半区间上D. H. Lehmer问题余项的一种均值 A mean value of the error term of D. H. Lehmer problem on half interval 1 first-author 刘晓莹，女，陕西汉中人，从事数论及其应用研究。 刘晓莹，女，陕西汉中人，从事数论及其应用研究。 1 corresp 徐哲峰，男，陕西西安人，教授，从事数论及其应用研究，zfxu@nwu.edu.cn。 徐哲峰，男，陕西西安人，教授，从事数论及其应用研究，zfxu@nwu.edu.cn。 zfxu@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 半区间上D. H. Lehmer问题余项的分布性质，有助于研究整数及其逆的分布。该文讨论了半区间上D. H. Lehmer问题余项 E （ a ， p ）在不完整区间上的一种均值分布性质。利用 E （ a ， p ）与Dirichlet L-函数的关系以及Dirichlet L-函数的一些均值性质，给出了 E （ a ， p ）一种均值的几个较强的渐近公式，结合文献[ 4 4 ]中的结论，可以看出这两种区间上D. H. Lehmer问题余项的相消性有显著的差异。该结果不仅扩充了半区间上D. H. Lehmer问题余项的研究内容，而且有助于该领域研究工作的进一步展开。 The distribution properties of the error term of the D. H. Lehmer problem on incomplete intervals are helpful for studying the distribution of integers and their inverse. This paper mainly discusses the mean value distribution properties of the error term of D. H. Lehmer problem E ( a , p ) on incomplete intervals. By using the relations between E ( a , p ) and Dirichlet L-function and some mean value properties of Dirichlet L-function, several strong asymptotic formulas of E ( a , p ) are given. Combined with the conclusion in reference [ 4 4 ], it can be seen that there are significant differences in the cancellation of the error of D. H. Lehmer problem in these two intervals. This result not only expands the research content of the error term of the D. H. Lehmer problem on incomplete intervals, but also contributes to the further development of relevant research work. 半区间上D. H. Lehmer问题 余项 Dirichlet L-函数 均值 相消性 D. H. Lehmer problem on half interval error term Dirichlet L-function mean value cancellation 设 q > 2和 a 是两个整数，且( a ， q )=1。对每一个满足1≤ b ≤ q 的整数 b ，存在唯一的 c 满足 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885277&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885295&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885297&type= 对于 q = p 为奇素数和 a =1, D．H.Lehmer建议研究 Q (1， p )的值或者至少给出它的一些非平凡结论 [ 1 1 ] 。对D. H. Lehmer提出的问题，Zhang W P [ 2 2 ] 证明了 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885298&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885315&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885305&type= 令 M ( a ， p )= Q ( a ， p )－ http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885307&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885321&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885327&type= ( p －1)，Zhang W P [ 3 3 ] 还证明了余项 M ( a ， p )的平方均值，并给出渐近公式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885340&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885345&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885333&type= Xu Z F等 [ 4 4 ] 研究了余项 M ( a ， p )的一种均值，证明了下面的渐近公式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885350&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885355&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885358&type= 而如果非负整数 s ≠2，3，4，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885359&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885361&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885365&type= 由此揭示了完整区间上D. H. Lehmer问题，即经典D. H. Lehmer问题余项相消性随参数 s 的不同有着非常显著的变化。 作为经典D. H. Lehmer问题的延伸，Xu Z F等 [ 5 5 ] 引入了半区间上的D. H. Lehmer问题。设 p 为奇素数，( a ， p )=1，令 N ( a ， p )表示同余式 bc ≡ a (mod p )满足1≤ b ， c ≤ http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885367&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885398&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885380&type= 和2 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885389&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885407&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885391&type= b + c 的解( b ， c )的个数。文献[ 5 5 ]给出了 N (1， p )的值，即 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885397&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885414&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885415&type= 令 E ( a ， p )= N ( a ， p )－ http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885432&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885437&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885419&type= ( p －1)，Xu Z F等 [ 5 5 ] 还研究了余项 E ( a ， p )的平方均值，给出了渐近公式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885439&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885443&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885442&type= 与文献[ 4 4 ]中关于完整区间上D. H. Lehmer问题余项类似，本文讨论半区间上D. H. Lehmer问题余项的一种均值，给出了定理1。 定理1 　设 p ≥3为素数，则有渐近公式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885460&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885463&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885466&type= 对于非负整数 s ≠0，1，3，4，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885470&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885474&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885477&type= 其中， Ο 常数只依赖于任意小的正数 ε ，而 Ο ε 常数依赖于 ε 和 s 。 从定理1的结论来看，对于不同的参数 s ，半区间上D. H. Lehmer问题余项的这种双参数均值的主项各不相同，但均不为0。而文献[ 4 4 ]中对于完整区间上D. H. Lehmer问题余项的相同形式的均值仅在 s =2，3，4时其主项不为0， s 取其他非负整数时主项均为0。从这些结论可以看出，这两种区间上D. H. Lehmer问题余项的相消性有显著的差异。 若干引理 1 为方便定理1的证明，本节先给出7个引理。 引理1 　设 p 为奇素数，则对满足( a ， p )=1的任意正整数 a ，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885479&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885495&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885497&type= 式中 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885498&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885505&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885507&type= 证明过程可参考文献[ 6 6 ]中的引理7.1。 引理2 　设 p 为素数， m ≥0为任意给定的整数，则有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885519&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885522&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885525&type= 和 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885527&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885528&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885517&type= 证明过程可参考文献[ 6 6 ]中的引理4.3。 引理3 　设 q > 4为一个奇数， χ 为模 q 的原特征且满足 χ (－1)=－1。则有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885535&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885537&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885550&type= 证明过程可参考文献[ 6 6 ]中的引理1.6。 引理4 　设 q ≥5为奇整数， χ 为模 q 的原特征且满足 χ (－1)=1，则下述等式成立。 