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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