Mean Value Estimation of The Sum-of-divisors Function

The sum-of-divisor function is one of the important number theory functions, and the study of the properties of the sum-of-divisor function can provide more methods for solving some number theory problems. Since the value of the function is irregular with the change of the independent variables, the mean estimation of the sum-of-divisor function is usually used to deduce its own properties. Therefore, it is meaningful to study the mean estimation of the sum-of-divisor function and make gradual improvements to the rest of the terms. Through more in-depth study of power sum functions, the important properties of sum-of-divisor function can be obtained, which provides new ideas for solving more number theory problems. Let is the sum of all factors of to the third power, be the integer part of . Using the Dirichlet hyperbolic principle, the following results are obtained: the asymptotic formula for Sum-of-Divisors function , and obtains the remainder of its asymptotic formula as .