Universality of L -Functions over function fields
aa r X i v : . [ m a t h . N T ] S e p UNIVERSALITY OF L -FUNCTIONS OVER FUNCTION FIELDS J. C. ANDRADE AND S. M. GONEKA
BSTRACT . We prove that the Dirichlet L -functions associated to Dirichletcharacters in F q [ x ] are universal. That is, given a modulus of high enough de-gree, L -functions with characters to this modulus can be found that approximateany given nonvanishing analytic function arbitrarily closely.
1. I
NTRODUCTION
Let s = σ + it and let ζ ( s ) denote the Riemann zeta function. In 1975 S. M.Voronin [14] proved the following remarkable “universality” theorem. Theorem 1 (Voronin) . Let D r be the open disc of radius < r < / centeredat s = 3 / and let F ( s ) be a function that is analytic in D r , continuous on D r ,the closure of D r , and nonvanishing on D r . Then for any ǫ > there exist realnumbers τ → ∞ such that max s ∈ D r | ζ ( s + iτ ) − F ( s ) | < ǫ. (1)This was soon extended in various directions by A. Good [6], S. M. Gonek [4],Bagchi [2, 3], Reich [10, 11], and Voronin [15, 16, 17, 18] himself. For instance,Gonek and Bagchi independently showed that the disc D r could be replaced byan arbitrary compact subset K in the strip < σ < whose complement isconnected. Another direction was to prove universality for Dirichlet L -functions,Hurwitz and Epstein zeta functions, and a variety of other Dirichlet series. Implicitin the proofs of these results, though not always stated, was that the set of transla-tion numbers τ has positive proportion. That is, the measure of the set of τ ∈ [0 , T ] satisfying (1) is ≥ cT for all sufficiently large T , where c is a positive constant.Another version of universality, first proved by Gonek and independently soonafter by Bagchi is Date : August 14, 2018.2010
Mathematics Subject Classification.
Primary 11G20, Secondary 14G10, 11M50.
Key words and phrases. function fields, hybrid formula, Dirichlet L -functions, universality. Theorem 2 (Gonek, Bagchi) . Let C be a compact subset in the strip < σ < whose complement is connected, and let F ( s ) be a nonvanishing function thatis analytic in the interior of C and continuous on C . Then for any ǫ > andsufficiently large integer q , there exists a Dirichlet character χ (mod q ) such thatthe Dirichlet L -function L ( s, χ ) satisfies max s ∈K | L ( s, χ ) − F ( s ) | < ǫ. (2)Here too it is implicit in the proofs that a positive proportion of the φ ( q ) char-acters χ (mod q ) satisfy (2).Over the years, a large number of papers on various aspects of universality haveappeared. Good general surveys are Matsumoto’s paper [9] and the monographsby Laurinˇcikas [8] and by Steuding [13]. The work of Liza Jones [7] on a randommatrix analog of Voronin’s theorem for characteristic polynomials associated tounitary, orthogonal and symplectic random matrices is also relevant. As far as weare aware, however, no one has extended the concept of universality to L -functionsover function fields. Our goal here is to prove such a result. It will be obvious tothose familiar with the area of universality that numerous extensions of our theoremare possible.We first introduce some of the basic notation for function fields. Let F q be afinite field with q elements, where q is odd, and let F q [ x ] be the polynomial ringover F q in the variable x . If f is a nonzero polynomial in F q [ x ] , we define the normof f to be | f | = q deg f . If f = 0 , we set | f | = 0 . A monic irreducible polynomial P is called a prime polynomial or simply a prime . The L -function correspondingto an odd Dirichlet character χ modulo Q is given by the Euler product L ( s, χ ) = Y P prime (1 − χ ( P ) | P | − s ) − Re ( s ) > , where s is a complex variable and Q is a monic polynomial. Multiplying out, weobtain the Dirichlet series representation L ( s, χ ) = X f monic χ ( f ) | f | s Re ( s ) > . It is sometimes convenient to work with the equivalent functions written in termsof the variable u = q − s , namely, L ( u, χ ) = Y P prime (1 − χ ( P ) u deg P ) − | u | < /q (3)and NIVERSALITY OF L -FUNCTIONS OVER FUNCTION FIELDS 3 L ( u, χ ) = X f monic χ ( f ) u deg f | u | < /q. (4)We refer to these pairs of equivalent expressions as the s -forms and u -forms of the L -function, respectively.It turns out that L ( u, χ ) is actually a polynomial of degree Q − (seeRosen [12], Proposition 4.3), and it satisfies a Riemann hypothesis (see Weil [19]).That is, all its zeros lie on the circle | u | = q − . It follows that we may write L ( u, χ ) = Q − Y j =1 (1 − α j u ) , where the α j = q e ( − θ j ) , j = 1 , . . . , Q − are the reciprocals of theroots u j = q − e ( θ j ) of L ( u, χ ) . (Throughout we write e ( x ) to denote e πix .) Inparticular, the restriction | u | < /q in (4) (but not in (3)) may be deleted.We may now state our first version of the universality theorem. Recall that if f ∈ F q [ x ] , Euler’s phi function φ ( f ) is the number of non-zero polynomials ofdegree less than deg( f ) and relatively prime to f . Theorem 3.
