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**Zentralblatt MATH**

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

**Zbl.No: ** 517.10048

**Autor: ** Erdös, Paul; Sárközy, András

**Title: ** Some asymptotic formulas on generalized divisor functions. I. (In English)

**Source: ** Studies in pure mathematics, Mem. of P. Turan, 165-179 (1983).

**Review: ** [This article was published in the book announced in Zbl 512.00007.]

For a given sequence A = **{**a_{1}ya_{2} < ...**}** of positive integers, define the counting function N_{A}(x) = **sum**_{a in A, A \leq x}l, the logarithmic counting function f_{A}(x) = **sum**_{a in A, A \leq x}a^{-l}, and the generalized divisor function \tau_{A}(n) = **sum**_{a|n, a in A}l; denote its maximum by D_{A}(x) = **max**_{1 \leq n \leq x}\tau_{A}(n). The authors' aim is to look for ``large'' values of D_{A}(x) and to check the truth of the ``natural'' conjecture **lim**_{x ––> oo}\frac{D_{A}(x)}{f_{A}(x)} = oo , if N_{A}(x) ––> oo. (*) However, the authors disprove this conjecture: There exists an infinite sequence A such that **lim**\sup_{x ––> oo}f_{A}(x)/ log log x > 0, but \sup D_{A}(x)/f_{A}(x) is finite.

Then the authors prove theorems giving large values for D_{A}(x), for example: If **lim**_{x ––> oo}f_{A}(x) = oo, then

**limsup**_{x ––> oo}D_{A}(x)/f_{A}(x) = oo. This theorem shows that a weakened version of conjecture (*) is true. **{**Parts II-IV of this series are published in J. Number Theory 15, 115-136 (1982; Zbl 488.10043), Acta Arith. 41, 395-411 (1982; Zbl 492.10037), Stud. Sci. Math. Hung. 15, 467-479 (1980; Zbl 512.10037)**}**.

**Reviewer: ** W.Schwarz

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

**Keywords: ** generalized divisor functions; sets of integers; lower bounds for divisor functions; large values

**Citations: ** Zbl.512.00007; Zbl.488.10043; Zbl.512.10037; Zbl.492.10037

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag