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On the number of divisors which are values of a polynomial

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Abstract

Let τ(n) be the number of positive divisors of an integer n, and for a polynomial P(X)∈ℤ[X], let

$$\tau_{P}(n)=\left\vert{\left\{{P(m)>0:\ m\in\mathbb{Z},P(m)\mid n}\right\}}\right\vert.$$

R. de la Bretèche studied the maximum values of τ P (n) in intervals. Here the following is proved: if P(X)∈ℤ[X] is not of the form a(X+b)k with a,b∈ℚ, and k∈ℕ then

$$\tau_{P}(n)\ll(\log n)\tau(n)^{3/5}.$$

This improves partially on La Bretèche’s results.

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Authors and Affiliations

  1. Number Theory Department, Alfred Renyi Institute of Mathematics, Realtanoda utca 13-15, H1053, Budapest, Hungary

    Katalin Gyarmati

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  1. Katalin Gyarmati
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Correspondence to Katalin Gyarmati.

Additional information

Research partially supported by Hungarian National Foundation for Scientific Research, Grants T043631, T043623 and T049693.

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Gyarmati, K. On the number of divisors which are values of a polynomial. Ramanujan J 17, 387–403 (2008). https://doi.org/10.1007/s11139-006-9010-8

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  • DOI: https://doi.org/10.1007/s11139-006-9010-8

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