Let U = { u : q − < | u | < q − } and let C be a compact subsetof U whose complement in U is connected. Let F ( u ) be a nonvanishing analyticfunction on the interior of C that is continuous on the boundary of C . Then for any ǫ > and monic polynomial Q of large enough degree, there exists a Dirichletcharacter χ (mod Q ) such that max u ∈C |L ( u, χ ) − F ( u ) | < ǫ. (5) Moreover, (5) holds for a positive proportion of the φ ( Q ) characters χ (mod Q ) . The theorem may also be formulated in terms of the variable s . Theorem 4.
Let U be the open rectangle with vertices , , i π log q , + i π log q and let C be a compact subset of U whose complement in U is connected. Let F ( s ) be a nonvanishing function that is analytic on the interior of C and continuous onthe boundary of C . Then for any ǫ > and monic polynomial Q of large enoughdegree, there exists a Dirichlet character χ (mod Q ) such that max u ∈ C | L ( s, χ ) − F ( s ) | < ǫ. (6) Moreover, (6) holds for a positive proportion of the φ ( Q ) characters χ (mod Q ) . J. C. ANDRADE AND S. M. GONEK
2. B
ASIC LEMMAS ON L - FUNCTIONS AND ARITHMETIC IN FUNCTION FIELDS
In this section we introduce several lemmas we require for the proof of Theo-rem 3. A version of our first lemma appears in [1]. The proof of the statement hereis virtually identical (see Remark 2 of [1]).
Lemma 5 (The hybrid formula for L ( u, χ ) ) . Let K ≥ be an integer and let P K ( u, χ ) = exp K X k =1 X f monicdeg f = k Λ( f ) χ ( f ) u k k ! , (7) where Λ( f ) = deg P if f = P n for some prime polynomial P , and Λ( f ) = 0 otherwise. Also set Z K ( u, χ ) = exp (cid:18) − Q − X j =1 (cid:18) X k>K ( α j u ) k k (cid:19)(cid:19) . (8) Then for χ = χ and | u | ≤ q − / , L ( u, χ ) = P K ( u, χ ) Z K ( u, χ ) . Lemma 6.
Let Q be a monic polynomial of degree greater than zero, χ a character (mod Q ) , and K ≥ an integer. If σ > , then uniformly for σ ≥ σ we have L ( s, χ ) = P K ( s, χ ) (cid:16) O (cid:16) deg QK q ( − σ ) K (cid:17)(cid:17) , (9) where P K ( s, χ ) = exp X ≤ deg P ≤ K [ K/ deg P ] X j =1 χ ( P j ) j | P | js ! , (10) Λ( f ) = deg P if f = P n for some prime polynomial P , and Λ( f ) = 0 otherwise.The constant implicit in the O -term depends at most on q and σ .Proof. It is not difficult to see that if u is replaced by q − s in (7), we obtain P K ( s, χ ) .Making this substitution in (8) also, and assuming that σ > , we find that NIVERSALITY OF L -FUNCTIONS OVER FUNCTION FIELDS 5 | log Z K ( u, χ ) | = (cid:12)(cid:12)(cid:12)(cid:12) Q − X j =1 X k>K ( α j u ) k k (cid:12)(cid:12)(cid:12)(cid:12) ≤ Q X k>K q ( − σ ) k k ≤ Q q ( − σ ) K K (1 − q − σ ) ≪ σ ,q deg Q q ( − σ ) K K .
Equation (9) follows from this. (cid:3)
Lemma 7.
