Maier proved that, for fixed $\lambda>1,$ $$ \limsup_{x\to\infty}\frac{\pi(x+\log^\lambda x)-\pi(x)}{\log^{\lambda-1}x}>1 $$ and in particular $$ \limsup_{x\to\infty}\frac{\pi(x+\log^\lambda x)-\pi(x)}{\log^{\lambda-1}x}>\frac{e^\gamma}{\lambda} $$ which is for $\lambda<e^\gamma$ is sharper. Has this been improved? Are upper bounds known? Is there an effective bound for the $\liminf$?
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2$\begingroup$ Have you tried finding the review of Maier's paper on MathSciNet to see whether there are any later reviews that refer to it? $\endgroup$– Gerry MyersonCommented Feb 23, 2015 at 22:51
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1$\begingroup$ ... or the same on Google Scholar $\endgroup$– StoppleCommented Feb 24, 2015 at 2:27
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$\begingroup$ Yes I tried MathSciNet (but not Google Scholar, maybe I'll try that later) -- but it's a heavily-cited paper and I might have missed something, or something new might have come up that isn't yet indexed. $\endgroup$– CharlesCommented Feb 24, 2015 at 2:40
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