We theoretically investigate the shear viscosity η\documentclass[12pt]{minimal}
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\begin{document}$$\eta $$\end{document}, as well as the entropy density s, in the normal state of an ultracold Fermi gas. Including pairing fluctuations within the framework of a T-matrix approximation, we calculate these quantities in the Bardeen–Cooper–Schrieffer (BCS)–Bose–Einstein condensation (BEC) crossover region. We also evaluate η/s\documentclass[12pt]{minimal}
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\begin{document}$$\eta / s$$\end{document}, to compare it with the lower bound of this ratio, conjectured by Kovtun, Son, and Starinets (KSS bound). In the weak-coupling BCS side, we show that the shear viscosity η\documentclass[12pt]{minimal}
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\begin{document}$$\eta $$\end{document} is remarkably suppressed near the superfluid phase transition temperature Tc\documentclass[12pt]{minimal}
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\begin{document}$$T_{\mathrm{c}}$$\end{document}, due to the so-called pseudogap phenomenon. In the strong-coupling BEC side, we find that, within the neglect of the vertex corrections, one cannot correctly describe η\documentclass[12pt]{minimal}
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\begin{document}$$\eta $$\end{document}. We also show that η/s\documentclass[12pt]{minimal}
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\begin{document}$$\eta / s$$\end{document} decreases with increasing the interaction strength, to become very close to the KSS bound, ħ/4πkB\documentclass[12pt]{minimal}
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\begin{document}$$\hbar /4\pi k_{\mathrm{B}}$$\end{document}, on the BEC side.
