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On the Universality for Functions in the Selberg Class

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On the Universality for Functions in the Selberg Class 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 <r< 4 , which is analytic in the interior, can be 1 approximated uniformly by shifts of the Riemann zeta-function in the strip 2 <σ<1. Reich [25] and Bagchi [1] improved Voronin’s result significantly. The strongest version 1 of Voronin’s theorem states: Suppose that K is a compact subset of the strip 2 <σ<1 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|ζ(s + iτ) − g(s)| < > 0; T s K →∞ T ∈ ∗Institut f¨ur Algebra und Geometrie, Fachbereich Mathematik, Johann Wolfgang Goethe- Universit¨at Frankfurt, Robert-Mayer-Str. 10, 60054 Frankfurt, Germany, [email protected] frankfurt.de 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 .
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