more morning speculation

since
\int_{0}^{\infty} \frac{dx}{\Gamma(x^{3})} \approx 0.753065231886\ldots

and
\int_{0}^{\infty} \frac{dx}{\Gamma(x^{2})} \approx 1.165261151780\ldots, then there is some number \mu between three and two such that

\int_{0}^{\infty} \frac{dx}{\Gamma(x^{\mu})}  = 1

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