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**Zentralblatt MATH**

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

**Zbl.No: ** 455.10031

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

**Title: ** Estimates for sums involving the largest prime factor of an integer and certain related additive functions. (In English)

**Source: ** Stud. Sci. Math. Hung. 15, 183-199 (1980).

**Review: ** Let P(n) denote the largest prime factor of an integer n \geq 2 and lrt \beta(n) = **sum**_{p|n}p, B(n) = **sum**_{pa||n}p^{a}, where as usual p^{a}||n means that p^{a} divides n, but p^{a+1} does not. The additive functions \beta, B and B_{1} have been studied recently in several works, including *K.Alladi* and *P.Erdös* [Pac. J. Math. 82, 295-315 (1979; Zbl 419.10042)], *L.-M.DeKoninck* and *A.Ivic* [Topics in arithmetical functions (Amsterdam 1980; Zbl 442.10032)] and *A.Ivic* [Arch. Math. 36, 57-61 (1980; Zbl 436.10019)]. The purpose of this work is to estimate sums of the form **sum**_{2 \leq n \leq x}f(n)/g(n) when f\neq g and f,g in **{**P(n),\beta(n),B(n),B_{1}(n)**}**. Some of these sums have been investigated by the aforementioned authors, and for those which were not following asymptotic formulas are obtained: **sum**_{2 \leq n \leq x}P(n)/B_{1}(n) = x+O(x log log x/ log x), (1)

**sum**_{2 \leq n \leq x}B_{1}(n)/P(n) = e^{\gamma}x log log x+O(x), (2)

**sum**_{2 \leq n \leq x}B_{1}(n)/B(n) = Dx+O(x log^{-1/3}x), (3)

**sum**_{2 \leq n \leq x}B_{1}(n)/\beta(n) = e^{\gamma}x log log x+O(x), (4) Here \gamma is Eulers's constant, D > 1 may be explicitly evaluated, and (1) remains true if P(n) is replaced by either \beta(n) or B(n). The proofs depend on six lemmas and utilize estimates for \psi(x,y), the number of positive integers \leq x all of whose prime factors are \leq y. The methods of proof allow one to improve the error term in

**sum**_{2 \leq n \leq x}B(n)/P(n) = x+O(x log log x/ log x), (5) to O(x/ log x); (5) was obtained in the aforementioned work of K. Alladi and P. Erdös. The formulas (1)-(4) improve the results of Ch. 6 of the above cited book by J.-M- De Koninck and A. Ivic.

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

**Keywords: ** sums involving largest prime factor; related additive functions; additive
functions; asymptotic formulas

**Citations: ** Zbl.419.10042; Zbl.442.10032; Zbl.436.10019

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