BAGCHI S THEOREM FOR FAMILIES OF AUTOMORPHIC FORMS

Transcription

1 BAGCHI S THEOREM FOR FAMILIES OF AUTOMORPHIC FORMS E. KOWALSKI Abstract. We rove a version of Bagchi s Theorem and of Voronin s Universality Theorem for the family of rimitive cus forms of weight 2 and rime level, and discuss under which conditions the argument will aly to a general reasonable family of automorhic L-functions.. Introduction The first universality theorem for Dirichlet series is Voronin s Theorem [8] for the Riemann zeta function, which states that for any r < /4, any continuous function ϕ defined and non-vanishing on the disc s 3/4 r in C, which is holomorhic in the interior, and any ε > 0, there exists t R such that max ζ(s + it) ϕ(s) < ε. s 3/4 r In other words, u to arbitrary recision, any function ϕ can be aroximated by some vertical translate of the Riemann zeta function. Bagchi, in his thesis [], rovided a clear concetual exlanation of this result, as the combination of two indeendent statements: Viewing translates of the Riemann zeta function by t [ T, T ] as random variables with values in a sace of holomorhic function on the disc, Bagchi roves that these random variables converge in law, as T +, to a natural random Dirichlet series, which is also exressed as a random Euler roduct; Comuting the suort of the limiting random Dirichlet series, and checking that it contains the sace of nowhere vanishing holomorhic functions on the disc, the universality theorem follows easily. The key ste, from our oint of view, is the first art, which we call Bagchi s Theorem. Indeed, once the convergence in law is known, it follows that there is some universality statement, with resect to the functions in the suort of the limiting random Dirichlet series. The second ste makes this suort exlicit. (This might be comared with Deligne s Equidistribution Theorem, as alied to families of exonential sums for instance: Deligne s Theorem shows that there is always some equidistribution of these sums.) The goal of this note is to give a first examle of a genuinely higher-degree statement of this tye, and to deduce the corresonding universality statement. We will also indicate a general rincile that should aly in many more cases. Key words and hrases. Modular forms, L-functions, Bagchi s Theorem, Voronin s Theorem, random Dirichlet series. Partially suorted by a DFG-SNF lead agency rogram grant (grant 20002L 53647).

2 Theorem. (Universality in level asect). For q rime 7, let S 2 (q) be the nonemty finite set of rimitive cus forms for Γ 0 (q) with weight 2 and trivial nebentyus. For f S 2 (q), let L(f, s) denote its Hecke L-function L(f, s) = n λ f (n)n s, normalized so that the critical line is Re(s) =. 2 For any real number r <, let D be the oen disc centered at 3/4 with radius r. Then 4 for any continuous function ϕ : D C which is holomorhic and non-vanishing in D and satisfies (.) ϕ(σ) > 0 for σ D R, we have lim inf q + S 2 (q) {f S 2(q) L(f, ) ϕ < ε} > 0 for any ε > 0, where the L norm is the norm on D. The main difference with revious results involving cus forms (the first one being due to Laurinčikas and Matsumoto [3]) is that we do not fix one such L-function L(f, s) and consider shifts (or twists) L(f, s + it) or L(f χ, s), but rather we average over the discrete family of rimitive forms in S 2 (q). It is also imortant to remark that the condition (.) is necessary for a function on D to be aroximated by L-functions L(f, s) with f S 2 (q). (We will give more general statements where the discs D are relaced with more general comact sets in the stri 2 < σ < ). We will rove this Theorem in Section 2, after stating the results generalizing the two stes of Bagchi s strategy for the zeta function. The roof of Bagchi s Theorem for this family is an analogue of a roof for the Riemann zeta function that is simler than Bagchi s roof (it avoids both the use of the ergodic theorem and any tightness or weak-comactness argument). In Section 5, we discuss very briefly how this strategy can in rincile be alied to very general families of L-functions, as defined in [8]. Acknowledgements. The simle roof of Bagchi s Theorem for the Riemann zeta funtion, that we generalize here to modular forms, was found during a course on robabilistic number theory at ETH Zürich in the Fall Semester 205; thanks are due to the students who attended this course for their interest and remarks, and to B. Löffel for assisting with the exercises (see [9] for a draft of the lecture notes for this course, esecially Chater 3). Thanks to the referee for carefully reading the text, and in articular for ointing out a number of confusing mistakes in references to the literature. Notation. As usual, X denotes the cardinality of a set. By f g for x X, or f = O(g) for x X, where X is an arbitrary set on which f is defined, we mean synonymously that there exists a constant C 0 such that f(x) Cg(x) for all x X. The imlied constant is any admissible value of C. It may deend on the set X which is always secified or clear in context. We write f g if f g and g f are both true. We assume q 7 to ensure this roerty; it also holds for q =. 2

