A remark on the number of distinct prime divisors of integers

Mehdi Hassani


We study the asymptotic formula for the sum $\sum_{n\leqslant x}\o(n)$ where $\o(n)$ denotes the number of distinct prime divisors of $n$, and we perform some computations which detect curve patterns in the distribution of a related sequence.


Prime number, arithmetic function

Full Text:



S. R. Finch, Mathematical constants, Encyclopedia of Mathematics and its Applications, 94, Cambridge University Press, Cambridge, 2003.

G. H. Hardy and S. Ramanujan, The normal number of prime factors of a number $n$, Quart. J. Math., 48(1917), 76--92.

J. Lee, The second central moment of additive functions, Proc. Amer. Math. Soc., 114(1992), 887--895.

J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math., 6 (1962), 64--94.

C. de la Vallee Poussin, Sur les valeurs moyennes de certaines fonctions arithmetiques, Annales de la

societe scientifique de Bruxelles, 22(1898), 84--90.

DOI: http://dx.doi.org/10.5556/j.tkjm.48.2017.1842

Sponsored by Tamkang University | ISSN 0049-2930 (Print), ISSN 2073-9826 (Online) | Powered by MathJax