A NOTE ON CHARACTER SUMS IN FINITE FIELDS. 1. Introduction

Transcription

1 A NOTE ON CHARACTER SUMS IN FINITE FIELDS ABHISHEK BHOWMICK, THÁI HOÀNG LÊ, AND YU-RU LIU Abstract. We prove a character sum estimate in F q[t] and answer a question of Shparlinski. Shparlinski [5] asks the following.. Introduction Problem. [5, Problem 4] Let m Z and χ be a non-principal character modulo m. Under the Generalized Riemann Hypothesis, obtain an estimate of the form N χk N /2 m o k= for any N, with an explicit expression for m o. Though it has never been explicitly written down, such a bound is presumably well-known among analytic number theorists due to its connection with upper bounds for L-functions. Indeed, let Ls, χ be the Dirichlet L-function associated with χ and s = σ + it. First we assume that χ is primitive. Then conditionally under GRH, we have the bound [3, Exercise 8, Section 3.2] log m t Ls, χ exp C log log m t for some absolute constant C > 0 and uniformly for /2 σ 3/2 and t 2. Using the bound and a standard contour integral one can show that N χk N /2 log m exp C 2 2 log log m k= for some absolute constant C 2 > 0, when χ is primitive. If χ is induced by a character χ modulo r with r m then N χk = k= N k= k,r= χ k = d m r µdχ d k N/d χ k. Bounding this trivially, together with the fact that the number of prime factors of n is O, it follows that we have a bound of type 2 as well when χ is not primitive. log The purpose of this note, however, is to obtain a bound similar to 2 in the polynomial ring F q [t]. Let Q F q [t] be a polynomial of degree n. A Dirichlet character χ modulo Q is a character on the multiplicative group F q [t]/q, which can be extended to a function on

2 2 ABHISHEK BHOWMICK, THÁI HOÀNG LÊ, AND YU-RU LIU all of F q [t] by periodicity and by setting χf = 0 for all f, Q. If Q is irreducible then χ is a character on the field F q n. Shparlinski private communication also asks an F q [t]-analog of Problem : Problem 2. Let Q F q [t], deg Q = n > 0 and χ be a non-principal character modulo Q. Let A d be the set of all monic polynomials of degree exactly d in F q [t]. Obtain an estimate of the form χf q d 2 +on f A d for any d, with an explicit expression for on. We prove the following explicit estimate. Theorem. Let Q F q [t], deg Q = n > 0 and χ be a non-principal not necessarily primitive character modulo Q. If n 0 4 log and log q, then we have χf q d d log + 2 e 8qn log 2 n. 3 Note that we have the analogies N q d and m q n between Z and F q [t]. The bound 2 corresponds to q d 2 exp Cn in F q [t]. Thus 3 is stronger than 2 when d is small e.g., n when d but weaker than 2 when d is large e.g., when d n. It is an interesting problem to see if in F q [t] we can achieve, or even beat 2 for all d. Nevertheless, our proof of 3 is much simpler than the proof in the integers and is potentially still useful in applications. The constants 8 and 0 4 in 3 can certainly be improved, but we do not attempt to do so to keep our estimates clean. We also record another similar character sum estimate, which might be of interest. Theorem 2. Let Q F q [t], deg Q = n > 0 and χ be a non-principal not necessarily primitive character modulo Q. Under the hypotheses of Theorem, we have µfχf q d d log + 2 e 8qn log 2 n. Here µ is the Möbius function on F q [t] defined by { µf = k, where k is the number of monic irreducible factors of f, if f is square-free, 0, otherwise. The difference between Theorems and Theorem 2 is that, while the former is trivial when d n, the latter is non-trivial for all d. It is useful to compare 3 with other character sum estimates in F q [t]. In [], the first two authors proved that for any 2 log q n r d n, we have χf qd ρ d q O d log d r 2 + O nq d r/2. 4 r

3 A NOTE ON CHARACTER SUMS IN FINITE FIELDS 3 Here ρu is the Dickman function, i.e., the unique continuous function satisfying ρu = u u u ρtdt for all u >, with initial condition ρu = for 0 u. We have the asymptotic estimate ρu = u u+o as u. The difference between the inequalities 4 and 3 is that while 4 is non-trivial in a wider range of d, 3 provides a better saving when d is large. Indeed, if one wants to match the bound in 3, the second term in RHS of 4 requires one to choose r close to d. But then the contribution of the first term in RHS of 4 becomes significant. Hence 3 does not follow from 4. One can also compare 3 with the analog of the Pólya-Vinogradov inequality in F q [t] [2, Proposition 2.], which states that for any d, n χf 2q Proofs of Theorems and 2 Similar to the integers, we deduce Theorems and 2 from a bound for L-functions near the critical line Proposition 3. We begin with some facts about L-functions in F q [t]. Throughout this paper, f will stand for a monic polynomial and P will stand for a monic, irreducible polynomial. Fix Q of degree n and a non-principal character χ modulo Q. Put Ad, χ = f A d χf. Then the L-function assuming Res > associated with χ is Ls, χ = f It is even more convenient to put Lz, χ = χf q s deg f = Ad, χq ds. d=0 Ad, χz d. 5 d=0 Clearly Ls, χ = Lq s, χ for any Res >. Since Ad, χ = 0 whenever d n [4, Proposition 4.3], Lz, χ is a polynomial of degree at most n, and in particular an entire function. We have the Euler product formula Lz, χ = P χp z deg P 6 whenever z < /q. In the same range of z, we also have Lz, χ = χp z deg P = µfχfz deg f. 7 P f

