On Certain Large Additive Functions

pα||n α. As usual p will denote primes, pα||n means that p divides n but p does not, and a function f(n) is additive if f(mn) = f(m)+f(n) whenever (m,n) = 1. From the pioneering works of Alladi and Erdős [1]-[2], P. Erdős’s perspicacity and insight have been one of the main driving forces in the research that brought on many results on summatory functions of large additive functions and P (n). The functions ω(n) and Ω(n) may be successfully investigated by various analytical methods. In fact, it is Erdős who in two classical works with M. Kac [14], [15] established the Gaussian distribution law for these functions. A general principle is that z is a multiplicative function whenever f(n) is an additive function. Thus from the Euler product representation (1.3) ∞ ∑