The Sum Of The Mixed Exponent And The Euler Function Value

In this thesis we investigate mixed exponential sums and the sum of values of the Euler function. The main results are as follows:1. For any integers m, n, k, q with k?2, q?3, we concern the following fourth power meanwhere , ? is a Dirichlet character modulo q and denotes the sum-mation over all a such that (a, q) = 1.In 2008, under the conditions (k,q) = 1 and (n,q) = 1, Liu obtained explicit formulas for the fourth power mean.In 2014, by avoiding the situation (k,q) = 1, Chen, Ai and Cai got explicit formulas for the fourth power mean.In this thesis, by removing the conditions (n, q) = 1 and (n, q) = 1 we also obtained explicit formulas for the fourth power mean.This result has been published in International Journal of Number Theory(DOI: 10.1142/S1793042117500841).2. For any integer k, letwhere ? is the Euler totient function. be the count function of Sk up to x.In 2008, Luca and Sankaranarayanan proved that In 2015, Balasubramanian, Luca and Ralaivaosaona improved the above result and proved that In addition, for any even integer k, using the above method the following result could be proved that In this thesis, we prove that for any odd number This result has been published in International Journal of Number Theory(DOI: 10.1142/S1793042117500671).3. Letwhere ? is the Euler totient function. Let A(x) denote the counting. function of A up to x.In 2015, Balasubramanian,Luca and Ralaivaosaona proved that In this thesis, we improve the above result and prove that This result has been published in Boletin de la Sociedad Matematica Mexicana(DOI: 10.1007/s40590-017-0164-8).