The Arithmetic Mean Of Estimates And Power Of The Remaining Formal Existence Theorem

In this paper, we discuss some problems about hybrid mean value of two special number theory functions and the existence of power residue elements or power residue normal elements over finite fields. The main achievements contained in this paper can be listed as follows:1. We make some asymptotic formulae ofΣept(n)dr(n) andΣep(n)σr(n), where d(n) is the number of the divisors of n, a(n) is the sum of the divisors of n, ep(n) is the number of prime divisors p of n, r is a positive integer and t is a positive real number. In some special cases, we get the explicit asymptotic formulae. Generally, we do the same thing ofΣf(ep(n))9(d(n)), where f(χ) is a rational polynomial and g(χ) is a polynomial.2. Assume f(χ) and g(χ) are polynomials over finite fields. Under some conditions we discuss the existence of power residue elements or power residue normal elements over finite fields:(1) there exists aχsuch that both f(χ) and g(χ) are power residue normal elements.(2) there exists aχsuch that both f(χ) and g(χ) are power residue elements.