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Sparsely totient numbers. (English) Zbl 0871.11060
{\it D. W. Masser} and {\it P. Shiu} [Pac. J. Math. 121, 407-426 (1986; Zbl 0538.10006)] called $n$ a sparsely totient number if $\phi(m)>\phi(n)$ 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 $\liminf P(n)/\log n=1$. Although they conjectured that $\limsup P(n)/\log n=2$, they only managed to prove that $P(n)\ll\log^2n$. Subsequently {\it 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 $P(n)\ll \log^\delta n$ holds for an exponent $\delta>122/65$. Making use of work of {\it E. Fouvry} and {\it 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 $v^\delta<x<v^2$ then there are $\gg x/v\log x$ primes $p$ in the interval $2v<p<3v$ with the fractional parts $\{x/p\}$ exceeding $1-x/16v^2$.
##### MSC:
 11N36 Applications of sieve methods 11N25 Distribution of integers with specified multiplicative constraints 11L07 Estimates on exponential sums
Full Text:
##### References:
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