eng Science and Education Publishing Turkish Journal of Analysis and Number Theory 2333-1232 2020-07-13 8 2 39 48 10.12691/tjant-8-2-4 TJANT2020824 article Myriam Amri myriam.amri@unileoben.ac.at 1 Khadija Mbarki 2 Montanuniversit?t Leoben Department Mathematik und Informationstechnologie, Austria Department of Mathematics, Faculty of Sciences of Monastir, Tunisia We examine the average order of some arithmetic functions written as sums over Euler function in arithmetic progression and in general over such that is a prime number, an integer and is a polynomial function with integer coefficients and a degree that is not constant modulo Our results are based on various estimates of rational exponential sums with the Euler Function in arithmetic progression which are due to William Banks and Igor E. Shparlinski. http://pubs.sciepub.com/tjant/8/2/4/tjant-8-2-4.pdf arithmetic functions exponential sums