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Sparsely totient numbers. (English) Zbl 0871.11060

D. W. Masser and P. Shiu [Pac. J. Math. 121, 407-426 (1986; Zbl 0538.10006)] called n a sparsely totient number if φ(m)>φ(n) for all m>n. Various interesting properties of such numbers were established. For example, if P(n) is the largest prime divisor of n, then lim infP(n)/logn=1. Although they conjectured that lim supP(n)/logn=2, they only managed to prove that P(n)log 2 n.

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 P(n)log δ n holds for an exponent δ>122/65.

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 δ=37/20 is admissible. The proof amounts to showing that if v δ <x<v 2 then there are x/vlogx primes p in the interval 2v<p<3v with the fractional parts {x/p} exceeding 1-x/16v 2 .

MSC:
11N36Applications of sieve methods
11N25Distribution of integers with specified multiplicative constraints
11L07Estimates on exponential sums