4 4 ABHISHEK BHOWMICK, THÁI HOÀNG LÊ, AND YU-RU LIU The Generalized Riemann Hypothesis states that all roots of Lz, χ have modulus q /2 or. In other words, we can write Lz, χ = D α i z 8 i= where α i = q /2 or for any q, for some D n. In particular, 7 remains valid when z < q /2. Our results are deduced from the following bound for Lz, χ near the circle z = q /2. Proposition 3. Let χ be a not necessarily primitive character modulo Q with deg Q = n. Let q 3/4 R < q /2. For any w = R, we have n 2 R L q 7 +, 9 where L = 2 log q n. Lw, χ exp Lw, χ exp n 2 R L q 7 + L q /2 R L q /2 R The correct choice of R will be made later in applications., 0 Proof. By taking the logarithmic derivatives of 6 and 8, we have two different expressions for L z,χ Lz,χ. Namely, we have L z, χ Lz, χ = a l z l where according to 8, while a l = a l = deg f=l l= D αi l i= Λfχf 2 according to 6. Here Λ is the von Mangoldt function on F q [t] defined by { deg P, if f = P Λf = k for some monic irreducible P and k, 0, otherwise. From we have and from 2 we have a l nq l/2 3 a l Λf = q l. 4 deg f=l

5 A NOTE ON CHARACTER SUMS IN FINITE FIELDS 5 Recall that L = 2 log q n. For l L we use the bound 3 and l < L we use the bound 4. Therefore, for any z, we have L L z, χ Lz, χ q l z l + nq l/2 z l. 5 Since L0, χ =, by integrating 5 along the line from 0 to w, we have l= l=l log Lw, χ L l= Rq l l + l=l n Rq/2 l. 6 l The second sum in 6 can be bounded by n L Rq /2 l n L RL q L 2 l=l Rq /2 n2 R L q L since q L 2 n. As for the first sum in 6, we bound it crudely by L Rq l Rq L Rq k l= k=0 since qr q /4 2 /4. By combining 7 and 8, we have log Lw, χ 7n 2 R L q + n2 R L q L. 7 Rq/2 n2 R L qr 7n2 R L q 8 Rq /2. from which both 9 and 0 follow. Our estimates above are crude. Even if we were more careful, this would result only in better constants, and the shape of the bounds would not be changed. Proof of Theorems and 2. In what follows, C R denotes the circle centered at 0 with radius R. Put R = q /2 ɛ, for some 0 < ɛ 4 to be chosen later. We also make the additional assumption that ɛ log q. From 9, for z C R, we have Lz, χ exp qn 2 R L 7 + L q /2 R exp qn 2ɛ 7 + L q ɛ exp qn 2ɛ Lɛ log q exp qn 2ɛ 7 +. ɛ

6 6 ABHISHEK BHOWMICK, THÁI HOÀNG LÊ, AND YU-RU LIU ɛ log q e ɛ log q The bound 9 follows from the fact that 2 if ɛ log q, which is a e x consequence of the fact that the function is increasing on 0,. From 5 we see that e x Ad, χ = Lz, χz d dz 2πi C R We now make the choice ɛ = We have qn 7 2ɛ + ɛ log = qn log 2 n max Lz, χ R d C R exp qn 2ɛ 7 + q /2+ɛd. 20 ɛ, then ɛ /4 if n 04. Also, by hypothesis ɛ 7 + log 8qn log 2 n if n 04. This implies Theorem. Using 7 instead of 5, Theorem 2 follows in exactly the same way. Acknowledgement. The authors would like to thank Micah Milinovich and Igor Shparlinski for helpful discussions and the referees for useful and detailed comments which help to improve the presentation of the paper. Part of this work was done when the second author was supported by the Fondation Mathématique Jacques Hadamard and visiting the Taida Institute for Mathematical Sciences, and he gratefully acknowledges their support and hospitality. References [] A. Bhowmick, T. H. Lê, On primitive elements in finite fields of low characteristic, Finite Fields and Their Applications [2] C-N. Hsu, Estimates for Coefficients of L-Functions for Function Fields, Finite Fields and Their Applications, Volume 5, Issue, January 999, Pages [3] H. L. Montgomery, R. C. Vaughan, Multiplicative Number Theory I: Classical Theory, Cambridge University Press, [4] M. Rosen, Number theory in function fields, volume 20 of Graduate Texts in Mathematics. Springer- Verlag, New York, [5] I. Shparlinski, Open Problems on Exponential and Character Sums, igorshparlinski/charsumprojects.pdf A. Bhowmick, Department of Computer Science, The University of Texas at Austin, TX address: bhowmick@cs.utexas.edu T. H. Lê, Department of Mathematics, The University of Mississippi, University, MS address: leth@olemiss.edu Y.-R. Liu, Department of Pure Mathematics, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G address: yrliu@math.uwaterloo.ca log q.