Let Q be a monic polynomial of degree greater than one. Then Euler’sphi function satisfies the bound φ ( Q ) ≫ | Q | log q (deg Q ) , (11) where log q is the logarithm to the base q .Proof. We have φ ( Q ) = | Q | Y P | Q (cid:16) − | P | (cid:17) . As a function of | Q | , this will clearly be smallest when Q is a product of all theprimes of degree , times all the primes of degree , and so on up to the primes ofdegree n , say. In that case we have φ ( Q ) | Q | = Y j ≤ n (cid:16) − q j (cid:17) π q ( j ) = exp (cid:18) X j ≤ n π q ( j ) log(1 − q − j ) (cid:19) . The sum in the exponential function equals − X j ≤ n (cid:16) q j j + O (cid:16) q j/ j (cid:17)(cid:17)(cid:16) q − j + O ( q − j ) (cid:17) = − X j ≤ n j + O ( q − / )= − log n + O (1) . Hence, φ ( Q ) | Q | ≫ n . (12)Now J. C. ANDRADE AND S. M. GONEK | Q | = Y j ≤ n q jπ q ( j ) = Y j ≤ n q j ( q j /j + O ( q j/ /j )) = q q n (1+ O ( q − )) . Thus, deg Q = q n (1 + O ( q − )) and log q (deg Q ) = n + O (1 / ( q log q )) . Usingthis in (12) we obtain the assertion of the lemma. (cid:3) Lemma 8.
Let N be a set of distinct polynomials and Q a monic polynomial. Ifthe norm of each element of N is less than | Q | , then X χ (mod Q ) (cid:12)(cid:12)(cid:12)(cid:12) X N ∈N a N χ ( N ) (cid:12)(cid:12)(cid:12)(cid:12) = φ ( Q ) X N ∈N | a N | . This follows from the orthogonality relations for Dirichlet characters in functionfields as presented in [12, Proposition 4.2].3. L
EMMAS ON DIOPHANTINE APPROXIMATION
The lemmas in this section are of similar nature as those proved in Gonek andMontgomery [5]. The first is exactly as in [5], the others are character analoguesof lemmas there.
Lemma 9.
Let K be a positive integer, and suppose that < δ ≤ / . There is atrigonometric polynomial f ( θ ) of the form f ( θ ) = K X k =0 c k e ( kθ ) (13) such that max θ | f ( θ ) | = f (0) = 1 and | f ( θ ) | ≤ e − πKδ for δ ≤ θ ≤ − δ . Let Q be a monic polynomial and let P Q be a finite set of primes P coprimeto Q . For given real numbers ≤ θ P < , P ∈ P Q , we want to show that if | Q | is large enough, there exist Dirichlet characters χ (mod Q ) such that (cid:13)(cid:13)(cid:13) arg χ ( P )2 π − θ P (cid:13)(cid:13)(cid:13) < δ ( P ∈ P Q ) . To this end we set g ( χ ) = Y P ∈ P Q (cid:12)(cid:12)(cid:12) f (cid:16) arg χ ( P )2 π − θ P (cid:17)(cid:12)(cid:12)(cid:12) , (14) NIVERSALITY OF L -FUNCTIONS OVER FUNCTION FIELDS 7 where f is defined as in Lemma 9. Lemma 10.
Let P Q be a fixed set of primes coprime to Q , and for P ∈ P Q letnumbers θ P ∈ [0 , be given. If g ( χ ) is defined as in (14) , and κ = R | f ( θ ) | dθ ,then X χ = χ (mod Q ) g ( χ ) = (cid:0) φ ( Q ) + O ( | P Q | K ) (cid:1) κ | P Q | . (15) Proof.
From (13) we see that Y P ∈ P Q f (cid:16) arg χ ( P )2 π − θ P (cid:17) = Y p ∈ P Q (cid:16) K X k =0 c k χ ( P k ) e ( − kθ P ) (cid:17) = X N ∈ N Q a N χ ( N ) , where N Q is the set of monic polynomials composed entirely of primes in P Q ,with multiplicities not exceeding K and a N = Y P ∈ P Q P k k N c k e ( − kθ P ) . Here the product is extended over all members of P Q , not just those dividing N .By Lemma 8, X χ (mod Q ) (cid:12)(cid:12)(cid:12)(cid:12) X N ∈ N Q a N χ ( N ) (cid:12)(cid:12)(cid:12)(cid:12) = φ ( Q ) X N ∈ N Q | a N | . We wish to remove the principal character from the sum on the left. By the Cauchy-Schwarz inequality, its contribution is (cid:12)(cid:12)(cid:12)(cid:12) X N ∈ N Q a N (cid:12)(cid:12)(cid:12)(cid:12) ≤ | N Q | X N ∈ N Q | a N | . The cardinality of N Q is clearly ≤ | P Q | K . Hence, X χ = χ (mod Q ) (cid:12)(cid:12)(cid:12)(cid:12) X N ∈ N Q a N χ ( N ) (cid:12)(cid:12)(cid:12)(cid:12) = ( φ ( Q ) + O ( | P Q | K )) X N ∈ N Q | a N | . Now
