一类二项指数和的四次幂均值 On the fourth power mean value of one kind two-term exponential sums 1 first-author 袁仁杰，男，陕西安康人，从事解析数论研究。 袁仁杰，男，陕西安康人，从事解析数论研究。 1 corresp 王婷婷，女，陕西西安人，教授，硕士生导师，从事解析数论研究，ttwang@nwsuaf.edu.cn。 王婷婷，女，陕西西安人，教授，硕士生导师，从事解析数论研究，ttwang@nwsuaf.edu.cn。 ttwang@nwsuaf.edu.cn 西北农林科技大学　理学院，陕西　杨凌　712100 西北农林科技大学　理学院，陕西　杨凌　712100 College of Science, Northwest A & F University, Yangling 712100, China College of Science, Northwest A & F University, Yangling 712100, China 张欢 指数和 C （ m ， n ， r ， s ； q ）的高次幂均值计算与上界估计方面的研究与诸多数论问题联系密切，例如华林问题等。设 p 为奇素数，关注参数 n =1，指数幂 r =4， s =2条件下的一类二项指数和的四次幂均值计算问题。利用解析方法，借助Dirichlet特征的奇偶性、正交性及特征和的性质，研究了形如 C （ m ，1，4，2； p ）的二项指数和的四次均值计算，给出了在素数 p ≡3 mod 4情况下上述二项指数和的一个精确的计算公式。同时，对于此类研究内容，该文也提出了一些有待解决的公开问题。 The calculation of the mean value of exponential sums C ( m , n , r , s ; q ) and its upper bound estimation is closely related to many number theory problems, such as Waring's problem. Let p be an odd prime, this paper focuses on the problem of the calculation of the fourth power mean value of one kind two-term exponential sums when parameter n =1 and exponent r =4, s =2. By utilizing the analytic methods, the odd-even property and orthogonality of Dirichlet character, and the properties of the character sums, we study the calculating problem of the fourth power mean of the two-term exponential sums for the form as C ( m , 1, 4, 2; p ), and give an exact calculating formula for it when the prime p ≡3 mod 4. At the same time, for such research content, some open problems in this field are also proposed. 二项指数和 四次均值 计算公式 解析方法 the two-term exponential sums fourth power mean computational formula analytic mathods 设 q 是一个正整数， m 、 n 是任意整数，则二项指数和定义为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885040&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885030&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885046&type= 其中： r > s > 0是整数； e ( y )=e 2πi y ，i 2 =－1。 指数和的估计与许多数论问题密切相关，如华林问题、素数分布问题等，在解析数论的发展中起着非常重要的作用。各种形式的指数和 C ( m ， n ， r ， s ； q )在不同参数下的高次均值计算问题也一直吸引了许多学者的兴趣，近年来有许多这方面的研究。例如，Zhang H等 [ 1 1 ] 证明了当 p 为奇素数时，有恒等式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885034&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885051&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885037&type= 式中， n 表示整数且满足( n ， p )=1。 Zhang W P等 [ 2 2 ] 研究了 C (1， m ，3，1； p )的六次均值，当 p 是奇素数且满足3 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885057&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885062&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885071&type= ( p －1)，得到了恒等式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885064&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885067&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885085&type= Chen L等 [ 3 3 ] 研究了一种新的形式 C ( m, 1, 4, 1; p )的四次均值，当 p 是奇素数时，得到了恒等式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885101&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885089&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885108&type= 和 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885112&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885095&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885097&type= 式中： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885121&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885125&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885123&type= 是一个整数； http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885137&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885146&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885144&type= 表示模 p 的勒让德符号； http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885159&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885153&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885161&type= 表示同余方程 x · a ≡1 mod p 的解。 事实上，对于常数 α = α ( p )也是有来源的。文献[ 4 4 ]中的定理4－11表明，如果 p 是一个素数且 p ≡1 mod 4，则有 p = α 2 + β 2 = α ( p )+ β ( p )。 其中： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885155&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885172&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885157&type= 是一个整数； http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885175&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885180&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885193&type= =－1。 近期工作中，Zhang W P等 [ 5 5 ] 对 C ( m ， n ，3，1； p )的六次均值做了研究，证明了对任意的奇素数 p 和满足( n ， p )=1的整数 n ，有恒等式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885183&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885199&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885203&type= 其中：4 p = d 2 +27· b 2 ； d 是由 d ≡1 mod 3唯一确定的。 此外，文献[ 6 6 - 14 14 6-14 ]中也对二项指数和的诸多性质进行了研究，此处不再赘述。 上述文献中只对 s =1情况下 C ( m ， n ， r ， s ； p )的均值问题做了一些研究，而对于 s ≥2的情况到目前为止还未看到相关研究。这是因为随着 s 的增大， C ( m ， n ， r ， s ； p )的均值计算难度会越来越大，对于原来的一些方法可能就不再适用。基于此，本文将尝试考虑参数为 r =4， s =2下，这种二项指数和的2 k 次幂均值的计算问题，即 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885206&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885208&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885210&type= 其中， k ≥2是一个整数。 