On the average number of cyclic subgroups of the groups $${\mathbb {Z}}_{n_1} \times {\mathbb {Z}}_{n_2}\times {\mathbb {Z}}_{n_3}$$ with $$n_1,n_2,n_3\le x$$

Abstract

Let Z n be the additive group of residue classes modulo n. Let c(n 1 , n 2 , n 3) denote the number of cyclic subgroups of the group Z n 1 × Z n 2 × Z n 3 , where n 1 , n 2 and n 3 are arbitrary positive integers. In this paper we obtain an asymptotic formula for the sum

ON THE WEIGHTED AVERAGE NUMBER OF CYCLIC SUBGROUPS OF Zl×Zm×Zn WITH lmn ⩽ x

“…Latter, Tóth and Zhai [13] improved the error term in (1.7) to O(x 3/2 (log x) 6.5 ) by means of multiple complex function theory and exponential sums theory. For r = 3, by using a multidimensional Perron's formula and the complex integration method, Tóth and Zhai [14] give the following asymptotic formula…”