An Asymptotic Formula for Estimating the Mean of A Class of Indices

Let , , with , . This article uses the exponential pair principle and Vinogradov method to first obtain some bound exponential sums of the type where , and then derives the asymptotic formula of . The asymptotic formula of is an important tool for studying the mean estimation of the exponential divisor functions and divisor functions. By using the asymptotic formula of , we can derive more accurate mean estimates of the divisor function and exponential divisor function. Therefore, with the help of the asymptotic formula of , we can have a deeper understanding of the divisor function and exponential divisor function, and then provide solutions for more number theory problems.