On Character Sums of Binary Quadratic Forms 1 2. Mei-Chu Chang 3. Abstract. We establish character sum bounds of the form.

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1 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 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. (.)

3 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 < 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 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 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 kv 2 2) (x ky 2 2)(u 2 + kv 2 ) (.4) (x u + ky v )(u 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 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 kv 2 2). Hence { u 2 + kv 2 = w u 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 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 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 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.

9 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), [Bu2], On the quadratic character of a olynomial, JLMS, 42, (967), [Bu3], A note on character sums of binary quadratic forms, JLMS, 43 (968), [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 address: mcc@math.ucr.edu