Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  435.10027
Autor:  Erdös, Paul; Hall, R.R.
Title:  Values of the divisor function on short intervals. (In English)
Source:  J. Number Theory 12, 176-187 (1980).
Review:  Let \tau(n) denote the number of divisors of n. The following two theorems are proved. I. For every fixed positive integer k one has

sumn < kmax{\tau(n),\tau(n+1),...\tau(n+k-1)} ~ kx log x,    x ––> oo.

This result still holds when k depends on x, limx ––> ook = oo, provided that k = o((x log x)3-2\sqrt{2}), x ––> oo. For the minimum taken by the divisor function on an interval of length k the problem turns out to be much more difficult. The sharpest result obtained here is the following. II. If k is a fixed positive integer and \alphak = k(21/k-1), then

\frac{c1(k)x(log x)\alphak}{(log log x)11k2} \leq sumn < xmax{\tau(n),\tau(n+1),...\tau(n+k+1)} \leq C2(k)x(log x)\alphak.

The proofs of the first theorem and the right hand inequality of the second theorem are elementary and presented in a series of six short lemmata. The left hand inequality of the second theorem is proved by an application of a lower bound form of the Selberg sieve.
Reviewer:  H.Jager
Classif.:  * 11N37 Asymptotic results on arithmetic functions
                   11A25 Arithmetic functions, etc.
                   11N35 Sieves
Keywords:  divisor function; short intervals; Selberg sieve

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