Inequalities For The Primes Counting Function

Abstract

The prime counting function inequality $\pi (x+y)<\pi (x)+\pi (y)$, which is known as Hardy-Littlewood conjecture, has been established for a variety of cases such as $\delta x\le y\le x$, where $0<\delta \le 1$, and $x\le y\le x\log x\log \log x$ as $x\to \infty$. The goal in note is to extend the inequality to the new larger ranges $\ge x{\log }^{-c}x\le y\le x$, where $c\ge 0$ is a constant, unconditionally; and for $\ge x^{1/2}{\log }^{3}x\le y\le x$, conditional on a standard conjecture.

Six Pages. Keyword: Distribution of prime; Prime in short interval; Hardy-Littlewood conjecture; Prime $k$-tuple

References in corpus (2)

• Estimates of Some Functions Over Primes without R.H.
• On some reasons for doubting the Riemann hypothesis