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Consider the following function :

$$f(x) = \sin^2(\frac{π\Gamma(x)}{2x})$$

Is following growth condition true ?:

$$\int_0^\infty \frac{f(x + iy) − f(x − iy)}{e^{2πy}-1} dy=o\left(\int_2^x f(t) dt\right) $$ ?

If yes , how to achieve it?

One can also ask for the following :

$$\int_0^\infty \frac{f(x + iy) − f(x − iy)}{e^{2πy}-1} dy=o(g_1(x))=O(g_2(x)) $$

As $x \to \infty$.

what are some possible candidates for $g_1(x)$ and $g_2(x)$ ?Also can we find $g_1(x)$ and $g_2(x)$ for which bounds are sharp?

See this MSE post for more details

Accurate (upto 4 decimals) values of functional upto $x=5$:

accurate (upto 4 decimals) values of functional upto x=5values of functional upto x=5

Important: Any computational analysis expert can help me to compute them for larger $x$ upto 100 or 1000 or provide me with graph for large $x$ Answers and comments from them are welcome

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  • $\begingroup$ What do you get from Stirling (the sum over poles of $\Gamma'/\Gamma$ is up to a $\log(z)$ term a power series in $z^{-1}$ which gives very good approximations of $\sin(\Gamma)$) $\endgroup$ – reuns Jun 20 at 22:05
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    $\begingroup$ Please avoid making several edits. $\endgroup$ – Aloizio Macedo Jun 29 at 10:12
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    $\begingroup$ Please explain the downvotes! $\endgroup$ – Bambi Jul 8 at 18:11

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