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Moments of Generalized Quadratic Gauss Sums Weighted by L-Functions1

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Moments of Generalized Quadratic Gauss Sums Weighted by L-Functions1 Journal of Number Theory 92, 304–314 (2002) doi:10.1006/jnth.2001.2715, available online at http://www.idealibrary.comon Moments of Generalized Quadratic Gauss Sums Weighted by L-Functions1 Zhang Wenpeng Research Center for Basic Science, Xi’an Jiaotong University, Xi’an, Shaanxi, People’s Republic of China Communicated by D. Goss Received May 5, 2000 The main purpose of this paper is using estimates for character sums and analytic methods to study the second, fourth, and sixth order moments of generalized quadratic Gauss sums weighted by L-functions. Three asymptotic formulae are obtained. © 2002 Elsevier Science (USA) Key Words: general quadratic Gauss sums; L-functions; asymptotic formula. 1. INTRODUCTION Let q \ 2 be an integer; q denotes a Dirichlet character modulo q. For any integer n, we define the general quadratic Gauss sums G(n, q;q)as q na2 G(n, q; q)= C q(a) e 1 2, a=1 q where e(y)=e2piy. This sum is important because it is a generalization of the classical quadratic Gauss sum. But about the properties of G(n, q;q), we know very little at present. The value of |G(n, q;q)| is irregular as q varies. One can only get some upper bound estimates. For example, for any integer n with (n, q)=1, from the general result of Cochrane and Zheng [8] we can deduce 1 w(q) |G(n, q;q)|[ 2 q 2, where w(q) denotes the number of distinct prime divisors of q; the case where q is prime is due to Weil [9]. 1 This work is supported by the N.S.F. and the P.N.S.F. of P. R. China. 304 0022-314X/02 $35.00 © 2002 Elsevier Science (USA) ⁄ All rights reserved. GENERALIZED QUADRATIC GAUSS SUMS 305 In this paper we show that G(n, q;p)enjoys many good weighted mean value properties. For convenience, in the following we always suppose that p denotes an odd prime, L(s, q) the Dirichlet L-function corresponding to the character q mod p, and that G(n; q) denotes the classical quadratic Gauss sum. We use estimates for character sums and the analytic methods to prove the following three results: Theorem 1. For any integer n with (n, p)=1, we have the asymptotic formula 3 2 2 2 C |G(n, q;p)| · |L(1, q)|=C · p +O(p 2 ln p), q ] q0 where r R2S2 R4S2 R2mS2 s C=D 1 2 m p 1+ + +···+ +··· 42 ·p2 44 ·p4 42m ·p2m ; is a constant, q ] q0 denotes the summation over all nonprincipal characters < 2m 2 modulo p, p denotes the product over all primes, and ( m )=(2m)!/(m!) . Theorem 2. For any integer n with (n, p)=1, we have the asymptotic formula 5 4 3 2 C |G(n, q;p)| · |L(1, q)|=3·C·p +O(p 2 ln p). q ] q0 Theorem 3. Let p be an odd prime with p — 3 mod 4. Then for any fixed positive integer n with (n, p)=1, we have the asymptotic formula 7 6 4 2 C |G(n, q;p)| · |L(1, q)|=10 ·C·p +O(p 2 ln p). q ] q0 Let n be any integer with (n, p)=1. Then using our methods we can easily deduce the identities n (p−1) 53p2 − 6p − 1+4 1 2 `p6, if p — 1 mod 4; 4 ˛ C |G(n, q;p)|= p q mod p (p − 1)(3p2 − 6p − 1), if p — 3 mod 4, 306 ZHANG WENPENG and C |G(n, q;p)|6=(p − 1)(10p3 − 25p2 −4p−1), if p — 3 mod 4, q mod p n where (p) is the Legendre symbol. For a general integer m \ 3, whether there exists an asymptotic formula for C |G(n, q;p)|2m and C |G(n, q;p)|2m |L(1, q)| q mod p q ] q0 is an unsolved problem. We believe that it is true and that we even have the following. Conjecture. For all positive integer m, C |G(n, q;p)|2m |L(1, q)| ’ C· C |G(n, q;p)|2m,pQ +., q ] q0 q mod p where C is the same as in Theorem 1. 