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==Asymptotic==
<math>\begin{align}\log f(x)&=-\frac{\log^2x}{2\log2}+\frac{\log x\cdot\log(-\log x)}{\log2}-\left(\frac12+\frac{1+\log\log2}{\log2}\right)\log x -\frac{\log^2(-\log x)}{2\log2}+\frac{\log\log 2\cdot\log(-\log x)}{\log2}\\&+\left(\frac{6\gamma ^2+12\gamma_1-\pi^2-6\log^2\log2}{12\log 2}-\frac{7\log 2}{12}-\frac{\log\pi}2\right)+\frac{\log^2(-\log x)}{2\log2\cdot\log x}-\frac{\log\log2\cdot\log(-\log x)}{\log2\cdot\log x}+O\!\left(\frac1{\log x}\right)
for <math>x\to0^+,</math> where <math>\gamma</math> is [[Euler's constant]], and <math>\gamma_1</math> is the [[Stieltjes constants|Stieltjes constant]]. Equivalently,
<math>\log f\!\left(2^{-n}\right)=-\frac{n^2\log2}2-n\log n+\left(1+\frac{\log2}2\right)n -\frac{\log^2n}{2\log2}+\left(\frac{6\gamma ^2+12\gamma_1-\pi^2}{12\log 2}-\frac{7\log 2}{12}-\frac{\log\pi}2\right)-\frac{\log^2n}{2n\log^22}+O\!\left(\frac1n\right)</math>
for <math>n\to\infty.</math>
==References==
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