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885552&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885553&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885541&type= 式中， χ 4 表示模为4的原特征。 证明过程可参考文献[ 6 6 ]中的引理1.4。 引理5 　设 q 和 r 是满足 q ≥2和( r ， q )=1的整数，有如下性质 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885558&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885561&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885564&type= 和 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885547&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885570&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885572&type= 式中： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885574&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885577&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885576&type= 表示所有模 q 的原特征的和； J ( q )表示模 q 的原特征的个数。 证明过程可参考文献[ 7 7 ]中的引理3。 引理6 　设 q > 2为奇整数， m ≥0为一个给定的整数，则对任意满足Re s > 1的复变量 s ，有恒等式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885590&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885596&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885598&type= 式中， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885600&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885603&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885615&type= 。 证明过程可参考文献[ 6 6 ]中的引理2.5。 引理7 　设 q > 2为奇整数， χ 为模 q 的特征， χ 4 表示模为4的原特征，则下述渐近公式成立。 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885604&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885620&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885639&type= 特别地，当 q = p 为素数且 k =2时，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885625&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885643&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885629&type= 和 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885631&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885633&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885655&type= 证明 　设 τ k ( n ) [ 8 8 - 10 10 8-10 ] 表示 k 次除数函数(即方程 n 1 ， n 2 ，…， n k = n 的所有正整数解的个数)，则对任意参数 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885651&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885661&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885658&type= 及模 q 的任意非主特征 χ ，利用Abel求和公式得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885674&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885664&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885677&type= 其中， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885683&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885688&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885685&type= 。可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885690&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885701&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885695&type= 分别估计式(1)中的每一项，利用引理5可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885705&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885707&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885715&type= 则有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885710&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885717&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885720&type= 式中， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885728&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885727&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885724&type= 表示对满足条件1≤ n ≤ N 和( n ，2 q )=1的所有正整数 n 求和。同样，对 n 1 和 n 2 做如下分类。 i) d ≤ n 1 ≤ N ， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885734&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885748&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885746&type= ≤ n 2 ≤ N ； ii) d ≤ n 1 ≤ N ，1≤ n 2 ≤ http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885738&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885753&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885752&type= －1； iii) 1≤ n 1 ≤ d －1, http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885754&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885759&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885756&type= ≤ n 2 ≤ N ； iv) 1≤ n 1 ≤ d －1, 1≤ n 2 ≤ http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885761&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885763&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885762&type= －1。 对前面3种情形，有估计式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885764&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885765&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885766&type= 及 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885767&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885768&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885769&type= 这里使用了估计 τ k ( n )≪ n ε 。对第4种情形1≤ n 1 ≤ d －1，1≤ n 2 ≤ http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885770&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885772&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885771&type= －1，同余式2 s n 2 ≡ n 1 mod d 的解为2 s n 2 = n 1 。因此 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885773&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885774&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885742&type= 结合引理6，可以得到 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885790&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885792&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885776&type= 因为( q ，2)=1， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885777&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885778&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885779&type= 所以有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885780&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885793&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885795&type= 其中，恒等式 ζ (2)= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885781&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885799&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885797&type= 。 