Title: An identity for a class of arithmetical functions of several variables
Abstract: Johnson [1] evaluated the sum<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="$\sum\nolimits_{d|n} {\left| {C\left( {d;r} \right)} \right|} $"><mml:msub><mml:mo>∑</mml:mo><mml:mrow><mml:mi>d</mml:mi><mml:mo>|</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:mi>C</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>d</mml:mi><mml:mo>;</mml:mo><mml:mi>r</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>|</mml:mo></mml:mrow></mml:mrow></mml:math>, where<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="$C\left( {n;r} \right)$"><mml:mi>C</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>;</mml:mo><mml:mi>r</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>denotes Ramanujan's trigonometric sum. This evaluation has been generalized to a wide class of arithmetical functions of two variables. In this paper, we generalize this evaluation to a wide class of arithmetical functions of several variables and deduce as special cases the previous evaluations.