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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 \(\phi(m)>\phi(n)\) 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 \(1-x/16v^2\).

MSC:

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