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On a problem of Erdös and Alladi

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Abstract

We give an estimate for the quantity Σ{f(n):n≦x, p(n)≦y}, wherep(n) denotes the greatest prime factor ofn andf belongs to a certain class of multiplicative functions. As an application, we show that for the Moebius function, (Σ{μ(n):n≦x, p(n)≦y}) (Σ{1:n≦x, p(n)≦y})−1 tends to zero, asx→∞, uniformly iny≧2, and thus settle a conjecture of Erdös.

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References

  1. Alladi, K. (Ed.): Number Theory. Proc. Third Matsci. Conf. Lecture Notes 938. Berlin-Heidelberg-New York: Springer. 1982.

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  2. Alladi, K.: Asymptotic estimates for sums involving the Moebius function II. Trans. Am. Math. Soc.272, 87–105 (1982).

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  3. Ennola, V.: On numbers with small prime divisors. Ann. Acad. Sci. Fenn. Ser. AI440 (1969).

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Authors and Affiliations

  1. Department of Mathematics, University of Illinois, 1409 West Green Street, 61801, Urbana, IL, USA

    Adolf Hildebrand

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  1. Adolf Hildebrand
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Supported by a grant from the Deutsche Forschungsgesellschaft.

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Hildebrand, A. On a problem of Erdös and Alladi. Monatshefte für Mathematik 97, 119–124 (1984). https://doi.org/10.1007/BF01653241

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  • DOI: https://doi.org/10.1007/BF01653241

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