关于经典高斯和及其一个新的恒等式 On the classical Gauss sums and its a new identity 1 first-author 张文鹏，男，陕西礼泉人，教授，博士生导师，从事基础数学的教学与研究，wpzhang@nwu.edu.cn。 张文鹏，男，陕西礼泉人，教授，博士生导师，从事基础数学的教学与研究，wpzhang@nwu.edu.cn。 wpzhang@nwu.edu.cn 1 corresp 高洁洁，女，陕西佳县人，从事基础数学的研究，gaojiejie@stumail.nwu.edu.cn。 高洁洁，女，陕西佳县人，从事基础数学的研究，gaojiejie@stumail.nwu.edu.cn。 gaojiejie@stumail.nwu.edu.cn 西北大学　数学学院，陕西　西安　710127 西北大学　数学学院，陕西　西安　710127 School of Mathematics, Northwest University, Xi'an 710127, China School of Mathematics, Northwest University, Xi'an 710127, China 亢小玉 经典高斯和是一类特征函数与指数函数加权的求和，这类和在数论问题的研究和实际应用中占有重要的位置。该文主要研究了一类特殊高斯和的方幂求和的计算问题。通过利用初等方法、解析方法以及特征和的性质得到了当 p 为一个奇素数且 p ≡1 mod 8时，经典高斯和 τ （ χ 8 ）的一类求和的精确计算公式，其中 χ 8 表示模 p 的8阶特征。该结果不仅扩充了经典高斯和的研究内容，而且有助于有关方面研究工作的进一步展开。 Classical Gauss sums is one kind sums of the exponential functions weighted by character functions, which occupies an important position in the study and practical application of number theory. In this paper, we focus on the problem of calculating formula of one kind square power mean of special Gauss sums. By using elementary methods, analytical methods and the properties of character sums, we obtain an exact calculating formula involving the classical Gauss sums τ ( χ 8 ) when p is an odd prime with p ≡1 mod 8, where χ 8 denotes an eight-order character modulo p . The results not only expand the research content of classical Gauss sums, but also contribute to the further development of relevant research work. 经典高斯和 初等方法 解析方法 计算公式 the classical Gauss sums elementary method analytic method calculating formula 设 q 为整数， χ 表示模 q 的Dirichlet特征，对任意整数 m ，经典高斯和 G ( m ， χ ； q )定义为： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884899&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884901&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884918&type= 其中 e ( y )=e 2πi y ，i 2 =－1。 特别当 m =1时，记 G (1， χ ； q )= τ ( χ )。这一和式在解析数论研究中具有重要的理论意义和应用价值。因此，许多学者研究了高斯和的各种性质，并取得了一系列重要的研究成果。高斯和最基本且最重要的性质之一是它的分离性质 [ 1 1 - 2 2 1-2 ] 。即当 χ 为模 q 的原特征或者( m ， q )=1时，有 G ( m ， χ ； q )= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884914&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884925&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884916&type= ( m )· τ ( χ )。此外，当 χ 为模 q 的原特征时，有恒等式| τ ( χ )|= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884928&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884937&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884930&type= 。关于高斯和的其他性质也有不少有趣的研究成果 [ 4 4 - 14 14 4-14 ] 。如Zhang W P等 [ 4 4 ] 或者Berndt.B.C及Evans.R.J [ 5 5 ] 中研究了3阶特征所对应的高斯和的性质，并证明了如下结论。 设 p 为奇素数且 p ≡1 mod 3， λ 是模 p 的任意3阶特征，则有恒等式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884938&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884950&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884940&type= 其中 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884953&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884957&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884956&type= 表示 λ 的共轭特征，并且 d 是由4 p = d 2 +27 b 2 且 d ≡1 mod 3唯一确定的。 Chen Z Y等 [ 6 6 ] 研究了4阶特征所对应的高斯和性质，并证明了如下结论。 对任意素数 p 满足 p ≡1 mod 4，有恒等式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884960&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884963&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884968&type= 其中 χ 4 是模 p 的4阶原特征， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884981&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884984&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884983&type= ， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884972&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884990&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884987&type= 表示模 p 的勒让德符号， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44884994&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885000&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885008&type= ≡1 mod p 。 这里值得指出的是在式(2)中常数 α ( p )有着特殊的意义。事实上当 p ≡1 mod 4时，一定可以表示成两个非零整数的平方和。即(参阅文献[ 3 3 ]中定理4-11) http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885003&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885016&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885007&type= 其中 r 是模 p 的任意二次非剩余。 Chen L [ 7 7 ] 对模 p 的6阶原特征的情况进行了讨论，并证明了对任意素数 p ≡1 mod 6以及6阶原特征 ψ mod p ，有恒等式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885025&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885021&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885039&type= 式中i 2 =－1， d 与式(1)中的定义相同。 纵观以上几个高斯和的结果，其特点是这些原特征的个数均为2，即欧拉函数 φ (4)= φ (3)= φ (6)=2。最近，Zhang W P等 [ 8 8 ] 研究了12阶特征的情况，证明了对任意素数 p ≡1 mod 12及12阶原特征 χ 12 mod p ，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885029&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885031&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885047&type= 其中 d 和 α ( p )的定义与式(1)与式(2)中相同。 由式(3)立刻推出恒等式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885050&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885053&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885056&type= 因为 φ (12)=4且当 p ≡1 mod 8时，模 p 的8阶原特征的个数也是4，即欧拉函数 φ (8)=4。所以我们自然会想到，对应的高斯和是否也存在一个类似于式(3)的恒等式，本文的主要目的就是研究这一问题，并利用初等方法以及经典高斯和的性质证明下面的结果。 定理1 　设 p 是一个奇素数且 p ≡1 mod 8，那么对模 p 的任意8阶特征 χ 8 ，有恒等式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885070&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885074&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885065&type= 式中 α ( p )的定义与式(2)相同， a · http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885081&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885087&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885098&type= ≡1 mod p 。 注释 　本文中的定理虽然给出了8阶特征高斯和的计算公式，但是其结论也有缺陷性，即表达式中含有多项式的二次特征和不能计算出具体值。对于其中包含的二次特征和，是否能计算出它的精确值是一个有趣的公开问题！ 此外，注意到 φ (5)=4，所以当 p ≡1 mod 5时，模 p 的5阶原特征个数也是4。对于5阶特征以及2 h 阶特征( h ≥4)对应的高斯和，是否也有一个类似于我们定理的计算公式，它是一个有意义的研究课题。建议有兴趣的读者与我们一起考虑！ 若干引理 1 为了完成定理的证明，需要以下引理。在下文中，将用到初等数论知识以及高斯和性质，这些内容在文献[ 1 1 - 3 3 1-3 ]中有详细的描述，这里不在赘述。 引理1 　设 p 是一个奇素数， χ 为模 p 的任意非主特征，则有恒等式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885105&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885091&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885110&type= 式中： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885093&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885128&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885116&type= 表示模 p 的勒让德符号。 证明 　事实上这一结论的一般情况可参阅文献[ 15 15 ]，这里给出这一特殊情况的简单证明。 