e Scientific World Journal, Article ID 139725, ages htt://dx.doi.org/10.1155/201/139725 Research Article A New Sum Analogous to Gauss Sums and Its Fourth Power Mean Shaofeng Ru 1 and Weneng Zhang 2 1 School of Economics & Management, Northwest University, Xi an 710127, China 2 Deartment of Mathematics, Northwest University, Xi an 710127, China Corresondence should be addressed to Weneng Zhang; wzhang@nwu.edu.cn Received 19 January 201; Acceted 12 May 201; Published 21 May 201 Academic Editor: Paolo Vannucci Coyright 201 S. Ru and W. Zhang. This is an oen access article distributed under the Creative Commons Attribution License, which ermits unrestricted use, distribution, and reroduction in any medium, rovided the original work is roerly cited. The main urose of this aer is to use the analytic methods and the roerties of Gauss sums to study the comutational roblem of one kind of new sum analogous to Gauss sums and give an interesting fourth ower mean and a shar uer bound estimate for it. 1. Introduction Let q 3 be an integer, and let χ be a Dirichlet character mod q. Thenforanyintegern, famous Gauss sum G(χ, n is defined as follows: G (χ, n q a1 χ (a e( na, (1 q where e(y e 2πiy. This sum lays a very imortant role in the study of analytic number theory; many famous number theoretic roblems are closely related to it. For examle, the distribution of rimes, Goldbach roblem, the estimate of character sums, and the roerties of Dirichlet L-functions are some good examles. It is clear that if χ is a rimitive Dirichlet character mod q, then we have G(χ, n χ(n G(χ, 1 χ(n τ(χ and τ(χ q.manyroertiesofg(χ, n and τ(χ can be found in [1 3]. In this aer, we introduce a new sum analogous to Gauss sums as follows: G (χ, b, c, m; q q 1 a0 χ (a 2 +ba+c e( ma q, (2 where χ is a character mod q, and b, c, and m are any integers with (m, q 1. The main urose of this aer is to study the roerties of G(χ, b, c, m; q, such as the following two roblems: (A giving an uer bound estimate of G(χ, b, c, m; q; (B the roblem of whether there exists a comutational formula for the 2kth ower mean q 1 q 1 c0a0 χ(a 2 +ba+c e( ma q 2k, k 2. (3 It seems that no one has studied these roblems yet; at least we have not seen any related results in the existing literature. The roblems are interesting, because there exists a close relationshi between the sum G(χ, b, c, m; q and the generalized Kloosterman sums; they can also hel us to further understand the roerties of hybrid mean value of an exonential sum and character of a olynomial. For general integer q>3,these two roblems seem to be hard to make rogress. But if q>2is a rime and k2, then we can rove the following two conclusions. Theorem 1. Let be an odd rime and χ any nonrincial character mod. Thenforanyintegersb, c, andm with (m, 1, one has the estimate χ(a 2 +ba+c e( ma a0 2. (
2 The Scientific World Journal Theorem 2. Let be an odd rime and χ any nonrincial character mod.thenforanyintegersb and m with (m, 1, one has the identity c0a0 χ(a 2 +ba+c e( ma { 23 3 2 3, if χ is the Legendre symbol mod ; 2 3 6 2, if χ is a nonreal character mod. (5 Some Notes. Ifχχ 0 is the rincial character mod, then note that χ 0 (a 2 +ba+c0or 1 and G(χ 0,b,c,m; 3. So,inthiscase,theresultistrivial;wedonotneedtodiscuss roblems (A and (B. For any integer k 3, whether there exists an exact comutational formula for the 2kth ower mean Lemma. Let be an odd rime and χ any nonrincial Dirichlet character mod. Thenforanyintegersb and c, one has the identity a0χ(a 2 +ba+c e( a r1 χχ 2 (r e( (c b2 r 16 r, where r denotes the solution of the congruence equation r x 1 mod. Proof. Since χ is a nonrincial Dirichlet character mod, from Lemma 3 and the roerties of Gauss sums we have (9 c0a0 χ(a 2 +ba+c e( ma 2k, (6 is an oen roblem. Furthermore, for general integer q > 3, whether there exists a nontrivial uer bound estimate for G(χ, b, c, m; q is also an interesting roblem. 2. Several Lemmas In this section, we will give several lemmas, which are necessary in the roof of our theorems. Hereinafter, we will use many roerties of character sums and Gauss sums; all of thesecanbefoundin[1, ]. So they will not be reeated here. First we have the following. Lemma 3. Let be an odd rime; then, for any integer c with (c, 1, one has the identity a0 e ( ca2 ( c τ(χ 2, (7 where χ 2 ( /denotes the Legendre symbol. Proof. From the roerties of Gauss sums and quadratic residue mod we have a0 e( ca2 1+ e( ca2 a1 1+ a1 a0 This roves Lemma 3. (1 + ( a e (ca e( ca + ( a e(ca a1 ( c ( a a1 e(a (c τ(χ 2. (8 a0 χ(a 2 +ba+c e( a 1 τ(χ χ (r e( r(a2 +ba+c e( a a0 r1 1 τ(χ χ (r a0 r1 e( ra2 + (br + 1 a+cr 1 τ(χ χ (r a0 r1 e( r(2a + r (br + 12 +cr r(br + 1 2 1 τ(χ χ (r u0 r1 e( ru2 + (c b 2 r 2 b r 1 τ(χ χ (r ( r τ(χ 2 r1 e( (c b2 r 2 b r τ(χ 2 τ(χ χ ( e( 2 b e( (c b2 r 16 r. χ (r ( r r1 (10
