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.Lehmerproblem on half interval

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]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],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.