3 We use standard robabilistic terminology: a robability sace (Ω, Σ, P) is a trile made of a set Ω with a σ-algebra Σ and a measure P on Σ with P(Ω) =. We denote by E(X) the exectation on Ω. The law of a random variable X is the measure ν on the target sace of X defined by ν(a) = P(X A). If A Ω, then A is the characteristic function of A. 2. Equidistribution and universality for modular forms in the level asect We will rove Theorem. by combining the results of the following two stes, each of which will be roved in a forthcoming section. Throughout, we assume that q is a rime number 7. Ste. (Equidistribution; Bagchi s Theorem) For q rime, we view the finite set S 2 (q) as a robability sace with the robability measure roortional to the harmonic measure where f S 2 (q) has weight f, f in terms of the Petersson inner roduct. We write E q ( ) or P q ( ) for the corresonding exectation and robability. Hence there exists a constant c q > 0 such that E q (ϕ(f)) = c q f, f ϕ(f) f S 2 (q) for any ϕ: S 2 (q) C. From the Petersson formula, it is known that c q /(4π) as q + (see, e.g., Iwaniec and Kowalski [7, Ch. 4] or Cogdell and Michel [3]). Let D be a relatively comact oen set in C such that D is invariant under comlex conjugation. Define H(D) to be the Banach sace of functions holomorhic on D and continuous and bounded on D, with the norm ϕ = su ϕ(s). s D This is a searable comlex Banach sace. Define also H R (D) to be the set of ϕ H(D) such that f( s) = f(s) for all s D (this is well-defined since C is assumed to be invariant under conjugation). Note that the L-function of f, restricted to D, is an element of H R (D) since the Hecke eigenvalues λ f (n) are real for all n. We define L q to be the random variable S 2 (q) H(D) maing f S 2 (q) to the restriction of L(f, s) to D. (This deends on D, but the choice of D will always be clear in the context.) If D is a comact subset of the stri 2 < Re(s) <, then we will show that L q converges in law to a random Dirichlet series. To define the limit, let (X ) be a sequence of indeendent random variables indexed by rimes, taking values in the matrix grou SU 2 (C) and distributed according to the robability Haar measure on SU 2 (C). Theorem 2. (Bagchi s Theorem for modular forms). Assume that D is a comact subset of the stri 2 < Re(s) <. Then, as q +, the random variables L q converge in law to the random Euler roduct L D (s) = det( X s ) = ( Tr(X ) s + 2s ) which is almost surely convergent in H(D), and belongs almost surely to H R (D). 3

4 Ste 2. (Suort of the random Euler roduct) To deduce Theorem. from Theorem 2., we need the following comutation of the suort of the limiting measure. Theorem 2.2. Suose that D is a disc with ositive radius and diameter a segment of the real axis, always with D contained in < Re(s) <. The suort of the law of the random 2 Euler roduct L D contains the set of functions ϕ H(D) such that ϕ(x) > 0 for x D R. Note that since D R is an interval of ositive length in R, the condition ϕ(x) > 0 for all x D R imlies by analytic continuation that ϕ H R (D), which by Bagchi s Theorem 2. is a necessary condition to be in the suort of L D. Ste 3. (Conclusion) The elementary Lemma 2.4 below, combined with Theorems 2. and 2.2, imlies Theorem. in the form (2.) lim inf P c q q( L(f, ) ϕ < ε) = lim inf q + q + f, f > 0 f S 2 (q) L(f, ) ϕ <ε for any function ϕ as in Theorem. and any ε > 0. We can easily deduce the natural density version from this: let A be the set of those f S 2 (q) such that L(f, ) ϕ < ε; then for any arameter η > 0, the definition of the harmonic measure on S 2 (q) gives S 2 (q) f, f ) ( = E q ( A (f) η c q S 2 (q) f A ( f, f )) P q (A) P q c q S 2 (q) < η. There exists δ > 0 such that the first term is δ > 0 for all q large enough by (2.); on the other hand, a result of Cogdell and Michel [3, Cor..6] and the classical relation between the Petersson norm and the symmetric square L-function at s = (see, e.g., [7, (5.0)]) imly that we can find η > 0 such that ( f, f ) lim P q q + c q S 2 (q) < η < δ 2. For this value of η, we obtain lim inf q + S 2 (q) f A ηδ 2 > 0. More recisely, the result of Cogdell-Michel is that for any η > 0, we have lim P q(l(sym 2 f, ) η) = F (log η) q + where F is the limiting distribution function for the secial value at of the symmetric square L-function of f S 2 (q). Since F (x) 0 when x, we obtain the result. Remark 2.3. It would also be ossible to argue throughout with the uniform robability measure on S 2 (q) ; the only change would be a slightly different form of Theorem 2., where the random variables (X ) would not be identically distributed (comare with the equidistribution theorems of Serre [6] and Conrey Duke Farmer [4]). 4

