Distribution of values of Hardy sums over Chebyshev polynomials

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.