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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 $\phi \left(m\right)>\phi \left(n\right)$ for all $m>n$. Various interesting properties of such numbers were established. For example, if $P\left(n\right)$ is the largest prime divisor of $n$, then $lim infP\left(n\right)/logn=1$. Although they conjectured that $lim supP\left(n\right)/logn=2$, they only managed to prove that $P\left(n\right)\ll {log}^{2}n$.

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 $P\left(n\right)\ll {log}^{\delta }n$ holds for an exponent $\delta >122/65$.

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 $\delta =37/20$ is admissible. The proof amounts to showing that if ${v}^{\delta } then there are $\gg x/vlogx$ primes $p$ in the interval $2v with the fractional parts $\left\{x/p\right\}$ exceeding $1-x/16{v}^{2}$.

MSC:
 11N36 Applications of sieve methods 11N25 Distribution of integers with specified multiplicative constraints 11L07 Estimates on exponential sums