Mean square estimates for the error terms of sums of the greatest common divisor

Abstract

Abstract We investigate the mean square behavior of error terms that arise in the summation of gcd ( n 1 , … , n k ) $\gcd (n_1,\ldots ,n_k)$ gcd left parenthesis n 1 comma ellipsis comma n Subscript k Baseline right parenthesis with n 1 … n k ≤ x $n_1\ldots n_k \le x$ n 1 ellipsis n Subscript k Baseline less than or equals x . We first analyze error terms related to certain Dirichlet series ζ 2 ( s ) ζ ( 2 s − 1 ) ζ M ( 2 s ) $\zeta ^2(s)\zeta (2s-1)\zeta ^M(2s)$ zeta squared left parenthesis s right parenthesis zeta left parenthesis 2 s minus 1 right parenthesis zeta Superscript upper M Baseline left parenthesis 2 s right parenthesis and obtain upper bounds for their mean squares, depending on the parameter M . We then extend our analysis to the general case with k ≥ 3 $k \ge 3$ k greater than or equals 3 , providing estimates for the mean square of the corresponding error terms.