5 Lemma 2.4. Let M be a searable comlete metric sace and (X n ) a sequence of random variables with values in M that converge in law to X. Let S be the suort of the law of X. Then for any x S, and any oen neighborhood U of x, we have lim inf P(X n U) > 0. n + Proof. By classical criteria for convergence in law, we have (2.2) lim inf n + P(X n U) P(X U) for any oen set U X (see, e.g., [2, Th. 2. (iv)]). Since x S, we have P(X U) > 0, hence the result. 3. Proof of Theorem 2. We begin with some reliminaries concerning the random Euler roduct L D. In fact, if will be convenient to view it as a holomorhic function on larger domains then D, in a way that will be clear below. For this urose, we fix a real number σ 0 such that 2 < σ 0 <, and such that the comact set D is contained in the half-lane S defined by Re(s) > σ 0. We recall that for ν 0, the d-th Chebychev olynomial is defined by U ν (2 cos(x)) = sin((ν + )x). sin(x) The imortance of these olynomials for us lies in their relation with the reresentation theory of SU 2 (C), namely ( ( ) e U ν (2 cos(x)) = Tr Sym ν ix ) 0 0 e ix for any x R, where Sym d is the d-th symmetric ower of the standard 2-dimensional reresentation of SU 2 (C). We define a sequence of random variables (Y n ) n by Y n = ν n U ν (Tr(X )). In articular, we have Y n Y m = Y nm if n and m are corime, and Y = Tr(X ) if is rime. The sequence (Y ) is indeendent and Sato-Tate distributed. Moreover, since U ν (t) ν + for all ν 0 and all t R, we have Y n ν n (ν + ) = d(n) n ε for n and ε > 0, where the imlied constant deends only on ε. Lemma 3.. () Almost surely, the random Euler roduct det( X s ) converges and defines a holomorhic function on S. In articular, L D converges almost surely to define an H(D)-valued random variable. 5

6 (2) Almost surely, we have det( X s ) = Y n n s n for all s S, and in articular L D coincides with the random Dirichlet series on the right. (3) For σ > /2 and u 2, we have ( E Y n n σ 2), n u where the imlied constant deends only on σ. Proof. () Let σ be a fixed real number such that 2 < σ < σ 0. By exanding, we can write where the random series log det( X s ) = Y s + g (s) g (s) converges absolutely (and surely) for Re(s) > /2. Since E(Y σ ) = 0 and E(Y 2 2σ ) = 2σ, Kolmogorov s Three Series Theorem (see, e.g., [4, Th. 0.III.2]) imlies that the random series Y σ converges almost surely. random series By well-known results on Dirichlet series, this means that the Y s converges almost surely to a holomorhic function on the half-lane S. This imlies the first statement by taking the exonential. The second follows by restricting to D since D is contained in S. (2) We first show that almost surely the random Dirichlet series L(s) = n Y n n s converges and defines a function holomorhic on S. The key oint is that the variables Y n for n squarefree form an orthonormal system: we have E(Y n Y m ) = δ(n, m) if n and m are squarefree numbers. Indeed, if n m, there is a rime dividing only one of n and m, say n, and then by indeendence we get E(Y n Y m ) = E(Y )E(Y n/ Y m ) = 0; and if n = m is squarefree then we have E(Yn 2 ) = E(Y 2 ) =. n Fix again σ such that < σ < σ 2 0. By the Rademacher Menchov Theorem (see, e.g. [9, Th. B.8.4]), the random series Y n n σ n 6

