On the theorem of Erdos-Kac

Abstract

Bu çalışmada, Olasılıksal Sayılar Teorisinin temel teoremi olarak bilinen Erdös-Kac teoreminin farklı ispatlarının yanısıra aynı konuyla ilgili birkaç çalışma verilmiştir. Öncelikle ana teoremi kompleks fonksiyon teorisi kullanmadan elementer metotlarla ispatlayarak hata terimi içermeyen bir asimptotik buluyoruz. İkinci kısımda, olasılıktan gelen bir teoremle, Riemann zeta-fonksiyonun ve genel Dririchlet serilerinin bazı özellikleri kullanılarak bulunan asimptotiğin yanısıra hata terimi de hesaplanmıştır. Son kısımda ise, konuyla ilgili elementer metotlar kullanılarak yapılan iki farklı çalışma incelenmiştir.

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.