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**Zentralblatt MATH**

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

**Zbl.No: ** 419.10006

**Autor: ** Erdös, Paul; Hall, R.R.

**Title: ** On the Möbius function. (In English)

**Source: ** J. Reine Angew. Math. 315, 121-126 (1980).

**Review: ** The function M(n,T) = **sum**{\mu(d): d| n,d \leq T} is studied in this paper. It is shown that M(n,T) is usually zero, in two senses. First, the density of the integers n such that M(n,T)\ne0 tends to zero as a function of T. Second, as previously conjectured by Erdös, for almost all n we have **sum****{**1/T: M(n,T)\ne0**}** = o(log n]. Both results are given in precise quantitative form, and are shown to be connected with other conjectures and unsolved problems, in particular with Erdös' conjecture that almost all integers n have two divisions d,d' such that d < d' < 2d.

**Classif.: ** * 11A25 Arithmetic functions, etc.

11N05 Distribution of primes

**Keywords: ** Möbius function; density

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag