Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  456.10024
Autor:  Erdös, Paul; Katai, I.
Title:  On the maximal value of additive functions in short intervals and on some related questions. (In English)
Source:  Acta Math. Acad. Sci. Hung. 35, 257-278 (1980); correction ibid. 37, 499 (1981).
Review:  Let g(n) be a non-negative strongly additive function, fk(n) = maxj = i,...,kg(n+j),

\rho(k,\epsilon) = \supx \geq 1 1/x card\left{n \leq x|fk(n) > (1+\epsilon)fk(0)\right}

\delta(k0,\epsilon) = \supx \geq 1 1/x card\left{n \leq x|\exists k, k > k0, fk(n) > (1+\epsilon)fk(0)\right}

and let

\theta(k,\epsilon) = lim\supx = oo 1/x card\left{n \leq x|fk(n) > (1+\epsilon)fk(0)\right}.

One notes that \theta(k,\epsilon) \leq \rho(k,\epsilon) and \supk \geq k0\rho(k,\epsilon) \leq \delta(k0,\epsilon). It is shown that if \rho(k,\epsilon) ––> 0 (k ––> oo) for all \epsilon > 0, then sump\frac{g(p)r}{p} < oo for every r \geq 1, thus considerably stregthening a result previously obtained by the authors [Acta Math. Acad. Sci. Hung. 33, 345-359 (1979; Zbl 417.10039)]. Among other results obtained (space limitations unfortunately preclude giving the complete list), we find the following: If g(p) = 1/p for promes p then

\supx \geq 1 1/x card\left{n \leq x|\exists k > k0,fk(n) > fk(0)+\lambdak\right} ––> 0, (k0 ––> oo),

where \lambdak = 3/ log log k, while if g(p) = 1(p\delta, 0 < \delta < 1, and \rho > 0 is an arbitrary constant, then

limk ––> oolimx ––> oo 1/x card\left{n \leq x|fk(n) > fk(0)+(log k)1-\delta-\rho\right} = 1.

In the paper reviewed above the authors stated erroneously that Theorem 1 is a consequence of Theorem 1'. In fact, the converse implication is true: Theorem 1 implies Theorem 1'. A proof of Theorem 1 is given.
Reviewer:  B.Garrison
Classif.:  * 11N37 Asymptotic results on arithmetic functions
                   11K65 Arithmetic functions (probabilistic number theory)
Keywords:  distribution of maximal value; strongly additive function

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