##
**Zentralblatt MATH**

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

**Zbl.No: ** 359.10038

**Autor: ** Alladi, K.; Erdös, Paul

**Title: ** On an additive arithmetic function. (In English)

**Source: ** Pac. J. Math. 71, 275-294 (1977).

**Review: ** Let n be a positive integer, n = **prod**_{i = 1}^{r}p_{i}^{\alpha1} in canonical form, and let A(n) = **sum**_{i = 1}^{r}\alpha_{1}p_{1}. Clearly A is an additive arithmetic function. Assume the primes p_{1} are arranged so that p_{1} \leq p_{2} < ... < p_{r}. Define P_{1}(n) = p_{r} and, in general, P_{k}(n) = P_{1}(n/P_{1}(n)... P_{k-1}(n)) for k \leq **sum**_{i = 1}^{r}\alpha_{1} and P_{k}(n) = 0 for k > **sum**_{i = 1}^{r}\alpha_{1}. If f and g are arithmetic functions such that **sum**_{n \leq x}f(n) ~ **sum**_{n < x}g(n), and if g is a well behaved function (e.g. polynomial, exponential), then g is referred to as the average order of f. It is proved that for all positive integers, we have **sum**_{n \leq x}P_{m}(n) ~ **sum**_{n \leq x}**{**A(n)-P_{1}(n)-... -P_{m-1}(n)**}** ~ k_{m}x^{1+(1/m)}/(log x)_{m} where k_{m} is a positive constant depending only on m. It follows almost immediately from this theorem that the average order of A(n) is \pi_{2}n/6 log n. Let A^*(n) = **sum**_{i = 1}^{r}p_{1}. Then the average order of A^*(n) is also \pi^{2}n/6 log n, and the average order of A(n)-A^*(n) is log log n. For any fixed positive integer M, the set of solutions to A(n)-A^*(n) = M has a positive natural density. Now A(n) = n if and only if n is a prime or n = 4. Call n a ''special number'' if n\equiv O(mod A(n)) and n\neq A(n), and let **{**l_{n}**}** be the sequence of special numbers. This paper's first author has previously proved that the sequence **{**l_{n}**}** is infinite [Srinivasa Ramanujan Commemoration Volume, Oxford Press, Madras, India, (1974) part II] . Denote by *L*(x) the number of \ell_{n} \leq x. It is shown that there exist positive constants c, c' such that

*L*(x) = O(xe^{-c\sqrt{log x log log x}}) and *L*(x) >> xe^{-c'\sqrt{log x log log x}}. Finally, let \alpha(n) = (-1)^{A(n)}. It is proved that there exists a positive constant c'' such that

**sum**_{1 \leq n \leq x}\alpha(n) = O(xe^{-{c''}\sqrt log x log log x}), and that **sum**_{n = 1}^{oo}\alpha(n)/n = 0.

**Reviewer: ** B.Garrison

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

11K65 Arithmetic functions (probabilistic number theory)

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag