Malaysian Journal of Mathematical Sciences, September 2016, Vol. 10, No. 3
Notes on Certain Arithmetic Inequalities Involving Two Consecutive Primes
Bellaouar Djamel
Corresponding Email: bellaouardj@yahoo.fr
Received date: -
Accepted date: -
Abstract:
Let $r$, $k$ be positive integers (parameters) with $r \geq 2$, and let $p_r$ be the $r$-th prime number. Let $W_k$ denote the set of positive integers $n$ for which the number of distinct prime factors of $n$ is greater or equal to $k$. By using the prime number theorem and Bertrand's theorem, we will determine arithmetic functions $f$, $g:\mathbb{N}\rightarrow \mathbb{N}$ for which $f(n)-\alpha_{r}g(n)$ has infinitely many sign changes on the set $W_k$, where $\alpha_{r}=\frac{p_{r}-1}{p_{r}}$. In the framework of internal set theory (for more details, see Nelson (1977)), some notions concerning nonstandard analysis as well as unlimited positive integers have been used.
Keywords: Arithmetic functions, prime number theorem, Bertrand's theorem, sign changes, internal set theory
Indexing
