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 $x=5$:
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
