Through various ad-hoc means I conjecture that for large x
$F\left( x \right)=\frac{2\pi }{{{x}^{2}}}\sum\limits_{n=1}^{\infty }{n\left( \frac{2\left( 1-{{\left( -1 \right)}^{n}}{{e}^{-1}} \right)}{{{n}^{2}}{{\pi }^{2}}+1} \right)\sin \left( \frac{{{x}^{2}}}{4\pi n} \right)}=O\left( \frac{2\left( {1-{e}^{-\tfrac{1}{4}{{x}^{2}}}} \right)}{{{x}^{2}}} \right)$
Can anyone derive this asymptotic behaviour, or perhaps correct it?