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On the difference between the number of prime divisors at consecutive integers

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Abstract

Suppose thatg(n) is equal to the number of divisors ofn, counting multiplicity, or the number of divisors ofn, a≠0 is an integer, andN(x,b)=|{n∶n≤x, g(n+a)−g(n)=b orb+1}|. In the paper we prove that sup b N(x,b)≤C(a)x)(log log 10 x )−1/2 and that there exists a constantC(a,μ)>0 such that, given an integerb |b|≤μ(log logx)1/2,x≥x o, the inequalityN(x,b)≥C(a,μ)x(log logx(−1/2) is valid.

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Authors and Affiliations

  1. Vladimir State Pedagogical University, USSR

    N. M. Timofeev

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  1. N. M. Timofeev
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Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 579–595, October, 1999.

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Timofeev, N.M. On the difference between the number of prime divisors at consecutive integers. Math Notes 66, 474–488 (1999). https://doi.org/10.1007/BF02679098

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  • DOI: https://doi.org/10.1007/BF02679098

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