On a sum involving the Euler function

Abstract We obtain reasonably tight upper and lower bounds on the sum ∑ n ⩽ x φ ( ⌊ x / n ⌋ ) , involving the Euler functions φ and the integer parts ⌊ x / n ⌋ of the reciprocals of integers. For slower growing arithmetic functions f we obtain asymptotic formulas for similar sums of f ( ⌊ x / n ⌋ ) . These are analogues of a series of previous results for sequences involving the integer part functions such Beatty and Piatetski–Shapiro sequences.