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So, I've understood the proof of A*x/logx < pi(x) < B*x/logx for (A,B) = (0.5,2), but how can I make this difference smaller? Does any one know the methods used by him and further by J.J. Sylvester? I'd like references in english for I found some papers in french, but could figure out anything. Thanks

Ps: pi(x) is the prime counting function

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Not sure if you are still searching but there is a great expository paper by Diamond, https://pdfs.semanticscholar.org/7147/23e8854b484a3ebcadfa518461af7eb98c73.pdf

Also, by Sylvester:

pages 530–545 of:

https://quod.lib.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=AAS8085.0003.001

pages 687–731 of:

https://archive.org/details/collectedmathem04sylvrich

Hope this helps!

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