7 over squarefree numbers converges almost surely. By elementary factorization and roerties of roducts of Dirichlet series (the roduct of an absolutely convergent Dirichlet series and a convergent one is convergent, see e.g. [6, Th. 54]) the same holds for Y n n σ. n As in (), this gives the almost sure convergence of the series defining L(s) to a holomorhic function in S. Restricting gives the H(D)-valued random variable L D. Finally, almost surely both the random Euler roduct and the random Dirichlet series converge and are holomorhic for Re(s) > σ 0. For Re(s) > 3/2, they converge absolutely, and coincide by a well-known formal Euler roduct comutation: for any rime and any x R, denoting ( ) e ix 0 t(x) = 0 e ix we have det( t(x) s ) = ( e ix s ) ( e ix s ) = ν 0 Tr Sym ν (t(x)) νs = ν 0 U ν (x) νs (comare the discussion of Cogdell and Michel in [3, 2]). By analytic continuation, we deduce that L D = L D almost surely as H(D)-valued random variables. (3) Since the random varibles Y n are real-valued, we have ( E Y n n σ 2) = (nm) E(Y ny σ m ). n u n,m u For given n and m, let d = (n, m) and n = n/d, m = m/d. Then by multilicativity and indeendence of the variables (Y ), we have E(Y n Y m ) = E(Y 2 d )E(Y n )E(Y m ). By the definition of Y, we have E(Y n ) = 0 if n is divisible by a rime with odd exonent, and similarly for E(Y m ). Hence we have E(Y n Y m ) = 0 unless both n and m are squares. Therefore ( E Y n n σ 2) n u d E(Y 2 d ) d 2σ since σ > /2 and E(Y n ) n ε for any ε > 0. m,n (mn) 2σ E(Y my n ) < + The key arithmetic roerties of the family S 2 (q) of modular forms that are required in the roof of Theorem 2. are the following: Proosition 3.2 (Local sectral equidistribution). As q +, the sequence (λ f ()) of Fourier coefficients of f S 2 (q) converges in law to the sequence (Y ). Proof. This is a well-known consequence of the Petersson formula, see e.g. [0, Pro. 8], [2, Aendix] or [3, Pro..9]; here restricting to rime level q and weight 2 also simlifies matters since this ensures that the old sace of S 2 (q) is zero. 7

8 Proosition 3.3 (First moment estimate). There exists an absolute constant A such that for any real number δ > 0 with δ < /2, and for any s C such that + δ Re(s), we 2 have E q ( L(f, s) ) ( + s ) A, where the imlied constant deends only on δ. Proof. This follows easily, using the Cauchy-Schwarz inequality, from the second moment estimate [, Pro. 5] of Kowalski and Michel (with = 0); although this statement is not formally the same, it is in fact a more difficult average (it oerates closer to the critical line). We now rove some additional lemmas. Lemma 3.4 (Polynomial growth). For any real number σ > σ 0, we have ( ) E Y n n s + s n uniformly for all s such that Re(s) σ > σ 0. Proof. We write L(s) = n Y n n s. This is almost surely a function holomorhic on the half-lane S. The series Y n n σ 0 n converges almost surely. Therefore the artial sums S u = n u are bounded almost surely. By summation by arts, it follows from the convergence of the series L(s) that for any s with real art Re(s) σ > σ 0, we have L(s) = (s σ 0 ) Y n n σ 0 + S u u s σ 0+ du, where the integral converges almost surely. Hence almost surely L(s) ( + s ) + S u u σ σ + du. Fubini s Theorem and the Cauchy-Schwarz inequality then imly by Lemma 3. (3). E( L(s) ) ( + s ) ( + s ) + + E( S u ) du u σ σ 0+ E( S u 2 ) /2 du u + s σ σ 0+ 8

