Research Article New Mixed Exponential Sums and Their Application

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1 Hindawi Publishing Cororation Alied Mathematics, Article ID 51053, ages htt://dx.doi.org/ /01/51053 Research Article New Mixed Exonential Sums and Their Alication Yu Zhan 1 and Xiaoxue Li 1 DeartmentofScience,HetaoCollege,Bayannur015000,China School of Mathematics, Northwest University, Xi an, Shaanxi 71017, China Corresondence should be addressed to Xiaoxue Li; lxx00701@163.com Received 6 May 01; Acceted 3 May 01; Published 19 June 01 Academic Editor: Ashraf Zenkour Coyright 01 Y. Zhan and X. Li. 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 introduce a new mixed exonential sums and then use the analytic methods and the roerties of Gauss sums to study the comutational roblems of the mean value involving these sums and give an interesting comutational formula and a shar uer bound estimate for these mixed exonential sums. As an alication, we give a new asymtotic formula for the fourth ower mean of Dirichlet L-functions with the weight of these mixed exonential sums. 1. Introduction Let q 3 be an integer, and let χ be a Dirichlet character mod q. Then, for any integer n, the famous Gauss sums G(χ, n are defined as follows: G (χ, n = q χ (a e( na q, (1 where e(y = e πiy. This sum and the other exonential sums (such as Kloostermansumslayveryimortantroleinthestudyof analytic number theory, and 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. From the roerties of characters mod q,weknownthat if χ is a rimitive character mod q, theng(χ, n = χ(n G(χ, 1 χ(n τ(χ,and τ(χ = q. Many other roerties of G(χ, n and τ(χ canalsobefoundin[1 ]. In this aer, we introduce new mixed exonential sums as follows: G (χ, c, m, n; q q 1 q 1 = χ(a +ab+b +c e( a=0, q where c, m,andn are any integers. We will study the arithmetical roerties of G(χ, c, m, n; q. About this roblem, it seems that none ( hasstudiedityet;atleastwehavenotseenanyrelatedresults before. The roblem is interesting, because this sum has a close relationshi with the general Kloosterman sums, and itisalsoanalogoustofamousgausssums,soitmusthave many roerties similar to these sums. It can also hel us to further understand and study Kloosterman sums and Gauss sums. The main urose of this aer is using the analytic method and the roerties of Gauss sums to study the fourth ower mean of G(χ, c, m, n; and its uer bound estimate androvethefollowingthreeconclusions. Theorem 1. Let be an odd rime; let χ be any nonrincial character mod. Then, for any integers c, m, andn, onehas the estimate 1 1 χ(a +ab+b +c e( a=0. (3 Theorem. Let be an odd rime; let χ be any nonrincial character mod.then,foranyintegersm and n, one has the identity c=0a=0 χ(a +ab+b +c e( ={ 35 8, if χ is the Legendre symbol mod ; ( 7, if χ is a comlex character mod. (

2 Alied Mathematics Theorem 3. Let be an odd rime. Then, for any integers c, m,andn with (c, m +n mn, =, one has the asymtotic formula 1 1 χ(a +ab+b +c e( χ mod a=0 L(1, χ = 5 7 π 3 +O( 5/ ln ln ex ( ln, where χ 0 is the rincial character mod, (l, m, n denotes the greatest common divisor of l, m,andn,andex(y = e y. In Theorem 1, we only discussed the case, in which there exist two variables. For general case (with k( 3variable, whether there exists a shar estimate for the sums is an interesting roblem. Let k 3; whether there exists an exact comutational formula for the kth ower mean, c=0a=0 is also an oen roblem.. Several Lemmas χ(a +ab+b +c e( (5 k, (6 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, Kloosterman sums, and Gauss sums; all of these can be found in [1, 5 7], so they willnotbereeatedhere.first,wehavethefollowing. Lemma. Let be an odd rime; then, for any integers c, m, and n, one has the identity 1 1 χ(a +ab+b +c e( a=0 1 = χ (r e( cr r(m +n mn, where r denotes the solution of the congruence equation r x 1 mod. Proof. If (n,, then, from the roerties of Gauss sums and quadratic residue mod,we have 1 a=0 e( na 1 =1+ e( na 1 =1+ (1 + ( a e (na (7 = 1 a=0 e( na 1 + ( a e(na =( n 1 ( a e(a =(n τ(χ, (8 where χ =( /denotes the Legendre symbol. Since χ is a nonrincial Dirichlet character mod, from (8, the roerties of Gauss sums, and comlete residue system mod,wehave 1 1 a=0 χ(a +ab+b +c e( 1 a=0 1 τ(χ e( τ(χ 1 χ (r a=0 χ (r τ(χ 1 1 a=0 1 τ(χ χ (r χ (r e( r(a +ab+b +c e( ra + (br + m a+rb +nb+cr e( r(a+b+rm +rb +rc + nb r(b+rm 1 1 a=0 e( ra +3r(b + (n m b + rc rm = τ(χ 1 τ(χ χ (r χ (r 1 e( 3r(b+3r (n m +rc rm 1r(n m = τ(χ 1 τ(χ χ (r χ (r 1 e( 3rb +rc r(m +n mn = τ 1 (χ τ(χ χ (3 χ (r e( cr r(m +n mn. (9

