Erdös-Kac teoremi üzerine

This study presents different proofs and applications of the celebrated Erdös-Kac theorem, named after Paul Erdös and Mark Kac, also known as the fundamental theorem of probabilistic number theory which states that if n is a randomly chosen large integer, then the number of distinct prime factors of n has approximately the normal distribution with mean and variance log log n.We first give the original proof from the authors which is so called elementary proof meaning that the complex functin theory is not used. This proof does not give an error term but rather gives an asymptotic result. In the second part we give the proof of A. Renyi and P. Turan which makes use of the standard tools of analytic number theory, Dirichlet series, contour integration. Although the latter method is not elementary, it is much simpler than the original proof and also helps us get an error term besides an asymptotic result.Finally we use the article " On the Normal Number of Prime Factors of phi (n)" by Paul Erdös and Carl Pomerance where "phi" is Euler's function. We also give the related result for the divisor function which counts the number of positive divisors for a given integer.