J. C. ANDRADE AND S. M. GONEK X N ∈ N Q | a N | = (cid:16) K X k =0 | c k | (cid:17) | P Q | = κ | P Q | , so we obtain the stated result. (cid:3) To apply Lemma 10 we need estimates for the parameter κ = Z | f ( θ ) | dθ. Since | f (0) | = (cid:12)(cid:12)(cid:12)(cid:12) K X k =0 c k (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( K + 1) K X k =0 | c k | , by Cauchy’s inequality, we find that K + 1 ≤ κ ≤ . (16)Suppose next that the primes in P Q are all of degree at most ρ . In order thatthe main term in (15) should be larger than the error term, we take ρ = log q (deg Q ) , K = h deg Q q (deg Q ) i , δ = (log q (deg Q )) deg Q . (17)Then | P Q | K ≪ ( q ρ /ρ ) K ≪ | Q | / . (18)The function g ( χ ) is large when the numbers (cid:13)(cid:13) arg χ ( P )2 π − θ P (cid:13)(cid:13) are small, but weneed a peak function that is positive only when all of these numbers are < δ . Wetherefore define h ( χ ) = Y P ∈ P Q (cid:12)(cid:12)(cid:12) f (cid:16) arg χ ( P )2 π − θ P (cid:17)(cid:12)(cid:12)(cid:12) − ε X P ∈ P Q Y P ∈ P Q P = P (cid:12)(cid:12)(cid:12) f (cid:16) arg χ ( P )2 π − θ P (cid:17)(cid:12)(cid:12)(cid:12) , (19)where ε = 4 q − πKδ . Note that if for some χ there is a prime P ∈ P Q for which NIVERSALITY OF L -FUNCTIONS OVER FUNCTION FIELDS 9 (cid:13)(cid:13)(cid:13) arg χ ( P )2 π − θ P (cid:13)(cid:13)(cid:13) > δ, then (cid:12)(cid:12)(cid:12) f (cid:16) arg χ ( P )2 π − θ P (cid:17)(cid:12)(cid:12)(cid:12) ≤ ε by Lemma 9, and so h ( χ ) ≤ . Lemma 11.
With f ( θ ) defined as in Lemma , h ( χ ) defined as in (19) , and param-eters chosen as in (17) , X χ = χ ( mod Q ) h ( χ ) = (1 + O (1 / deg Q )) κ | P Q | φ ( Q ) . (20) Proof.
By Lemma 10 and (18) we see that the sum above is = (cid:0) φ ( Q ) + O ( | Q | / ) (cid:1) κ | P Q | + O (cid:0) ε | P Q | φ ( Q ) κ | P Q |− (cid:1) . Since | P Q | ≪ q ρ /ρ ≤ deg Q , /κ ≪ deg Q by (16) and (17), and ε < (deg Q ) − ,it follows that this last error term is ≪ φ ( Q ) κ | P Q | / deg Q . Using (11) to estimatethe first O -term, we obtain (20). (cid:3) By means of these lemmas we see that there are characters χ ( mod Q ) at whichthe primes P with deg P ≤ ρ and P coprime to Q behave as we want. However,for our application to universality we also need to know that for these “good” char-acters, the primes of larger degree, say those with ρ < deg P ≤ z , do not alwaysmake too large a contribution. To eliminate this possibility we establish one finallemma. Lemma 12.
Let g ( χ ) be defined as in (14) where P Q is the set of all primes with deg P not exceeding ρ that are coprime to Q . For the primes with ρ < deg P ≤ z ,let the functions b P ( s ) have the property that | b P ( s ) | ≤ / | P | σ , where σ > .Then X χ ( mod Q ) g ( χ ) (cid:12)(cid:12)(cid:12)(cid:12) X ρ< deg P ≤ z b P ( s ) χ ( P ) (cid:12)(cid:12)(cid:12)(cid:12) ≪ φ ( Q ) κ | P Q | q ρ (1 − σ ) ρ . Proof.
