Consider the following integral:
$$S(q)=\int_{x=2}^q\sin^2\left(\frac{π\Gamma(x)}{2x}\right)dx$$
And consider the functions :
$$R(q)=\frac{q}{\log(q)}$$
$$T(q)=\int_2^q\frac{1}{\log(x)}dx$$
I want to compare them with each other ( at least numerically for large interval of value )
If graph for very large intervals (upto atleast $10^4$) possible please add (please add three graphs in one axis system , so that I can compare)
(Does numerics suggest $S(q) \sim R(q)$ or $T(q)$? )
See ; Related : https://math.stackexchange.com/q/3570663/702232
Note : Can't calculate the first integral on Mathematica for large values
LevinRule
, I can't think of a straight-forward way of efficiently computing it. Maybe someone else can think of a way, but I think your best bet might be to figure out analytically how to make the integrand nicer. $\endgroup$LogIntegral[]
is built-in? Additionally: perhaps you could explore usingNDSolve[]
for your $S(q)$; e.g.sol = NDSolveValue[{s'[q] == Sin[Exp[Log[Pi/2] + LogGamma[q] - Log[q]]]^2, s[2] == 0}, s, {q, 2, 10}]
. $\endgroup$NIntegrate[Sin[(π Gamma[x])/(2 x)]^2, {x, 2, 10^4}, Method -> "LevinRule", MinRecursion -> 20, MaxRecursion -> 50]
gives out an answer without warnings, but I can't really judge its validity. $\endgroup$