一类广义二项指数和的四次均值 On the fourth power mean of one kind generalized two-term exponential sums 1 first-author 孟园园，女，陕西西安人，博士生，从事数论及其应用研究，yymeng@stumail.nwu.edu.cn。 孟园园，女，陕西西安人，博士生，从事数论及其应用研究，yymeng@stumail.nwu.edu.cn。 yymeng@stumail.nwu.edu.cn 西北大学　数论及其应用研究中心，陕西　西安　710127 西北大学　数论及其应用研究中心，陕西　西安　710127 Research Center for Number Theory and its Applications, Northwest University, Xi'an 710127, China Research Center for Number Theory and its Applications, Northwest University, Xi'an 710127, China 张欢 关于指数和的研究在数论问题和实际应用中都具有十分重要的意义。该文主要对广义二项指数和的性质进行了深入地探讨。利用初等方法、解析方法以及同余方程解的个数研究了一类广义二项指数和四次均值的计算问题，并给出了一个精确的计算公式和渐近公式。同时，该研究结果也为高次二项指数和的计算问题提供了一些新的思路和有效方法，具有创新性。 The research on exponential sums is very important in number theory and practical application. This paper mainly discusses the properties of generalized two-term exponential sums. By means of elementary method, analytical method and the number of solutions of congruence equation, this paper studies the calculation of the fourth power mean of a kind of generalized two-term exponential sums, and gives an exact calculation formula and asymptotic formula. At the same time, the research results of this paper also provide some new and effective ideas and methods for the calculation of high-th two-term exponential sums, which is innovative. 广义二项指数和 四次均值 初等方法 计算公式 渐近公式 the generalized two-term exponential sums fourth power mean elementary method calculating formula asymptotic formula 设 p 是一个奇素数， χ 是模 p 的任意狄利克雷特征。对任意整数 k > h ≥1， m 以及 n 且( n ， p )=1，广义二项指数和 S ( m ， n ， k ， h ， χ ； p )定义为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884991&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884992&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885009&type= 其中： e ( y )=e 2πi y ；i 2 =－1。 广义二项指数和在解析数论研究中起着非常重要的作用，许多重要的数论问题都与之密切相关，如素数分布以及华林问题等。因此，不少数论专家和学者对 S ( m ， n ， k ， h ， χ ； p )的各种性质进行了研究，取得了一系列有意义的研究成果。例如，Zhang H等 [ 1 1 ] 证明了对于任意奇素数 p ，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885011&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885013&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885017&type= 式中， n 为整数且满足( n ， p )=1。 Zhang W P等 [ 2 2 ] 利用初等和解析的方法得到了如下结论。 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885005&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885024&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885020&type= 式中， p 为满足3 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885028&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885045&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885043&type= ( p －1)的任意奇素数。 最近，Zhang W P等研究了 S ( m ， n ，3，1， χ 0 ； p )的六次幂均值计算问题 [ 3 3 ] ，并证明了对任意奇素数 p 和整数 n 且( n ， p )=1，有恒等式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885033&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885035&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885036&type= 式中：4 p = d 2 +27· b 2; d 是由 d ≡1 mod 3唯一确定。 另一方面，以Chen L等研究了二项指数和 S ( m ，1，4，1， χ 0 ； p )的四次均值的计算问题 [ 4 4 ] ，并得到了计算公式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885068&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885059&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885061&type= 式中， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885075&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885080&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885077&type= 是一个整数，满足如下性质(见参考文献[ 5 5 ]定理4～定理11)。 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885082&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885100&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885103&type= 式中： r 是模 p 的二次非剩余； http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885090&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885114&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885092&type= 表示勒让德符号。 一些与指数和相关的结论可参阅文献[ 6 6 - 15 15 6-15 ]，这里不再一一列举。 从式(1)～式(4)可以看出，所有这些论文的内容都只涉及 S ( m ， n ， k ， h ， χ ； p )中的 h =1。通过对文献的检索，到目前为止还没有发现关于广义二项指数和 S ( m ， n ， k ，2， χ ； p )的四次均值的相关研究内容。这说明当 k > h =2时，研究工作是具有相当难度的，不易得到理想的结果。 本文利用初等和解析方法以及一些同余方程解的个数等性质研究了2 k 次幂均值的计算问题，并给出了当 p ≡3 mod 4以及 k =2时该式的一个精确的计算公式。 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885096&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885118&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885129&type= 即就是证明下面的结果。 定理1 　设 p 是素数， p ≡3 mod 4，那么，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885134&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885138&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885141&type= 式中，模 p 的六次特征定义为存在模 p 的特征 χ 1 ，使得 χ = http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885148&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885164&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885160&type= 。 由定理1可推出以下推论。 推论1 　设 p 是素数， p ≡3 mod 4，那么对模 p 的任意非主偶特征 χ ，有渐近公式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885166&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885170&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885188&type= 注1 　本文只讨论了 p ≡3 mod 4的情况。如果 p ≡1 mod 4，暂时不能得到一个令人满意的结果。因为无法得到下式的精确值或平非凡的上界估计。 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885191&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885178&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885196&type= 式中， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885184&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885204&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885200&type= 表示勒让德符号。 另一方面，假定 χ 是模 p 的一个非主偶特征。否则其均值为零。事实上，如果 χ 是模 p 的一个奇特征，即 χ (－1)=－1，那么可以得到 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885220&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885222&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885225&type= 所以有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885213&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885215&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885234&type= 此外，考虑这种形式的四次均值的主要原因是研究单个四次均值不能得到一个精确的计算结果。这种形式的和抵消了在单个均值中无法计算的部分。 若干引理 1 为了完成定理1的证明，需要借助3个简单的引理。这些引理的证明需要一些初等数论和解析数论的知识 [ 5 5 - 6 6 5-6 ] 。 引理1 　设 p 是一个奇素数， χ 为模 p 的任意非主偶特征，有恒等式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885236&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885238&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885255&type= 证明 　由模 p 的同余方程解的性质可知 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885257&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885244&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885262&type= 若 p ≡3 mod 4，那么，对于任意整数1≤ a ， b ≤ p -1，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885278&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885271&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885269&type= 。所以同余式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885288&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885290&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885292&type= 当且仅当 a 2 ≡1 mod p 或者 b 2 ≡1 mod p 时成立。