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arXiv · 1808.02366

Inequalities For The Primes Counting Function

N. A. Carella
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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 \leq y \leq x$, where $0< \delta \leq 1$, and $x \leq y\leq x \log x \log \log x$ as $ x \to \infty$. The goal in note is to extend the inequality to the new larger ranges $\geq x \log^{-c}x\leq y \leq x$, where $c\geq 0$ is a constant, unconditionally; and for $\geq x^{1/2} \log^3x\leq y \leq x$, conditional on a standard conjecture.

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N. A. Carella. 2018-08-03. Inequalities For The Primes Counting Function. https://arxiv.org/abs/1808.02366

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