D. W. Masser and P. Shiu [Pac. J. Math. 121, 407-426 (1986; Zbl 0538.10006)] called a sparsely totient number if for all . Various interesting properties of such numbers were established. For example, if is the largest prime divisor of , then . Although they conjectured that , they only managed to prove that .
Subsequently G. Harman [Glasg. Math. J. 33, 349-358 (1991; Zbl 0732.11049)] made significant progress by means of the estimation of exponential sums to prove that holds for an exponent .
Making use of work of E. Fouvry and H. Iwaniec [J. Number Theory 33, 311-333 (1989; Zbl 0687.10028)] on exponential sums, the authors make further improvement by showing that is admissible. The proof amounts to showing that if then there are primes in the interval with the fractional parts exceeding .