Let N Q and a N be defined as in the proof of Lemma 10, and let the func-tions C M ( s ) be determined by the identity (cid:16) X N ∈ N Q a N χ ( N ) (cid:17)(cid:16) X x< deg P ≤ z b P ( s ) χ ( P ) (cid:17) = X M C M ( s ) χ ( M ) . If we assume the norm of Q is greater than the norm of every M , then by Lemma 8, X χ ( mod Q ) (cid:12)(cid:12)(cid:12) X M C M ( s ) χ ( M ) (cid:12)(cid:12)(cid:12) dt = φ ( Q ) X M | C M ( s ) | . We note that a monic polynomial M has at most one decomposition M = N P with N ∈ N Q and ρ < deg P ≤ z . In the main term we have X M | C M ( s ) | = (cid:16) X N ∈ N Q | a N | (cid:17)(cid:16) X ρ< deg P ≤ z | b P ( s ) | (cid:17) . Here the sum over N is κ | P Q | , and the sum over P is ≪ X ρ< deg P ≤ z | P | − σ ≪ q ρ (1 − σ ) ρ . Thus we have the result. (cid:3)
4. L
EMMAS ON APPROXIMATING ANALYTIC FUNCTIONS BY D IRICHLETPOLYNOMIALS
We remind the reader that a multiset is a set in which each element may berepeated a finite number of times. Our next lemma is a minor modification ofLemma 2.2 of Gonek [4].
Lemma 13.
Let Λ be an infinite multiset of real numbers whose counting functionsatisfies N Λ ( x ) = X λ ∈ Λ λ ≤ x ≪ e x , (21) and for any fixed c > N Λ ( x + c/x ) − N Λ ( x ) ≫ e x x . (22) Let C be a simply connected compact set in the strip / < σ < σ < andsuppose that F ( s ) is continuous on C and analytic in the interior of C . Then foreach µ > there is a ρ > µ such that if ρ ≥ ρ , there are numbers θ λ ∈ [0 , such that NIVERSALITY OF L -FUNCTIONS OVER FUNCTION FIELDS 11 max s ∈ C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F ( s ) − X λ ∈ Λ µ Let Q be as above. Let C be a simply connected compact set in thestrip / < σ < σ < and suppose that f ( s ) is continuous on C and analyticin the interior of C . Then for each µ > there is a ρ > µ such that if ρ ≥ ρ ,there are numbers θ P ∈ [0 , such that for all Q ∈ Q , max s ∈ C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( s ) − X µ< | P |≤ ρP ∤ Q =1 e ( θ P ) | P | s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ q − µ/ . Here ρ depends at most on σ , σ , C, Λ , f ( s ) , and µ , while the implied constantin the last inequality depends on at most σ , σ , C , and Λ . PROOF OF T HEOREM C is a compact subset of the open rectangle U with vertices , , i π log q , + i π log q whose complement in U is connected. Also, F ( s ) isanalytic on the interior of C , continuous on the boundary, and nonvanishing on C .It follows that there is an analytic branch of logarithm of F ( s ) on the interior of C that is continuous on the boundary of C . We denote it by f ( s ) . We wish to showthat for every ǫ > and monic polynomial Q of large enough degree, a positiveproportion of the φ ( Q ) Dirichlet characters χ (mod Q ) satisfy max s ∈ C | L ( s, χ ) − e f ( s ) | < ǫ. Let σ = 1 / d , where d = min s ∈ C ( σ − ) > . By Lemma 6 with theparameters K and Q large and to be chosen later, we have NIVERSALITY OF L -FUNCTIONS OVER FUNCTION FIELDS 13 log L ( s, χ ) = log P K ( s, χ ) + O (cid:16) deg QK q − dK (cid:17) , (24)where P K ( s, χ ) is defined in (10) and the implied constant depends on σ and q .Next let ≤ µ < ρ with ρ < K and write log P K ( s, χ ) = X deg P ≤ µ [ K/ deg P ] X j =1 χ ( P j ) j | P | js + X µ< deg P ≤ ρ χ ( P ) | P | s + X ρ ≤ deg P ≤ K χ ( P ) | P | s + X µ ≤ deg P ≤ K [ K/ deg P ] X j =2 χ ( P j ) j | P | js = f ( s, χ ) + f ( s, χ ) + f ( s, χ ) + f ( s, χ ) . (25)Clearly, for s ∈ C we have f ( s, χ ) ≪ X µ< deg P ≤ K | P | σ = X µ The first author is grateful to the Leverhulme Trust (RPG-2017-320) for its support through the research project grant “Moments of L -functionsin Function Fields and Random Matrix Theory". Both authors would like to thankthe Universities of Rochester and Exeter for their hospitality during the course ofthis work. R EFERENCES [1] J. C. Andrade, S. 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Weil, Sur les Courbes Algébriques et les Variétés qui s’en Déduisent (Hermann, Paris,1948).D EPARTMENT OF M ATHEMATICS , U NIVERSITY OF E XETER , N ORTH P ARK R OAD , E XETER EX4 4QF, UK E-mail address : [email protected] D EPARTMENT OF M ATHEMATICS , U NIVERSITY OF R OCHESTER , R OCHESTER , NY 14627,USA E-mail address ::