Bounds on pi(x) without the RH?

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Bounds on pi(x) without the RH?

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Bounds on pi(x) without the RH?

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Apr7-09, 12:40 PM	#1
CRGreathouse CRGreathouse is Offline: Posts: 2,939 Recognitions: Homework Helper Science Advisor	Bounds on pi(x) without the RH? Title says it all. I see a few claims online $LaTeX Code: \\pi(x) = \\operatorname{Li}(x)+O\\left(x\\exp\\left(-\\frac{A(\\ln x)^{3/5}}{(\\ln \\ln x)^{1/5}}\\right)\\right)$ for some constant A $LaTeX Code: \\pi(x) = \\operatorname{li}(x)+O\\left(xe^{-\\sqrt{\\ln x}/15}\\right)\\!$ but no sources and no idea how old these are (or if they've been improved). Anyone know?

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error term, pi(n), pi(x), prime-counting, remainder term

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