of Corollary 3. References [] B. Bagchi, The statistical behaviour and universality properties of the Riemann zetafunction and other allied Dirichlet series, Ph.D. thesis, Indian Statistical Institute, Calcutta, 98.

19 8 H.-Z. Li & J. Wu [2] K.-L. Chung, A course in probability theory, Third edition, Academic Press, Inc., San Diego, CA, 200, xviii+49 pp. [3] J. Cogdell & P. Michel, On the complex moments of symmetric power L-functions at s =, IMRN ), [4] S. Gelbart & H. Jacquet, A relation between automorphic representations of GL2) and GL3), Ann. Sci. École Norm. Sup. 4) 978), [5] H. Iwaniec, Topics in Classical Automorphic Forms, Graduate Studies in Mathematics, vol. 7, American Mathematical Society, Providence, Rhode Island, 997. [6] A. Kačěnas & A. Laurinčikas, On Dirichlet series related to certain cusp forms, Liet. Mat. Rink ), in Russian); = Lithuanian Math. J ), MR j:049) [7] H. Kim, Functoriality for the exterior square of GL 4 and symmetric fourth of GL 2, Appendix by Dinakar Ramakrishnan, Appendix 2 by Henry H. Kim and Peter Sarnak, J. Amer. Math. Soc ), [8] H. Kim & F. Shahidi, Functorial products for GL 2 GL 3 and functorial symmetric cube for GL 2 with an appendix by C.J. Bushnell and G. Henniart), Ann. of Math ), [9] H. Kim & F. Shahidi, Cuspidality of symmetric power with applications, Duke Math. J ), [0] Y.-K. Lau & J. Wu, A density theorem on automorphic L-functions and some applications, Trans. Amer. Math. Soc ), [] A. Laurinčikas, Limit Theorems for the Riemann Zeta-Function, Kluwer, Dordrecht, 996. [2] A. Laurinčikas, On limit distribution of the Matsumoto zeta-function II, Liet. Mat. Rink ), in Russian); = Lithuanian Math. J ), [3] A. Laurinčikas & K. Matsumoto, The universality of zeta-function attached to certain cusp forms, Acta Arith ), [4] A. Laurinčikas & K. Matsumoto, The joint universality of twisted automorphic L- functions, J. Math. Soc. Japan, ), [5] A. Laurinčikas, K. Matsumoto & J. Steuding, The universality of L-function associated with new forms in Russian), Izv. Ross. Akad. Nauk Ser. Mat ), no., 83 98; translation in Izv. Math ), no., [6] K. Matsumoto, The mean values and the universality of Rankin-Selberg L-functions, in: Number theory Turku, 999), 20 22, de Gruyter, Berlin, 200. [7] A. Perelli, General L-functions, Ann. Mat. Pura Appl. 4) ), [8] R.A. Rankin, An Ω-result for the coefficients of cusp forms, Math. Ann ), [9] W. Rudin, Functional Analysis, Second edition, International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 99, xviii+424 pp. [20] Z. Rudnick & P. Sarnak, Zeros of principal L-functions and random matrix theory, Duke Math. J ), [2] J.-P. Serre, Cours d arithmétiques, French) Deuxième édition revue et coorigée. Le Mathématicien, No. 2. Presses Universitaires de France, Paris, pp.

20 The universality of symmetric power L-functions and their Rankin-Selberg L-functions 9 [22] G. Tenenbaum, Introduction to analytic and probabilistic number theory, Translated from the second French edition 995) by C. B. Thomas, Cambridge Studies in Advanced Mathematics 46, Cambridge University Press, Cambridge, 995. xvi+448 pp. [23] S. M. Voronin, A theorem on the universality of the Riemann zeta-function, Izv. Akad. Nauk SSSR Ser. Mat ), in Russian); = Math. USSR-Izv ), Department of Mathematics, Shanghai Jiaotong University, 954 Hua Shan Road, Shanghai , P. R. of China lihz@sjtu.edu.cn Institut Elie Cartan, UMR 7502 UHP CNRS INRIA, Université Henri Poincaré Nancy ), Vandœuvre lès Nancy, France wujie@iecn.u-nancy.fr