Consider the following functional :

$$ I(x,s) =\int_0^\infty\mathrm dy \frac{F(x + \mathrm iy, s) − F(x −\mathrm iy, s)}{\mathrm e^{2πy}-1}, $$ where $ F(z, s) = \dfrac{\sinh(\sin^2[π\Gamma(z)/(2z)])}{z^s} $.

Let us restrict $s\in[0,1]$

Can we get sharp numerical asymptotic of $I(x)$ as as $x\rightarrow \infty$?

Also, can we get quantitative upper and lower bound estimations on the functional ?

Moreover , if someone can construct a generalized function such that: $$ F_*(z, s) = \dfrac{\phi(\sin^2[π\Gamma(z)/(2z)])}{z^s} $$

(1) $\phi(x) =0$ if $x$ is zero ; and finite otherwise

(2)$I_*(x,s)\rightarrow c$ as $x\rightarrow\infty$

Here $c$ is a constant .

Related: Attempt at new proof of infinitude of primes

  • $\begingroup$ Any updates on the question? $\endgroup$ – Bambi Mar 17 at 11:18

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