On two sums related to the Lehmer problem over short intervals

In this article, we study sums related to the Lehmer problem over short intervals, and give two asymptotic formulae for them. The original Lehmer problem is to count the numbers coprime to a prime such that the number and the its number theoretical inverse are in different parities in some intervals. The numbers which satisfy these conditions are called Lehmer numbers. It prompts a series of investigations, such as the investigation of the error term in the asymptotic formula. Many scholars investigate the generalized Lehmer problems and get a lot of results. We follow the trend of these investigations and generalize the Lehmer problem.