Consider the function $F(x)$ defined in following manner:
$F(n)$ is finite (likely $F(x)\in[0,1]$) if $n$ is prime and zero otherwise:
It has to satisfy following conditions:
(1) $F(x)$ is analytic on $x\in [m,\infty] $ for some positive $m$
(and it is non decreasing at prime values )(optional condition)
(2)$$\lim_{y→∞}|F(x ± iy)|e^{−2πy }= 0$$ uniformly in $x (\geqslant m)$ on every finite interval
(3)$$\int_0^\infty |F(x + iy) − F(x − iy)|e^{−2πy} dy$$ exists for every $x ≥ m$ and tends to zero when $x → ∞$
I've got a function almost satisfying above conditions:
$$F(x)= \sin^2(\frac{π\Gamma(x)}{2x})$$
The above function satisfies condition (1) and (2) but doesn't seem to satisfy the 3rd condition .
Question : How to construct a function that satisfies all above conditions?