3 Alied Mathematics 3 For any nonrincial character χ mod, we have τ(χ =.So,from(9 and noting that χ (3,wehave 1 1 χ(a +ab+b +c e( a=0 1 = This roves Lemma. χ (r e( cr r(m +n mn. (10 Lemma 5. Let be an odd rime; let χ be any nonrincial character mod. Then, for any integers m and n, one has the estimate 1 χ (a e(. (11 Proof. Since χ is a nonrincial character mod, if m and n,then 1 χ (a e( 1 = χ (a =0. (1 If m and (, n or n and (, m or (mn,, then, from the results of Weil [8], Malyšev [7], and Estermann [6], with some minor modifications, we can deduce the estimate 1 χ (a e( (m,n, 1/ =, (13 where (m,n,denotes the greatest common divisor of m, n, and. Now Lemma 5 follows from (1and(13. Lemma 6. Let be an odd rime; then, for any integer n with (n,, one has the calculating formula 1 1 m=1 χ(a e ( , if χ is the rincial character mod ; { 3 3 8, = if χ is the Legendre s symbol mod ; ( 7, { { if χ is a non-real character mod. Proof. See [9]orCorollaryof[10]. (1 Lemma 7. Let be an odd rime and let χ be the Dirichlet character mod.then one hasthe estimate 1 χ mod χ (a L(1,χ Proof. See Lemma 5 of [11]. ln ln =O( ex ( ln. (15 3. Proof of the Theorems In this section, we will comlete the roof of our theorems. First we rove Theorem 1. Infact,fromLemmas and 5, we may immediately deduce the estimate 1 1 χ(a +ab+b +c e( a=0 1 = χ (r e( cr r(m +n mn. (16 This roves Theorem 1. Theorem follows from Lemmas and 6. Infact,from thesetwolemmas,wehave c=1a=0 1 = χ(a +ab+b +c e( 1 c=1 χ(r e ( cr r(m +n mn ={ 35 8, if χ is the Legendre symbol mod ; ( 7, if χ is a comlex character mod. (17 This roves Theorem. Now, we rove Theorem 3. Note that the asymtotic formula is χ mod L(1,χ = 5 7 π +O(ex ( ln ln ln, (18 and the estimate for Kloosterman sums (see [6] is as follows: 1 e( (m,n, 1/ ; (19 from Lemmas and 7 and the method of roving the theorem in [11], we have 1 1 χ(a +ab+b +c e( χ mod a=0 L(1,χ = 1 χ mod L(1,χ χ (r e( cr r(m +n mn

4 Alied Mathematics = = 1 1 s=1 χ mod 1 e( c (r s (r s (m +n mn χ (rs L(1,χ 1 s=1 χ mod =( 1 e( cs (r 1 s (r 1 (m +n mn χ (r L(1,χ χ mod L(1,χ [6] T. Estermann, On Kloostermann s sums, Mathematica,vol.8,.83 86,1961. [7] A.V.Malyšev, A generalization of Kloosterman sums and their estimates, Vestnik Leningrad University, vol.15,no.13,.59 75,1960(Russian. [8] A. Weil, On some exonential sums, Proceedings of the National Academy of Sciences of the United States of America, vol. 3,. 0 07, 198. [9] W. Zhang, On the fourth ower mean of the general Kloosterman sums, Indian Pure and Alied Mathematics, vol.35,no.,.37,00. [10] J. H. Li and Y. N. Liu, Some new identities involving Gauss sums and general Kloosterman sums, Acta Mathematica Sinica, vol.56,no.3,.13 18,013(Chinese. [11] W. Zhang, Y. Yi, and X. He, On the k-th ower mean of Dirichlet L-functions with the weight of general Kloosterman sums, Number Theory, vol. 8, no., , O( 3/ r= χ mod χ (r L(1,χ = 5 7 π 3 +O( 5/ ln ln ex ( ln. (0 This comletes the roof of Theorem 3. Conflict of Interests The authors declare that they have no conflict of interests. Authors Contribution Zhan Yu obtained the theorems and comleted the roof. Li Xiaoxue corrected and imroved the final version. Both authors read and aroved the final aer. Acknowledgments The authors would like to thank the referee for his very helful and detailed comments, which have significantly imroved the resentation of this aer. This work is suorted by the P. S. F. (013JZ001 and N. S. F. ( of China. References [1] T. M. Aostol, Introduction to Analytic Number Theory, Sringer,NewYork,NY,USA,1976. [] C. D. PanandC. B. Pan, Goldbach Conjecture, Science Press, Beijing, China, 199. [3] H. Davenort, Multilicative Number Theory, vol. 1966, Markham, [] K.F.IrelandandM.I.Rosen,A Classical Introduction to Modern Number Theory,vol.8,Sringer,NewYork,NY,USA,198. [5] S. Chowla, On Kloosterman s Sum, vol.0ofvolume 1967 of Forhandlinger,F.Bruns,1968.

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