On Character Sums of Binary Quadratic Forms 2 Mei-Chu Chang 3 Abstract. We establish character sum bounds of the form χ(x 2 + ky 2 ) < τ H 2, a x a+h b y b+h where χ is a nontrivial character (mod ), 4 +ε < H <, and a, b < ε/2 H. As an alication, we obtain that given k Z\{0}, x 2 + k is a quadratic non-residue (mod ) for some x < 2 e +ɛ. Introduction. Let k be a nonzero integer. Let be a large rime and let H. We are interested in the character sum x,y χ(x2 + ky 2 ), where χ (mod q) is a nontrivial character, and x and y run over intervals of length H; say a x a + H and b y b + H, and a and b are less than ɛ H. The trivial bound for this character sum is H 2, and we seek an uer bound of the form H 2 δ for some δ > 0. Burgess [Bu3] considered such character sums, and obtained the desired H 2 δ estimate rovided H 3 +ɛ. Moreover, in the case that x 2 + ky 2 is irreducible (mod ) (i.e., k is a quadratic non-residue (mod )), Burgess obtained such cancelation in the wider range H 4 +ɛ. In this aer we obtain a corresonding result in the case that x 2 + ky 2 is reducible (mod ) ( i.e., k is a quadratic residue (mod )). More recisely, we rove Theorem. Given ε > 0, there is τ > 0 such that if is a sufficiently large rime and H is an integer satisfying we have 4 +ε < H <, (0.) 2000 Mathematics Subject Classification.Primary L40, L26; Secondary A07, B75. 2 Key words. character sums, quadratic residues, Burgess 3 Research artially financed by the National Science Foundation.
2 a x a+h b y b+h χ(x 2 + ky 2 ) < τ H 2 (0.2) for any nontrivial character χ (mod ) and arbitrary a, b < ε/2 H. Our argument is a variant of Burgess well-known method [Bu]. Following [Bu2], this estimate for binary forms allows us to deduce ( ) Corollary. Given k Z\{0}, we have = for some < x < 2 e +ɛ, for all ɛ > 0. x 2 +k In [Bu2], Burgess established this statement for x < 2 3 e +ɛ and for x < 2 e +ɛ rovided x 2 + k is assumed irreducible (mod ). Thus, both the theorem and the corollary are only new if k is a quadratic residue (mod ). The other case was treated by Burgess based on the following aroach. Recall that if k is a quadratic non-residue (mod ), then χ(x 2 +ky 2 ) is a character (mod ) of x + ωy with ω = k. Estimate (0.2) is then equivalent to bounding a character sum χ (z) (0.3) z B where B = {x + ωy : a x a + H, b y b + H} and χ is a nontrivial multilicative character of F 2. In [Bu5], Burgess established the desired bound for (0.3) assuming H 4 +ɛ. A more general result along these lines was obtained by A. Karacuba [Ka], for boxes B F n of the form B = {x 0 +x ω+ +x n ω n : r i x i r i +H, for i = 0,..., n }. Here ω is a root of a given olynomial of degree n, which is irreducible (mod ) and assuming again H > 4 +ɛ. Notation. [a, b] := {i Z : a i b}. x y means x y (mod ).. Estimate of the character sums. Denote f(x, y) = x 2 + ky 2, k Z\{0} and assume f(x, y) (x + λy)(x λy), λ F. (.)