9 We now consider some elementary aroximation statements of the L-functions and of the random Dirichlet series by smoothed artial sums. For this, we fix once and for all a smooth function ϕ: [0, + [ [0, ] with comact suort such that ϕ(0) =, and we denote ˆϕ its Mellin transform. We also fix T and a comact interval I in ]/2, [ such that the comact rectangle R = I [ T, T ] C is contained in S and contains D in its interior. We then finally define δ > 0 so that min{re(s) s R} = 2 + 2δ. Lemma 3.5. For N, define the H(D)-valued random variable L (N) D = ( n ) Y n ϕ n s. N n We then have E( L D L (N) D ) N δ for N, where the imlied constant deends on D. Proof. We again write L(s) = Y n n s n when we wish to view the Dirichlet series as defined and holomorhic (almost surely) on S. For any s in the rectangle R, we have almost surely the reresentation (3.) L(s) L (N) (s) = L(s + w) ˆϕ(w)N w dw 2iπ by standard contour integration. 2 We also have almost surely for any v in D the Cauchy formula L D (v) L (N) D (v) = (L(s) L (N) (s)) ds 2iπ R s v, where the boundary of R is oriented counterclockwise. The definition of the rectangle R ensures that s v for v D and s R, and therefore L D L (N) D R ( δ) L(s) L (N) (s) ds. Using (3.) and writing w = δ + iu with u R, we obtain L D L (N) D N δ L( δ + iu + s) ˆϕ( δ + iu) ds du. R R Therefore, taking the exectation, and using Fubini s Theorem, we get E( L D L (N) D ) N δ E ( L( δ + iu + s) ) ˆϕ( δ + iu) ds du R R N δ E ( L( δ + iu + σ + it) ) ˆϕ( δ + iu) du. su s=σ+it R R 2 Here and below, it is imortant that the almost surely roerty holds for all s, which is the case because we work with random holomorhic functions, and not with articular evaluations of these random functions at secific oints s. 9

10 We therefore need to bound E ( L( δ + iu + σ + it) ) ˆϕ( δ + iu) du. R for some fixed σ+it in the comact rectangle R. The real art of the argument δ+iu+σ+it is σ δ + δ by definition of δ, and hence 2 E( L( δ + iu + σ + it) ) + δ + iu + σ + it + u uniformly for σ+it R and u R by Lemma 3.4. Since ˆϕ decays faster than any olynomial at infinity, we conclude that E ( L( δ + iu + σ + it) ) ˆϕ( δ + iu) du R uniformly for s = σ + it R, and the result follows. We roceed similarly for the L-functions. Lemma 3.6. For N and f S 2 (q), define L (N) (f, s) = ( n ) λ f (n)ϕ n s, N n and define L (N) q D. We then have for N and all q. to be the H(D)-valued random variable maing f to L (N) (f, s) restricted to E q ( L q L (N) q ) N δ Proof. For any s R, we have the reresentation (3.2) L(f, s) L (N) (f, s) = 2iπ ( δ) L(f, s + w) ˆϕ(w)N w dw. and for any v with Re(v) > /2, Cauchy s theorem gives L(f, v) L (N) (f, v) = (L(f, s) L (N) (f, s)) ds 2iπ s v, R where the boundary of R is oriented counterclockwise. As in the revious argument, we deduce that L q L (N) q L(f, s) L (N) (f, s) ds R for f S 2 (q). Taking the exectation with resect to f and changing the order of summation and integration leads to ( ) ( E q L q L (N) q E q L(f, s) L (N) (f, s) ) ds R ( (3.3) su E q L(f, s) L (N) (f, s) ). s R Alying (3.2) and using again Fubini s Theorem, we obtain ( E q L(f, s) L (N) (f, s) ) ( ) N δ ˆϕ( δ + iu) E q L(f, δ + iu + σ + it) du R 0

11 for s R. Since σ δ + δ, we get 2 (3.4) E q ( L(f, δ + iu + σ + it) ) ( + u ) A by Proosition 3.3, where A is an absolute constant. Hence ( (3.5) E q L(f, s) L (N) (f, s) ) N δ ˆϕ( δ + iu) ( + u ) A du N δ. We conclude from (3.3) that as claimed. R E q ( L q L (N) q ) N δ, Proof of Theorem 2.. A simle consequence of the definition of convergence in law shows that it is enough to rove that for any bounded and Lischitz function f : H(D) C, we have E q (f(l q )) E(f(L D )) as q + (see [2,. 6, (ii) (iii) and (.),. 8]). To rove this, we use the Dirichlet series exansion of L D given by Lemma 3. (2). Let N be some integer to be chosen later. Let L (N) q = n λ f (n)n s ϕ ( n N ) (viewed as random variable defined on S 2 (q) ) and L N = ( n ) Y n n s ϕ N n be the smoothed artial sums of the Dirichlet series, as in Lemmas 3.6 and 3.5. We then write E q (f(l q )) E(f(L)) E q (f(l q ) f(l (N) q )) + E q (f(l (N) q )) E(f(L (N) )) + E(f(L (N) ) f(l)). Since f is a Lischitz function on H(D), there exists a constant C 0 such that for all x, y H(D). Hence we have f(x) f(y) C x y E q (f(l q )) E(f(L)) CE q ( L q L (N) q )+ E q (f(l (N) q )) E(f(L (N) )) + CE( L (N) L ). Fix ε > 0. Lemmas 3.6 and 3.5 together show that there exists some N such that for all q 2 and E q ( L q L (N) q ) < ε E( L (N) L ) < ε.

