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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 704.11032

**Autor: ** Erdös, Paul; Ivic, Aleksandar

**Title: ** The distribution of values of a certain class of arithmetic functions at consecutive integers. (In English)

**Source: ** Number theory. Vol. I. Elementary and analytic, Proc. Conf., Budapest/Hung. 1987, Colloq. Math. Soc. János Bolyai 51, 45-91 (1990).

**Review: ** [For the entire collection see Zbl 694.00005.]

The problem considered is that of the distribution of values of certain arithmetic functions, especially of the values at consecutive integers. The main motivation is the function a(n), which counts the number of non- isomorphic Abelian groups of order n. One of the main results is **sum**_{n \leq x; a(n) = a(n+1)} 1 = Ax+O(x^{3/4} log^{4}x). The result can be extended to nonnegative integer-valued arithmetic functions with squarefull kernel.

A great part of the paper deals with the functions C(x) and D(x), which denote the number of distinct values a(n) for n \leq x and the number of n \leq x such that n = a(m) for some m, respectively. For these functions lower bounds are given. The conjecture C(x) = \exp (log^{ ½+o(1)}x) and a similar one for D(x) are proved if a certain conjecture involving the partition function is assumed.

Finally it is proved that there are infinitely many n such that the values a(n+1),...,a(n+t) are all distinct for t = [C(log n/ log log n)^{ ½}] (C > 0).

**Reviewer: ** E.Krätzel

**Classif.: ** * 11N37 Asymptotic results on arithmetic functions

11N45 Asymptotic results on counting functions for other structures

**Keywords: ** asymptotic results; arithmetic functions; values at consecutive integers; number of non-isomorphic Abelian groups

**Citations: ** Zbl 694.00005

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