Baker, Roger C.; Harman, Glyn Sparsely totient numbers. (English) Zbl 0871.11060 Ann. Fac. Sci. Toulouse, VI. Sér., Math. 5, No. 2, 183-190 (1996). D. W. Masser and 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 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 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 \(v^\delta<x<v^2\) then there are \(\gg x/v\log x\) primes \(p\) in the interval \(2v<p<3v\) with the fractional parts \(\{x/p\}\) exceeding \(1-x/16v^2\). Reviewer: P.Shiu (Loughborough) MSC: 11N36 Applications of sieve methods 11N25 Distribution of integers with specified multiplicative constraints 11L07 Estimates on exponential sums Keywords:sparsely totient numbers; largest prime divisor; estimates on exponential sums; fractional parts Citations:Zbl 0538.10006; Zbl 0732.11049; Zbl 0687.10028 PDFBibTeX XMLCite \textit{R. C. Baker} and \textit{G. Harman}, Ann. Fac. Sci. Toulouse, Math. (6) 5, No. 2, 183--190 (1996; Zbl 0871.11060) Full Text: DOI Numdam EuDML Online Encyclopedia of Integer Sequences: Sparsely totient numbers; numbers n such that m > n implies phi(m) > phi(n). References: [1] Baker, R.C.) . - The greatest prime factor of the integers in an interval, Acta Arithmetica47 (1986), pp. 193-231. · Zbl 0553.10035 [2] Baker, R.C.) and Harman .- Numbers with a large prime factor, Acta Arith.73 (1995), pp. 119-145. · Zbl 0834.11037 [3] Baker, R.C.), Harman and Rivat, J.) .- Primes of the form [nc], J. of Number Theory, 50 (1995), pp. 261-277. · Zbl 0822.11062 [4] Fouvry, E.) and Iwaniec, H.) . - Exponential sums with monomials, J. Number Theory33 (1989), pp. 311-333. · Zbl 0687.10028 [5] Harman, G.) .- On the distribution of αp modulo one, J. London Math. Soc.27 (1983), pp. 9-18. · Zbl 0504.10018 [6] Harman, G.) . - On sparsely totient numbers, Glasgow Math. J.33 (1991), pp. 349-358. · Zbl 0732.11049 [7] Iwaniec, H.) and Laborde, M.) .- P2 in short intervals, Ann. Inst. Fourier, Grenoble, 31 (1981), pp. 37-56. · Zbl 0472.10048 [8] , H.-Q.) .- The greatest prime factor of the integers in an interval, Acta Arith., 65 (1993), pp. 301-328. · Zbl 0797.11071 [9] Masser, D.W.) and Shiu, P.) . - On sparsely totient numbers, Pacific J. Math.121 (1986), pp. 407-426. · Zbl 0538.10006 [10] , J.) .- P2 dans les petits intervalles, Séminaire de Théorie des Nombres de Paris (1989-90), Birkhaüser. · Zbl 0743.11050 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.