The differentiation of Li(t) gives 1logt
In mathematica using D[LogIntegral[t], t] it confirms this as one would expect.
But I'm having difficulty integrating:
∫∞a1log(t)dt
I'm not sure by hand how to do this and it certainly won't compute in mathematica; yet it differentiates Li(t). I can see that with limit a set to zero there would be a problem with it going to infinity as it approaches the y-axis, something like a=2 would not present that problem.
A similar question was posed here:
Convergence or Divergence using Limits
but only established that it is divergent and did not explain that it came from Li(t) and what would be needed to get back.
Further I cannot integrate ∫∞ae−stlog(t)dt
again with a=0 this would be a problem I suppose, but a=2 should be okay.
Answer
∫∞adtlogt does not converge. logt<t→1logt>1t and as ∫∞adtt diverges, so does the (∗)
The second integral converges for any s>0;a>1, but I can't find a closed form and I think It doesn't exist
Hope this helps
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