当 s =2时，如果 a 通过模 p 的简化剩余系时，而 a 2 不完全通过模 p 的简化剩余系，那么，这部分内容无法直接使用高斯和或者三角恒等式的性质来研究。这无疑对 C ( m, 1, 4, 2; p )的均值计算增加了难度，研究起来会很复杂。本文借助初等和解析方法，利用模 p 特征的正交和奇偶的性质将问题进行转化，再结合模 p 的特征和的性质给出式(1)在 k =2和 p ≡3 mod 4情况下的一个简洁的计算公式，即定理1。 定理1 　令 p ≡3 mod 4是一个素数。则有恒等式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885212&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885214&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885231&type= 注1 　在定理1中，只考虑了 p ≡3 mod 4这种情况。如果 p ≡1 mod 4并且 k =2时，此情况下的问题研究非常复杂，使用本文的方法无法对其进行研究，故对其确切的计算公式目前还无法给出。 对于素数 p ≡3 mod 4和整数 k ≥3的情况下，式(1)是否存在一个精确的计算公式？这是一个公开的问题，有待于进一步的研究。 若干引理 1 要证明定理1，需要用到以下5个引理。其中，引理的证明需要用到初等或者解析数论中的一些知识，详见文献[ 4 4 , 15 15 - 16 16 15-16 ]，这里不再一一列举。 引理1 　令 p ≡3 mod 4是奇素数。若 χ 为任意模 p 的非实偶特性，那么有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885217&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885249&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885237&type= 若 χ 为模 p 的任意非实奇特征，那么有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885253&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885240&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885242&type= 式中， χ 2 表示模 p 的勒让德符号。 证明 　利用模 p 的简化剩余系以及经典Gauss和的相关性质将上述均值展开，得到 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885246&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885264&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885267&type= 如果 χ 是任一模 p 的偶特征，那么 χ (－1)=1。又 χ 2 (－1)=－1，则有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885270&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885284&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885286&type= 如果 χ 是任一模 p 的奇特征，那么， χ (-1)，则有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885274&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885276&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885294&type= 注意到 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885310&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885302&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885313&type= ，那么，结合式(2)～式(4)，引理1得证。 引理2 　令 p ≡3 mod 4是一个素数，则有恒等式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885303&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885318&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885324&type= 和 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885328&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885338&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885331&type= 证明 　首先，利用三角和的性质，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885332&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885334&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885353&type= 以上过程中借助了恒等式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885356&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885369&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885373&type= 注意到 τ 2 ( χ 2 )=－ p ，类似地，又有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885375&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885366&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885384&type= 由式（5）和式(6)，引理2得证。 引理3 　令p≡3 mod 4是一个奇素数，有恒等式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885386&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885388&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885403&type= 和 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885393&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885395&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885413&type= 证明 　根据模 p 特征的相关性质，则有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885429&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885418&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885435&type= 又注意到 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885421&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885426&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885423&type= ， p |( a 2 -1)当且仅当 a ≡± 1 mod p ，运用式(7)有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885441&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885458&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885462&type= 根据特征的性质，与式(7)类似的，又有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885452&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885454&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885471&type= 运用式(9)，则有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885475&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885489&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885492&type= 那么，由式(8)和式(10)，引理3得证。 引理4 　令 p ≡3 mod 4，有恒等式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885481&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885484&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885499&type= 证明 　设 χ 0 为模 p 的主特征，根据引理1～3的结果，容易得到 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885500&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885508&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885521&type= 引理4得证。 