这种情况下注意到 χ （-1）=1，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885309&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885311&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885314&type= 若 p ≡1 mod 4，那么 a 2 + b 2 ≡0 mod p ，1≤ a ， b ≤ p -1，有2( p -1)个解。阿 a 2 -1≡0 mod p 有2个解。这种情况下，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885317&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885319&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885326&type= 结合式(5)～式(7)，引理1得证。 引理2 　设 p 是一个奇素数。则对于模 p 的任意非主偶特征 χ ，有恒等式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885320&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885339&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885343&type= 证明 　若3 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885347&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885337&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885335&type= ( p －1)，令 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885352&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885372&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885368&type= 为模 p 勒让德符号。则由模 p 的简化剩余系的性质可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885362&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885376&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885383&type= 由经典高斯和的性质可知 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885381&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885399&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885402&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885392&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885394&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885409&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885428&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885416&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885436&type= 同样，也有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885438&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885427&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885445&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885448&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885450&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885464&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885467&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885455&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885473&type= 结合式（8）~式（14），得到 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885488&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885491&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885494&type= 若3|( p －1)，设整数 r 满足 r ≠1， r 3 ≡1 mod p 。若 χ ( r )≠1，则 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885483&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885503&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885501&type= 或者 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885509&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885511&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885512&type= 若 χ ( r )=1，令 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885515&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885534&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885531&type= ， Ψ 是模 p 的六阶本原特征。由模 p 的六阶本原特征的性质和式(8)~式(14)的证明思路，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885548&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885538&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885551&type= 结合式(15)~(17)，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885540&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885542&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885557&type= 引理2得证。 引理3 　设 p 是一个奇素数， p ≡3 mod 4。对于模 p 的任意非主偶特征 χ ，有恒等式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885559&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885562&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885566&type= 证明 　对于素数 p, p ≡3 mod 4, http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885568&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885573&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885580&type= =－1，由模 p 的简化剩余系的性质，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885584&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885575&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885587&type= 引理3得证。 定理1的证明 1 现在应用第1节中的3个引理来完成定理1的证明。对任何整数 n ，显然有三角恒等式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885591&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885609&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885612&type= 对任意整数 n 且( n ， p )=1，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885601&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885602&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885619&type= 由模 p 的简化剩余系的性质，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885607&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885623&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885640&type= 同样，也有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885626&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885628&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885645&type= 由引理3的证明方法也有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885634&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885653&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885650&type= 和 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885668&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885671&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885662&type= 若 χ 不是模 p 的一个六次非主偶特征，结合式(18)~(21)，引理1~3，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885675&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885666&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885680&type= 若 χ 是模 p 的一个六次非主偶特征，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885687&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885689&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885700&type= 结合式(22)和式(23)，定理1得证。 结语 1 本文的主要结果是给出了一类二项指数和四次均值的一个精确的计算公式。 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885702&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885703&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885697&type= 式中： p ≡3 mod 4; χ 是模 p 的一个非主偶特征。 同时，研究的结果也为高次二项指数和均值的计算问题提供了一些新的有效方法，这些工作将对相关问题的研究发挥积极的作用。 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 - 81 . 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 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 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 APOSTOL T M . Introduction to analytic number theory [M]. New York : Springer-Verlag , 1976 . 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New York : Springer-Verlag , 1974 . Basic number theory 作者对导师张文鹏教授的悉心指导表示衷心感谢！ 10.16152/j.cnki.xdxbzr.2023-03-015 O156.7 国家自然科学基金（12126357） 2022-11-15 2023-06-25 2023-06-26 2023 39049468 西北大学学报（自然科学版） 3 53