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**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 **sum**_{n < k}**max****{**\tau(n),\tau(n+1),...\tau(n+k-1)**}** ~ kx log x, x ––> oo. This result still holds when k depends on x, **lim**_{x ––> oo}k = 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 \alpha_{k} = k(2^{1/k}-1), then

\frac{c_{1}(k)x(log x)^{\alphak}}{(log log x)^{11k2}} \leq **sum**_{n < x}**max****{**\tau(n),\tau(n+1),...\tau(n+k+1)**}** \leq C_{2}(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

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag