Monday, 25 April 2016

calculus - Integral largeint10left(fracoperatornamelixxright)adx

Let lix denote the logarithmic integral
lix=x0dtlnt.

Consider the following parameterized integral:
I(a)=10(lixx)adx.




Can we find a closed form for this integral?




We can find some special values of this integral:
I(0)=1,I(1)=1,I(2)=π26,I(3)?=7ζ(3)2
The last value was suggested by numeric computations, and I do not yet have a proof for it.





Can we prove the conjectured value of I(3)?




One could expect that I(4) might be a simple rational (or at least algebraic) multiple of π4 but I could not find such a form.




Can we find closed forms for I(4),I(5) and other small integer arguments?


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