Recall also that by Weil s theorem χ(x 2 + a) < c log (.2) max a F x I for χ a nontrivial character (mod ) and I [, ) an interval. Hence χ(f(x, y)) < c (log )H, (.3) a x a+h b y b+h and we may therefore assume H < 3 4. 3 Proof of the theorem. We let a = b = 0. The modification needed in the argument below to treat the situation a, b < ɛ/2 H are straightforward and left to the reader. As mentioned earlier, the basic technique is Burgess. Let and We introduce arameters Here the inequality is because of (0.). Thus for all 0 u, v D, 0 t M S := 0 x,y H χ(f(x, y)) = δ = ε 4. (.4) M = [ δ ] (.5) D = 3 H 3 < 2δ H. (.6) 0 x,y H Taking some subset D [0, D] 2, it follows that S 0 x,y H (u,v) D χ(f(x + ut, y + vt)) + O( δ H 2 ). (.7) M χ(f(x + ut, y + vt)) + O( δ H 2 ). (.8) t= Assume u + λv, u λv 0 for (u, v) D. By (.2), the first term in (.8) equals
4 where w = = 0 x,y H (u,v) D F w M t= (( x + λy )( x λy )) χ u + λv + t u λv + t M χ((ξ + t)(ζ + t)), t= { (x, y, u, v) [0, H] 2 D : x + λy u + λv = ξ and (.9) x λy u λv = ζ}. Let [ 0 ] r =. (.0) ε To estimate (.9), we follow the usual aroach of alying Hölder s inequality with suitable exonent 2r Z + and Weil s theorem later. Thus, (.9) is bounded by ( (w ) 2r 2r ) ( 2r M which is bounded by ( ) ( ( ) w r w 2 2r t i t= ( ξ χ((ξ + t)(ζ + t)) 2r) 2r, χ (ξ + t )... (ξ + t r ) ) ) 2 2r (ξ + t r+ )... (ξ + t 2r ) Here i =,..., 2r, t i [, M] and ξ F such that ξ + t r+,..., ξ + t 2r are nonzero. Now by Weil s theorem, (.9) is bounded by (H2 D ) r ( w 2 ) 2r ( r 2r M r 2 + M 2r (2r 2 ) 2 ) 2r. (.) (From the definition of w, we have w = H 2 D.) By (.4), (.5) and (.0), if ( 0 ) 40 3ε >, (.2) ε then > r 2r M r 2. Therefore, by (.8) and (.) (after canceling M), we have ( ) S < H 2 (H 2 D ) r w 2 2r 2r + O( δ H 2 ). (.3) Our next aim is to estimate w 2, which is the number of solutions of the following system of equations in F.
5 x + λy x 2 + λy 2 u + λv u 2 + λv 2 x λy x 2 λy 2, u λv u 2 λv 2 when x i, y i [0, H] and (u i, v i ) D for i =, 2. Define D = { [ D ] 2 (u, v) 2, D } : (u, v) = (u, k) = (v, k) =, u ± λv 0. (Here (u, v) denotes gcd(u, v).) Hence D D 2. Multilying and adding the equations in the above system, we get by (.2) (x 2 + ky 2 )(u 2 2 + kv 2 2) (x 2 2 + ky 2 2)(u 2 + kv 2 ) (.4) (x u + ky v )(u 2 2 + kv 2 2) (x 2 u 2 + ky 2 v 2 )(u 2 + kv 2 ). (.5) We imose on H, D the condition HD 3 < 8k. (.6) 2 Hence (.5) holds in Z and we have (x u + ky v )(u 2 2 + kv 2 2) = (x 2 u 2 + ky 2 v 2 )(u 2 + kv 2 ). (.7) Fix u, u 2, v, v 2 and let = gcd(u 2 + kv 2, u 2 2 + kv 2 2). Hence { u 2 + kv 2 = w u 2 2 + kv 2 2 = w 2, (.8) where (w, w 2 ) =. log a Since a rational integer a has at most log D factorizations of log log a the form x + yλ in Q(λ) with x, y [0, D], the equation u 2 + kv 2 = a has at most ex ( c log(d+ a ) log log(d+ a )) solutions (u, v) [0, D] 2. Therefore, given w, w 2,, the system (.8) has < ε solutions (u, v, u 2, v 2 ). It follows from (.7) that
6 for some t Z, satisfying { x u + ky v = tw x 2 u 2 + ky 2 v 2 = tw 2 (.9) t HD w + w 2. (.20) Let x, y, x 2, y 2 be some solution (other than x, y, x 2, y 2 ) of (.9) and (.4) with secified u, v, u 2, v 2 and t. Then { (x x )u = k(y y )v (x 2 x 2)u 2 = k(y 2 y 2 )v 2. (.2) Since (u, kv ) = = (u 2, kv 2 ), we get for some s, s 2 Z satisfying x x = s kv y y = s u x 2 x 2 = s 2 kv 2 y 2 y 2 = s 2 u 2. (.22) s i H D for i =, 2. (.23) Substituting (.22) in (.4) and (.8) yield the following equation in s, s 2 w 2 ( (x + s kv ) 2 + k(y s u ) 2) w ( (x 2 + s 2 kv 2 ) 2 + k(y 2 s 2 u 2 ) 2). Hence w w 2 (s 2 s 2 2) + 2w 2 (x v y u )s 2w (x 2v 2 y 2u 2 )s 2 0 (.24) and there are obviously at most c H solutions in (s D, s 2 ) satisfying (.23) and (.24). Summarizing, we showed that for given, w, w 2, the system of equations (.4) and (.5) has at most ε HD w + w 2 H D (.25) solutions in x, y, x 2, y 2. Notice that by (.8), ( w + w 2 ) D 2. Summing (.25) over, w, w 2 we obtain
