I have been using
Manipulate[Plot[{(LogIntegral[x])^(1/2),
(((x*E^s)/Log[x*E^s]) ((((Log[Log[x*E^s]])^(w - 1))/((w - 1)!))))/
RiemannR[x]}, {x, 2, 5000}, PlotStyle -> {Blue,Red}, ImageSize -> 700],
{w, 33.34, 40, 0.01}, {s, 43.2, 50, 0.01}]
in Mathematica to play with the plot
I am trying to find the max values of w and s where the red curve is at no point greater in value that the blue curve. Is there a better way of doing this? What would the mathematical approach be?
I am guessing I will have to find the derivative of $$ {{\rm Li}^{1/2}\left(x\right)\,y \left[\log\left(\log\left(y\right)\right)\right]^{z} \over {\rm R}\left(x\right)\log\left(y\right)\left(z!\right)}$$ where ${\rm R}$ is the Riemann counting function, but am at a bit of a loss as to how to proceed.
(I have started w and s at a reasonable estimate - but when values become much higher than 1000, manipulation is not really feasible.)
NB My best guess so far at the relationship between s and w is s=[N[Log[((w/5) + 1)!]], but this is clearly way off.