This paper mainly studied the distribution of values of Hardy sums involving Chebyshev polynomials. By using the method of analysis and the arithmetic properties of Hardy sums and Chebyshev polynomials of the first kind, we obtained a sharp asymptotic formula for the hybrid mean value of Hardy sums $S_5(h,q)$ involving Chebyshev polynomials of the first kind. In addition, we also gave the value of Hardy sums $S(h,q)$ and $S_3(h,q)$ involving Chebyshev polynomials. Finally, we found the reciprocal formulas of $S_3(h,q)$ and $S_4(h,q)$ involving Chebyshev polynomials of the first kind.

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.