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**Zentralblatt MATH**

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

**Zbl.No: ** 186.35804

**Autor: ** Erdös, Pál; Katai, I.

**Title: ** On the sum **sum** d_{4}(n) (In English)

**Source: ** Acta Sci. Math. 30, 313-324 (1969).

**Review: ** Let d(n) denote the number of divisors of n, and d_{k}(n) be the k-fold iterate of d(n), i. e. d_{1}(n) = d(n) and d_{k}(n) = d(d_{k-1}(n)) for k \geq 2. It was conjectured by Bellman and Shapiro that the relation **sum**_{n \leq k} d_{k}(n) = c_{k}(1+o(1))x log_{k}x holds, where log_{k} denotes the k-fold iterate of logarithm function. This was proved previously for k = 2 by the authors independently, for k = 3 by Kátai. Here the authors prove the case k = 4. The cases k \geq 5 seem to be very difficult.

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

**Index Words: ** number theory

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