2. SOME LEMMAS In order to complete the proof of the theorems, we need the following lemmas. Lemma 1. For any odd prime p, we have the estimate p−1 C : C q(a) |L(1, q)|:=O(p ln p). a=1 q ] q0 Proof. Let N=p3/2, q be a nonprincipal character mod p, and ; A(q, y)= N<n[ y q(n). Then by Abel’s identity and the Po´lya–Vinogradov inequality we have q +. q C (n) F A( ,y) L(1, q)= + 2 dy n [ N n N y q(n) ln p = C +O 1 2. n [ N n p GENERALIZED QUADRATIC GAUSS SUMS 307 So that q(n) ln p (1) |L(1, q)|=: C :+O 1 2. n [ N n p On the other hand, let r(n) be a multiplicative function defined by R2aS a r(pa)= and r(1)=1, 4a where p is a prime and a is any positive integer. For this number-theoretic function r(n), it is easily proved that (see Lemma 1 of [6]) n C r(d) · r 1 2=1 d|n d and q(n) r(n) 2 q(nm) r(m) r(n) (2) 1 C 2 = C C n [ N n m [ N n [ N mn q(n) q(n) r(n, N) = C + C , n [ N n N<n[ N2 n where n r(n, N)= C r(d) · r 1 2. d|n d n d, d [ N From (2), Cauchy’s inequality, and the orthogonality relationships for character sums p−1, if n — 1 mod p; C q(n)=˛ q mod p 0, otherwise, and p−1 p−1, if q=q0 ; C q(a)=˛ a=1 0, otherwise, 308 ZHANG WENPENG we have the estimates p−1 q(n) q(n) r(n) 2 (3) C : C q(a) 1: C : − : C : 2: a=1 q n [ N n n [ N n 1 p−1 2 2 q(n) q(n) r(n) 2 [ p1/2 · 5 C : C q(a) 1: C : − : C : 2:6 a=1 q n [ N n n [ N n 1 2 q(n) q(n) q(n) r(n, N) 2 [ p·5C 1: C : − : C + C :2 6 q n [ N n n [ N n N<n[ N2 n 1 2 q(n) r(n, N) 2 [ p·5C : C : 6 q N<n[ N2 n 1 r(m, N) · r(n, N) 2 [ p3/2 · 1 C C 2 N<m[ N2 N<n[ N2 mn m — n mod p ° p·ln p and p−1 q(n) r(n) 2 p−1 r(m) r(n) (4) C : C q(a) : C : :=(p−1) C C C a=1 q n [ N n a=1 m [ N n [ N mn am — n mod p r(m) 2 =(p−1)·1 C 2 ° p ln p. m [ N m Combining (1), (3), and (4) we obtain p−1 C : C q(a) |L(1, q)| : a=1 q ] q0 p−1 q(n) = C : C q(a) : C :+O(ln p) : a=1 q ] q0 n [ N n p−1 q(n) = C : C q(a) : C ::+O(p ln p) a=1 q n [ N n p−1 q(n) q(n) r(n) 2 ° C : C q(a) 1: C : − : C : 2: a=1 q n [ N n n [ N n p−1 q(n) r(n) 2 + C : C q(a) : C : :+p ln p a=1 q n [ N n ° p·ln p. This proves Lemma 1. L GENERALIZED QUADRATIC GAUSS SUMS 309 Lemma 2. For any odd prime p, we have the asymptotic formula 1 1 CŒ 2 |L(1, q)|=2 · C · p+O(p · ln p), q(−1)=1 where R2S2 R4S2 R2nS2 . r2(n) r s C= C =D 1 2 n n2 1+ + +···+ +··· n=1 p 42 ·p2 44 ·p4 42n ·p2n ;− is an absolute constant, and q(−1)=1 denotes the summation over all nonprincipal even characters mod p. Proof. Let N=p3/2. Note the orthogonality relationship for character sums 1 2 (p − 1), if n — ±1mod p; C q(n)=˛ q(−1)=1 0, otherwise. By (1), (2), and the method of proving Lemma 1 we have q(n) CŒ |L(1, q)|= CŒ : C :+O(ln p) q(−1)=1 q(−1)=1 n [ N n q(n) r(n) 2 = C : C : +O(ln p) q(−1)=1 n [ N n q(n) q(n) r(n) 2 + C 1: C : − : C : 2 q(−1)=1 n [ N n n [ N n 1 r(m) · r(n) = (p−1)· C C +O(ln p) 2 m [ N n [ N mn m — ±n mod p q(n) · r(n, N) +O 1C : C :2 q N<n[ N2 n 1 . r2(n) C 2 = p· 2 +O(ln p) 2 n=1 n (n, p)=1 1 2 q(n) · r(n, N) 2 +O 1p1/2 1C : C : 2 2 q N<n[ N2 n 310 ZHANG WENPENG . 2 1 r (n) 1 C 2 = p· 2 +O(p · ln p) 2 n=1 n 1 1 = · C · p+O(p 2 · ln p).
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