12 We fix such a value of N. By Proosition 3.2 (and comosition with a continuous function), the random variables L (N) q (which are Dirichlet olynomials) converge in law to L (N) as q +. We deduce that we have E q (f(l q )) E(f(L)) < 4ε for all q large enough. This finishes the roof. 4. Proof of Theorem 2.2 For the comutation of the suort of the random Dirichlet series L(s), we aly a trick to exloit the analogous result known for the case of the Riemann zeta function. We denote ŜU 2 the roduct of coies of the unit circle indexed by rimes, so an element (x ) of ŜU 2 is a family of matrices in SU 2 (C) indexed by. The assumtions on D in Theorem imly that there exists τ such that /2 < τ < and r > 0 such that D = {s C s τ r} {s C /2 < Re(s) < }. Lemma 4.. Let N be an arbitrary ositive real number. The set of all series Tr(x ), (x s ) ŜU 2 >N which converge in H(D) is dense in the subsace H R (D). In the roof and the next, we allow ourselves the luxury of writing sometimes ϕ(s) instead of ϕ. Proof. Bagchi [, Lemma 5.2.0] roves (using results of comlex analysis due to Bernstein, Polyá and others) that the set of series >N e iθ s, θ R that converge in H(D) is dense in H(D) (recisely, he roves this for N =, but the same roof alies to any value of N). If ϕ H R (D) and ε > 0, we can therefore find real numbers (θ ) such that ϕ(s) e iθ < ε 2 s 2. >N It follows then that ϕ( s) e iθ, < ε 2 s 2, >N hence (since ϕ H R (D)) that ϕ(s) e iθ + e iθ < ε, s >N 3 These assumtions could be easily weakened, as has been done for Voronin s Theorem. 2

13 which gives the result since is the trace of a matrix in SU 2 (C). We will use this to rove: Proosition 4.2. The suort of the law of ( ) e iθ 0 e iθ + e iθ = Tr 0 e iθ log L D (s) = log det( X s ) in H(D) is H R (D). Proof. Since X SU 2 (C), the function log L(s) is almost surely in the sace H R (D). Since the summands are indeendent, a well-known result concerning the suort of random series (see, e.g., [9, Pro. B.8.7]) shows that it suffices to rove that the set of convergent series log det( x s ), (x ) ŜU 2, is dense in H R (D). Denote L(s; (x )) this series, when it converges in H(D). We can write log det( x s ) = Tr(x ) + g(s; (x s )) where s g(s; (x )) is holomorhic in the region Re(s) > /2. Indeed g(s; (x )) = ( log + Tr(x ) k ). (k+2)s k 0 Fix ϕ H R (D) and let ε > 0 be fixed. There exists N such that (4.) s ( log + ) Tr(x ) k (k+2)s < ε >N k 0 for any (x ) ŜU 2. Now take x = SU 2 (C) for N and define ϕ = ϕ + N Tr(x ) s = ϕ + 2 N which belongs to H R (D). By Lemma 4., there exist x for > N in SU 2 (C) such that Tr(x ) ϕ s (s) < ε. >N s, The left-hand side is the norm of log L(s; (x )) g(s; (x )) N Tr(x ) s ϕ (s) = log L(s; (x )) ϕ(s) g(x; (x )), and by (4.), we obtain This imlies the lemma. log L(s; (x )) ϕ(s) < 2ε. 3

