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An answer to a previous question (see here) regarding maximum prime gaps states that "Bertrand's postulate gives that $p_k−p_{k−1}≤p_{k−1}$. A result of Baker, Harman, & Pintz can be used to improve this to $$p_k−p_{k−1}≪p^{0.525}_{k−1}\quad(1).$$ What exactly is the argument used to obtain (1) from what I interpret to be the quoted Baker, Harman and Pintz result which is stated in their paper as

Combining all our estimates we conclude that, for all large $x$, $$\pi(x+x^{0.525})-\pi(x) \geq \frac{9}{100} \frac{x^{0.525}}{\log x},\quad(2)$$ where $\pi(x)$ is the prime-counting function. My problem is while $\pi(x)$ is monotone increasing with $x$ the difference on the left hand side (LHS) of (2) is not monotone.

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kodlu
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1 Answers1

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The right-hand side of $(2)$ is greater than $1$ for sufficiently large $x$. Thus $]x,x+x^{0.525}]$ contains at least one prime, so the prime gap following $p_{k-1}$ is bounded by $p_{k-1}^{0.525}$. This is what $(1)$ expresses (if I correctly interpret the $\ll$ symbol as "asymptotically bounded by").

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