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 = {a1ya2 < ...} of positive integers, define the counting function NA(x) = suma in A, A \leq xl, the logarithmic counting function fA(x) = suma in A, A \leq xa-l, and the generalized divisor function \tauA(n) = suma|n, a in Al; denote its maximum by DA(x) = max1 \leq n \leq x\tauA(n). The authors' aim is to look for ``large'' values of DA(x) and to check the truth of the ``natural'' conjecture

limx ––> oo\frac{DA(x)}{fA(x)} = oo , if NA(x) ––> oo.    (*)

However, the authors disprove this conjecture: There exists an infinite sequence A such that lim\supx ––> oofA(x)/ log log x > 0, but \sup DA(x)/fA(x) is finite.
Then the authors prove theorems giving large values for DA(x), for example: If limx ––> oofA(x) = oo, then

limsupx ––> ooDA(x)/fA(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

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