The Scientific World Journal 3 For any nonrincial character χ mod, we have τ(χ.so,from(10 and noting that χ( e( 2 b/ 1, we have So without loss of generality we can assume that m1.now we rove Theorem 1.FromLemmas and 5 we have a0χ(a 2 +ba+c e( a r1 This roves Lemma. χχ 2 (r e( (c b2 r 16 r. (11 Lemma 5. Let be an odd rime and χ any Dirichlet character mod.thenforanyintegersm and n, one has the estimate a0χ(a 2 +ba+c e( a r1 χχ 2 (r e( (c b2 r 16 r 2 (c b 2, 16, 1/2 2. (15 a1 χ (a e( ma + na 2 (m, n, 1/2, (12 where (m,n, denotes the greatest common divisor of m, n, and. Proof. ThisestimateisbyWeil[5], Chowla [6], Malyshev [7], and Estermann [8] with some minor modifications. This roves Theorem 1. Now we rove Theorem 2. From the roerties of a comlete residue system mod we know that if c asses through a comlete residue system mod, thenc b 2 also asses through a comlete residue system mod. So from Lemma we have Lemma 6. Let be an odd rime; then, for any integer n with (n, 1, one has the calculating formula m1a1 χ(a e ( 2 3 3 2 { 3 1, 3 { 3 8 2, { 2 (2 7, ma + na Proof. See [9]orCorollary2of[10]. if χ is the rincial character mod ; if χ is the Legendre ssymbolmod ; if χ is a nonreal character mod. (13 c0 a0χ(a 2 +ba+c e( a c0 r1 m0r1 m0r1 χχ 2 (r e ( (c b2 r 16 r χχ 2 (r e ( χχ 2 (r e ( mr 16 r. (16 3. Proof of the Theorems In this section, we will comlete the roof of our theorems. First note that if a asses through a reduced residue system mod,then ma also asses through a reduced residue system mod,if(m, 1. From these roerties we have χ(a 2 +ba+c e( ma a0 a0χ(m 2 a 2 + mba+c e( a χ 2 (m a0χ(a 2 +mba+cm 2 e( a a0χ(a 2 +mba+cm 2 e( a. (1 If χ χ 2 is the Legendre symbol, then χχ 2 χ 0 is the rincial character mod,sofrom(16andlemma6 we have c0 a0χ(a 2 +ba+c e( a m0r1 1+ χχ 2 (r e ( m1r1 e( 1+2 3 3 2 3 2 3 3 2 3. (17
The Scientific World Journal If χ isnotarealcharactermod,thenfrom(16andlemma6 we have c0 a0χ(a 2 +ba+c e( a m0r1 χχ 2 (r e ( 2 (r e ( r1χχ r + m1r1 χχ 2 (r e ( 2 + 2 (2 7 2 2 ( 3. (18 [7] A. V. Malyshev, A generalization of Kloosterman sums and their estimates, Vestnik Leningrad University,vol.15,.59 75, 1960 (Russian. [8] T. Estermann, On Kloostermann s sums, Mathematica,vol.8,.83 86,1961. [9] W. Zhang, On the fourth ower mean of the general Kloosterman sums, Indian Pure and Alied Mathematics, vol.35,no.2,.237 22,200. [10] L. Jianghua and L. Yanni, Some new identities involving Gauss sums and general Kloosterman sums, Acta Mathematica Sinica, vol. 56,. 13 18, 2013 (Chinese. Now combining (1, (17, and (18 we may immediately deduce the identity c0a0 χ(a 2 +ba+c e( ma { 23 3 2 3, if χ is the Legendre symbol mod ; 2 3 6 2, if χ is a nonreal character mod. (19 This comletes the roof of Theorem 2. Conflict of Interests The authors declare that there is no conflict of interests regarding the ublication of this aer. Acknowledgments The authors would like to thank the referee for the very helful and detailed comments, which have significantly imroved the resentation of this aer. This work is suorted by the P.S.F. (2013JZ001 and N.S.F. (11371291 of China. References [1] T. M. Aostol, Introduction to Analytic Number Theory, Sringer,NewYork,NY,USA,1976. [2] K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Sringer, New York, NY, USA, 1982. [3] M.B.Nathanson,Additive Number Theory, the Classical Bases, vol. 16 of Graduate Texts in Mathematics, Sringer, 1996. [] E. Alkan, Values of Dirichlet L-functions, Gauss sums and trigonometric sums, Ramanujan Journal,vol.26,no.3,.375 398, 2011. [5] A. Weil, On some exonential sums, Proceedings of the National Academy of Sciences, vol. 3,. 20 207, 199. [6] S. Chowla, On Kloosterman s Sum, Det Kongelige Norske Videnskabers Selskabs Forhandlinger,vol.0,.70 72,1967.
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