A note on arithmetic Diophantine series

We consider an asymptotic analysis for series related to the work of Hardy and Littlewood (1923) on Diophantine approximation, as well as Davenport. In particular, we expand on ideas from some previous work on arithmetic series and the RH. To accomplish this, Mellin inversion is applied to certain infinite series over arithmetic functions to apply Cauchy’s residue theorem, and then the remainder of terms is estimated according to the assumption of the RH. In the last section, we use simple properties of the fractional part function and its Fourier series to state some identities involving different arithmetic functions. We then discuss some of their individual properties, such as convergence, as well as implications related to known work.