International Journal of Pure and Applied Mathematics Volume 119 No. 2 2018, 369-374 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: 10.12732/ijpam.v119i2.9 P A ijpam.eu SOME ARITHMETIC FUNCTIONS AND THEIR MEANS G. Sudhaamsh Mohan Reddy 1 § , S. Srinivas Rau 2 , B. Uma 3 1,2 Department of Mathematics Faculty of Science and Technology The Icfai Foundation for Higher Education, Hyderabad (Deemed to be University under section 3 of UGC Act, 1956), INDIA 3 CTW, Military College Secundrerabad, 500015, INDIA Abstract: We give four formulas for d(n 2 ) in terms of arithmetical functions ω, µ, d3 and d: d(n 2 )=  k|n d(k)µ 2 ( n k )=  k 2 |n µ(k)d3( n k 2 )=  k 2 |n (  m|n d(m))µ(k)=  k|n 2 ω(k) . We estimate the partial sum  n≤x d(n 2 )=( 1 ζ (2) + ◦(1))xlog 2 x using Tauberian theorem. Similarly, we estimate the partial sums for 1 d(n) , logn d(n) , and σ(n) d(n) . AMS Subject Classification: 11M06, 11M41, 11M45 Key Words: arithmetical functions, divisor functions, Tauberian theorem 1. Introduction We consider some arithmetical functions whose mean is not finite; in such cases the order of growth of the partial sums is determined. Our examples are (i) d(n 2 ) (ii) 1 d(n) (iii) logn d(n) (iv) σ(n) d(n) . Our main tools are Abel’s summation formula and a generalized Tauberian theorem, due to H. Delange. Received: April 24, 2017 Revised: July 6, 2018 Published: July 6, 2018 c  2018 Academic Publications, Ltd. url: www.acadpubl.eu § Correspondence author