14 Using comosition with the exonential function and a lemma of Hurwitz (see, e.g., [7, 3.45]) on zeros of limits of holomorhic functions, we see that the suort of the limiting Dirichlet series L D in H(D) is the union of the zero function and the set of functions ϕ H R (D) such that ϕ(σ) > 0 for σ D R. In articular, this roves Theorem Generalizations It is clear from the roof that Bagchi s Theorem should hold in considerable generality for any family of L-functions. Indeed, the crucial ingredients are the local sectral equidistribution (Proosition 3.2), and the first moment estimate (Proosition 3.3). The first result is a qualitative statement that is understood to be at the core of any definition of family of L-functions (this is exlained in [8], but also aears, with a different terminology, for the families of Conrey Farmer Keating Rubinstein Snaith [5] and Sarnak Shin Temlier [5]); it is now known in many circumstances (indeed, often in quantitative form). The moment estimate is tyically derived from a second-moment bound, and is also definitely exected to hold for a reasonable family of L-functions, but it has only been roved in much more restricted circumstances than local sectral equidistribution. However, it is very often the case that one can at least rove (using local sectral equidistribution) a weaker statement: for some σ such that /2 < σ <, the second moment of the L-functions satisfies the analogue of Proosition 3.3; an analogue of Bagchi s Theorem then follows at least for comact discs in the region σ < Re(s) <. As far as universality (i.e., Theorem 2.2) is concerned, one may exect that (using tricks similar to the roof of Theorem 2.2) only two different cases really occur, deending on whether the coefficients of the L-functions are real (as in our case) or comlex (as in the case of vertical translates of a fixed L-function). References [] B. Bagchi: Statistical behaviour and universality roerties of the Riemann zeta function and other allied Dirichlet series, PhD thesis, Indian Statistical Institute, Kolkata, 98; available at library.isical. ac.in/jsui/handle/0263/4256 [2] P. Billingsley: Convergence of robability measures, Wiley (968). [3] J. Cogdell and P. Michel: On the comlex moments of symmetric ower L-functions at s =, Internat. Math. Res. Notices (2004), [4] B. Conrey, W. Duke and D. Farmer: The distribution of the eigenvalues of Hecke oerators, Acta Arith. 78 (997), [5] J.B. Conrey, D. Farmer, J. Keating, M. Rubinstein and N. Snaith: Integral moments of L-functions, Proc. Lond. Math. Soc. 9 (2005) [6] G.H. Hardy and M. Riesz: The general theory of Dirichlet s series, Cambridge Tracts in Math. 8, C.U.P. 95. [7] H. Iwaniec and E. Kowalski: Analytic number theory, Colloquium Publ. 53, American Math. Soc [8] E. Kowalski: Families of cus forms, Pub. Math. Besançon, 203, [9] E. Kowalski: Arithmetic randonnée: an introduction to robabilistic number theory, ETH Zürich Lecture Notes, [0] E. Kowalski, Y-K. Lau, K. Soundararajan and J. Wu: On modular signs, Math. Proc. Cambridge Phil. Soc. 49 (200), doi:0.07/s x [] E. Kowalski and Ph. Michel: The analytic rank of J 0 (q) and zeros fo automorhic L-functions, Duke Math. J. 00 (999),

15 [2] E. Kowalski, A. Saha and J. Tsimerman: Local sectral equidistribution for Siegel modular forms and alications, Comositio Math. 48 (202), [3] A. Laurinčikas and K. Matsumoto: The universality of zeta-functions attached to certain cus forms, Acta Artih. 98 (200), [4] D. Li and H. Queffélec: Introduction à l étude des esaces de Banach; Analyse et robabilités, Cours Sécialisés 2, S.M.F, [5] P. Sarnak, S-W. Shin and N. Temlier: Families of L-functions and their symmetry, in Families of automorhic forms and the trace formula, Simons Symosia, Sringer 206, [6] J-P. Serre: Réartition asymtotique des valeurs rores de l oérateur de Hecke T, J. American Math. Soc. 0 (997), [7] E.C. Titchmarsh: The theory of functions, 2nd edition, Oxford Univ. Press, 939. [8] S.M. Voronin: Theorem on the universality of the Riemann zeta function, Izv. Akad. Nauk SSSR, 39 (975); ; translation in Math. USSR Izv. 9 (975), ETH Zürich D-MATH, Rämistrasse 0, 8092 Zürich, Switzerland address: kowalski@math.ethz.ch 5