引理5 　令 p ≡3 mod 4，有恒等式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885524&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885526&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885516&type= 证明 　结合引理1～3的结果，可推出 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885530&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885532&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885536&type= 引理5得证。 定理1的证明 1 首先，由模 p 特征的正交性，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885549&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885539&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885554&type= 其次，由引理4和引理5，可以得到 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885555&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885556&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885544&type= 最后，结合式(11)和式(12)，得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885563&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885546&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885569&type= 如果 p ≡3 mod 4，注意到恒等式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885579&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885582&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885583&type= 由式(13)和式(14)，可以推导出恒等式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885585&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885589&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885593&type= 综合以上结果，定理1得证。 结语 1 本文的主要研究成果是给出了二项指数和 C ( m ，1，4，2； p )的四次幂均值，即对任意素数 p 满足 p ≡3 mod 4，有恒等式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885595&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885610&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885616&type= 同时，也提出了几个关于这类二项指数和的2 k 次幂均值( k ≥2)的有趣的公开问题。 参考文献 ZHANG H , ZHANG W P . The fourth power mean of two-term exponential sums and its application [J]. Mathematical Reports , 2017 , 19 ( 1 ): 75 - 83 . The fourth power mean of two-term exponential sums and its application ZHANG W P , HAN D . On the sixth power mean of the two-term exponential sums [J]. Journal of Number Theory , 2014 , 136 : 403 - 413 . On the sixth power mean of the two-term exponential sums CHEN L , WANG X . A new fourth power mean of two-term exponential sums [J]. Open Mathematics , 2019 , 17 ( 1 ): 407 - 414 . A new fourth power mean of two-term exponential sums 张文鹏 , 李海龙 . 初等数论 [M]. 3 版. 西安 : 陕西师范大学出版社 , 2013 . 初等数论 ZHANG W P , MENG Y Y . On the sixth power mean of the two-term exponential sums [J]. Acta Mathematica Sinica(English Series) , 2022 , 38 ( 3 ): 510 - 518 . On the sixth power mean of the two-term exponential sums ZHANG H , ZHANG W P . On the fourth power mean of the two-term exponential sums [J]. The Scientific World Journal , 2014 , 2014 : 1 - 5 . On the fourth power mean of the two-term exponential sums DUAN R , ZHANG W P . On the fourth power mean of the generalized two-term exponential sums [J]. Mathematical Reports , 2020 , 22 ( 3 ): 205 - 212 . On the fourth power mean of the generalized two-term exponential sums 王婷婷 . 广义二项指数和的四次幂均值 [J]. 陕西师范大学学报(自然科学版) , 2021 , 49 ( 5 ): 6 - 12 . 广义二项指数和的四次幂均值 WANG T T . On the fourth power mean of the generalized two-term exponential sum [J]. Journal of Shaanxi Normal University (Natural Science Edition) , 2021 , 49 ( 5 ): 6 - 12 . On the fourth power mean of the generalized two-term exponential sum CHEN Z Y , ZHANG W P . On the fourth-order linear recurrence formula related to classical Gauss sums [J]. Open Mathematics , 2017 , 15 ( 1 ): 1251 - 1255 . On the fourth-order linear recurrence formula related to classical Gauss sums CHEN L . On classical Gauss sums and some of their properties [J]. Symmetry , 2018 , 10 ( 11 ): 625 . On classical Gauss sums and some of their properties ZHANG W P , YUAN X D . On the classical Gauss sums and their some new identities [J]. AIMS Mathematics , 2022 , 7 ( 4 ): 5860 - 5870 . On the classical Gauss sums and their some new identities 吕星星 . 三次Gauss和及二项指数和的混合均值 [J]. 数学学报(中文版) , 2019 , 62 ( 2 ): 225 - 232 . 三次Gauss和及二项指数和的混合均值 LYU X X . On the hybrid power mean of three-th Gauss sums and two-term exponential sums [J]. Acta Mathematica Sinica(Chinese Series) , 2019 , 62 ( 2 ): 225 - 232 . On the hybrid power mean of three-th Gauss sums and two-term exponential sums HAN D . A hybrid mean value involving two-term exponential sums and polynomial character sums [J]. Czechoslovak Mathematical Journal , 2014 , 64 ( 1 ): 53 - 62 . A hybrid mean value involving two-term exponential sums and polynomial character sums ZHANG W P , LIU H N . On the general Gauss sums and their fourth power mean [J]. Osaka Journal of Mathematics , 2005 , 42 ( 1 ): 189 - 199 . On the general Gauss sums and their fourth power mean APOSTOL T M . Introduction to analytic number theory [M]. New York : Springer-Verlag , 1976 . Introduction to analytic number theory IRELAND K , ROSEN M . A classical introduction to modern number theory [M]. New York : Springer-Verlag , 1982 . A classical introduction to modern number theory 10.16152/j.cnki.xdxbzr.2023-03-016 O156.4 国家自然科学基金（12126357）；陕西省自然科学基金（2022JM-013） 2022-11-15 2023-06-25 2023-06-26 2023 39049479 西北大学学报（自然科学版） 3 53