类似地，可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885800&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885801&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885802&type= 则从式(1)～(3)可以得到 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885803&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885783&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885804&type= 结合估计 [ 11 11 ] http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885805&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885806&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885807&type= 由Cauchy不等式，可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885808&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885809&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885810&type= 则有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885811&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885812&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885813&type= 类似地，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885814&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885815&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885816&type= 取 N = q 2 k, ε < http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885817&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885819&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885818&type= ，结合式(4)～(7)，有渐近公式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885820&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885821&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885822&type= 引理7得证。 定理1的证明 1 首先，利用引理1～4，可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885823&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885824&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885825&type= 注意到，当 χ (－1)=－1时， τ ( χ ) τ ( http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885826&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885828&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885827&type= )=－ p [ 12 12 - 13 13 12-13 ] ，而当 χ (－1)=1时，有 τ ( χχ 4 ) τ ( http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885829&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885831&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885830&type= χ 4 )=－4 p [ 14 14 - 15 15 14-15 ] 。从而 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885832&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885833&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885834&type= 当 s ≥6时， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885835&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885836&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885837&type= 注意到 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885838&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885839&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885840&type= 其次，利用引理2和引理7，则有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885841&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885842&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885843&type= 当 s =3时，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885844&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885845&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885846&type= 和 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885847&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885848&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885849&type= 类似地，当 s =0，1，4时，可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885850&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885851&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885852&type= 和 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885853&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885854&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885855&type= 而当 s =2，5时，可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885856&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885857&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=44885858&type= 定理1证毕。 结语 1 本文主要研究了半区间上D．H.Lehmer问题余项 E ( a ， p )在不完整区间上的一种均值分布性质。利用 E ( a ， p )与Dirichlet L-函数的关系以及Dirichlet L-函数的一些均值性质，得到对于不同的参数 s ，半区间上D. H. Lehmer问题余项的这种双参数均值的主项各不相同，但均不为0，再结合文献[ 4 4 ]中的结论可以看出，这两种区间上D. H. Lehmer问题余项的相消性有显著的差异。 参考文献 GUY R K . Unsolved problems in Number Theory [M]. New York : Springer-Verlag , 1994 : 105 - 136 . Unsolved problems in Number Theory ZHANG W P . Aproblem of D. H. Lehmer and its generalization(II) [J]. Composition Mathematica , 1992 , 91 ( 1 ): 47 - 56 . Aproblem of D. H. Lehmer and its generalization(II) ZHANG W P . A problem of D. H. Lehmer and its mean square value formula [J]. Japanese Journal of Mathematics , 2003 , 29 ( 1 ): 109 - 116 . A problem of D. H. Lehmer and its mean square value formula XU Z F , ZHANG W P . A problem of D. H. Lehmer and its mean value [J]. Mathematische Nachrichten , 2008 , 281 ( 4 ): 596 - 606 . A problem of D. H. Lehmer and its mean value XU Z F , ZHANG W P . Onaproblem of D. H. Lehmer over short intervals [J]. Journal of Mathematical Analysisand Applications , 2006 , 320 ( 2 ): 756 - 770 . Onaproblem of D. H. Lehmer over short intervals 徐哲峰 , 张文鹏 . Dirichlet特征及其应用 [M]. 北京 : 科学出版社 , 2008 : 144 - 151 . Dirichlet特征及其应用 ZHANG W P . On a Cochrane sum and its hybrid mean value formula [J]. Journal of Mathematical Analysis and Applications , 2002 , 267 ( 1 ): 89 - 96 . On a Cochrane sum and its hybrid mean value formula 徐哲峰 . 数论中一些和式的算术性质研究 [D]. 西安 : 西北大学 , 2007 . 数论中一些和式的算术性质研究 张天平 . 关于数论中一些著名和式的均值研究 [D]. 西安 : 西北大学 , 2005 . 关于数论中一些著名和式的均值研究 刘华宁 . 关于一些算术函数的均值 [D]. 西安 : 西北大学 , 2004 . 关于一些算术函数的均值 ZHANG W P , YI Y . On the 2 k -th power mean value of Dirichlet L-functions with the weight of general Kloosterman sums [J]. Journal of Number Theory , 2000 , 84 ( 2 ): 199 - 213 . On the 2k-th power mean value of Dirichlet L-functions with the weight of general Kloosterman sums APOSTOL T M . Introduction to analytic number theory [M]. New York : Springer-Verlag , 1976 : 129 - 132 . Introduction to analytic number theory 华罗庚 . 数论导引 [M]. 北京 : 科学出版社 , 1979 : 45 - 86 . 数论导引 BURGESS D A . Mean value of character sums [J]. Mathematika , 1986 , 33 ( 1 ): 1 - 5 . Mean value of character sums BURGESS D A . Mean value of character sums(II) [J]. Mathematika , 1987 , 34 ( 1 ): 1 - 7 . Mean value of character sums(II) 10.16152/j.cnki.xdxbzr.2023-03-017 O156.2 国家自然科学基金（11971381） 2022-11-15 2023-06-25 2023-06-26 2023 39049492 西北大学学报（自然科学版） 3 53