对模 p 的任一非主特征，由经典高斯和的性质，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885130&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885133&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885136&type= 另一方面，对任意整数 b 且( b ， p )=1，由恒等式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885143&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885145&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885158&type= 也有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885151&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885163&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885167&type= 结合式(4)和(5)，有恒等式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885169&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885174&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885176&type= 注意到 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885194&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885198&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885181&type= ，并在式(6)中以 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885187&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885207&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885218&type= 代替 χ 立刻推出 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885224&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885211&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885228&type= 于是证明了引理1。 引理2 　设 p 是一个奇素数且 p ≡1 mod 8， χ 8 为模 p 的8阶原特征，则有恒等式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885232&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885248&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885235&type= 式中 χ 4 = http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885251&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885241&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885254&type= 为模 p 的4阶原特征。 证明 　注意到 p ≡1 mod 8，有 τ ( χ 2 )= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885243&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885247&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885245&type= ，在引理1中取 χ = χ 8 可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885279&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885281&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885283&type= 类似也有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885272&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885287&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885289&type= 注意到 χ 4 (2)· χ 8 (－1)= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885293&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885299&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885308&type= 4 (2)· http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885301&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885306&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885316&type= 8 (－1)，结合式(7)及(8)，可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885325&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885329&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885330&type= 或者 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885342&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885346&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885349&type= 于是证明了引理2。 引理3 　设 p 是一个奇素数且 p ≡1 mod 8，则有恒等式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885351&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885370&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885374&type= 证明 　由勒让德符号的性质可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885363&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885377&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885379&type= 应用高斯和的性质，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885382&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885400&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885404&type= 同理可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885406&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885396&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885411&type= 结合式(9)，(10)及(11)，可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885430&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885417&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885434&type= 于是证明了引理3。 定理的证明 1 有了上一节的几个引理，容易完成定理1的证明。事实上 τ ( χ 4 )· τ ( http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885424&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885446&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885444&type= 4 )= p ，由式(2)可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885449&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885461&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885465&type= 由式(12)及引理3立刻推出 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885468&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885472&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885476&type= 注意到 χ 4 (2) χ 8 (－1)=±1及 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885478&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885496&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885480&type= ，有恒等式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885486&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885504&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885502&type= 于是完成了定理1的证明。 结语 1 本文主要利用经典高斯和的性质研究了一类特殊高斯和方幂和的计算问题，并给出了它们的一个精确计算公式。即证明了如下结论：设 p 为素数满足 p ≡1 mod 8， χ 8 为模 p 的任意8阶特征，那么有恒等式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885520&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885523&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885513&type= 其中 α ( p )由式(2)给出， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885514&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885533&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885529&type= 表示勒让德符号。 显然这一研究结果是有意义的，它不仅扩充了经典高斯和的研究内容，而且有助于有关方面研究工作的进一步展开！ 参考文献 APOSTOL T M . 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Die Nullstellen der Kongruenz zeta funktionen in gewissen zyklischen Fällen 【主持人语】 张文鹏 指数和的各种性质研究是解析数论的重要内容之一，其研究工作不仅具有重要的理论意义，同时具有广泛的应用价值！特别是在近代解析数论的发展中，许多重大的数论难题之所以能够取得实质性进展都与指数和的估计密切相关，如素数分布、哥德巴赫问题、孪生素数问题、华林问题、丢番图逼近等内容都是如此。此外，我们也知道，单个指数和的值分布并不规则，没有规律性，因而很难刻画其深刻的内在性质！然而，关于其高次均值或加权均值的计算，通常却能得到精确的恒等式或较强的渐近公式。这一现象对于计算和估计指数和的上界有着十分重要的意义。因此，对于Gauss和、二项指数和以及Dedekind和等一些特殊和式的均值问题研究已经成为当下数论研究的热门内容之一！本专题主要针对高斯和、二项指数和、推广的Dedekind DC和、D.H.Lehmer问题等均值的计算问题进行了研究，并给出了一系列新的并且有理论意义的恒等式或渐近公式！通过对这些问题的探索，必将对该领域相关问题的研究起到积极的推动作用。 10.16152/j.cnki.xdxbzr.2023-03-012 O156.7 国家自然科学基金项目（12126357） 2022-12-20 2023-06-25 2023-06-26 2023 39049416 西北大学学报（自然科学版） 3 53