**Sankhya: The Indian Journal of Statistics**

1995, Volume 57, Series A, Pt. 1, pp. 29--32

**ABSOLUTE CONTINUITY OF THE DISTRIBUTIONS OF ADDITIVE ARITHMATIC FUNCTIONS**

By

GUTTI JOGESH BABU, *Pennsylvania State University*

SUMMARY. It is well known that the distribution of a real valued additive arithmetic function f, if it exists, is either purely discrete, purely continuous singular, or purely absolutely continuous. In this paper the distribution of f is shown to be absolutely continuous, if f(p) does not approach zero "too fast", as $p \rightarrow \infty$ through primes.

*AMS (1980) subject classification.* Primary 11N60, 11K65; secondary 60E10; 11N64.

*Key words and phrases.* Singular distribution, characteristic function, Plancherel theorem.