IJPAM: Volume 18, No. 3 (2005)

LOWER BOUNDS FOR THE SUM
DIVISOR FUNCTION

Barbara Medryk
Department of Didactical Mathematics and Number Theory
Faculty of Mathematics, Computer Sciences and Econometrics
University of Zielona Góra
Ul. Professor Szafrana 4a, Zielona Góra, 65-516, POLAND
e-mail: B.Medryk@wmie.uz.zgora.pl

Abstract. Let $\sigma (n)$ be the sum divisor function and let denote the square-free kernel of positive integer . We prove that for every , such that $% 2\leq k\leq r=\omega (n)$ we have (*) $\sigma (n)>(\sqrt[r]{n}+% \sqrt[r]{n_{0}})^{r}\geq (\sqrt[k]{n}+\sqrt[k]{n_{0}})^{k},$ where $n_{0}=\frac{n}{s(n)}$ and $\omega (n)$ is the number of distinct prime divisor of . Moreover, we prove that for infinitely many we have $\sigma (n)>\frac{6}{\pi ^{2}}% e^{\gamma }n\log \log n$ , where $\gamma \approx 0.57721$ is Euler's constant.