7 ε H 2 Therefore D 2 w + w 2 D2 w + w 2 < ε H 2 D2 D 2 < ε H 2 D 2. w 2 < ε H 2 D 2, (.26) rovided H, D satisfy (.6). Substitute (.26) in (.3). By (0.) and (.6), we have S < H 2 ε2 /5. With some small modification of the roof of the theorem, we can also obtain the following more general statement. Theorem. Given ε > 0, there is τ > 0 such that if is a sufficiently large rime and H is an integer satisfying we have a x a+h b y b+h 4 +ε < H <, χ(x 2 + ky 2 ) < τ H 2 for any nontrivial character χ(mod ) and arbitrary a, b < ε/2 H. 2. An alication to quadratic non-residues. In this section we will rove the corollary. Let φ(x) = x 2 + k, with k Z\{0} and let be a large rime. Assume ( φ(x) ) = for x H. (2.) The roblem of estimating H = H() was considered in Burgess aer [Bu2]. (See also [Bu4].) We distinguish the following two cases. Case. k = l 2, l Z. Hence φ(x) = (x + l)(x l). In this case H < 4 e ( ) ( ) +ɛ. Indeed, if x+l x l = for x H, taking x = l, 3l, 5l..., gives that ( ) ( ) = constant for y < H2l. Hence =, which contradicts 2l y y
8 Burgess theorem [Bu] on the existence of quadratic non-residues in short intervals [, 4 e +ɛ ]. Case 2. k is not a square. We will follow the argument in [Bu2] with some adjustment. We may assume k > 0, since the case k < 0 is similar. (See [Bu2].) For readers convenience we state below the lemmas we use from [Bu2]. (See Lemmas, 3 in [Bu2].). For x, y Z, there exists a reresentation of n = x 2 + ky 2 r n = u 2 (vi 2 + k) α i, (2.2) i= for some r N, ositive integers u, v,..., v r all n and α i = ±. 2. Given < β < e, there is a constant M = M(β) > 0 such that if ( x 2 + ky 2 ) = for x 2 + ky 2 H, then for H sufficiently large and any rime > H β we have ( x 2 + ky 2 ) > MH β, x 2 +ky 2 H β where the sum is over all airs x, y (not necessarily integers) for which x + y k is an integer of Q( k). Since for k 3 (mod 4), x + y k is an algebraic integer of Q( k) if and only if x, y Z, the sum is over all x, y Z with x 2 + ky 2 H β. For k (mod 4) the ring of algebraic integers is generated by + k. Burgess showed that the inequality holds when 2 the sum is over all x, y Z such that x 2 + 4ky 2 H β. In both cases, the roofs of the theorem are identical, so we give only the former here. It follows from our assumtion (2.) and (2.2) that ( x 2 + ky 2 ) = if x 2 + ky 2 H. (2.3) Hence we may aly Burgess second lemma and get a contradiction, if we show that ( x 2 + ky 2 ) = O( δ H β ). (2.4) x 2 +ky 2 H β We divide the region enclosed by the ellise x 2 + ky 2 = H β into squares of length h with h > /4+ɛ. For those squares comletely lying in the ellise, we use Theorem to estimate the character sum.
For the others, we count the number of lattice oints and use the trivial bound. According to Theorem, it follows that (2.4) will hold rovided H β 2 > 4 +ɛ for some ɛ > 0. Therefore H < 2 e. Hence the corollary is roved. Acknowledgement. The author would like to thank the referee for helful comment. 9 References [Bu] D.A. Burgess, On character sums and rimitive roots, Proc LMS (3), 2, (962), 79-92. [Bu2], On the quadratic character of a olynomial, JLMS, 42, (967), 73-80. [Bu3], A note on character sums of binary quadratic forms, JLMS, 43 (968), 27-274. [Bu4], Direchlet characters and olynomials, Trudy Math Inst-Steklov, 32, (973). [Bu5] Character sums and rimitive roots in finite fields, Proc. London Math. Soc. (3) 7 (967), -25. [Ka] A. A. Karacuba, Estimates of character sums, Izv. Akad Nauk SSR, Ser Mat Tom, 34, (970), N. Deartment Of Mathematics, University Of California, Riverside, CA 9252 E-mail address: mcc@math.ucr.edu