Some Steps in Introduction to Section 13.10

We first obtain $\log n\sim x.$

Thus $2^{\pi (x)}\approx 2^{x/\log x}\approx 2^{\log n/\log \log n}.$

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$$2^{a\log n}=e^{\log 2^{a\log n}}=e^{a\log n\log 2}={\left(e^{a\log n}\right)}^{\log 2}={\left(e^{\log n^a}\right)}^{\log 2}={\left(n^a\right)}^{\log 2}=n^{a\log 2}.$$

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We have $$2^{a\log n}=n^{a\log 2}.$$ Thus $$2^{\log n/\log \log n}=2^{\stackrel{\text{Consider this }a}{\overbrace{(1/\log \log n)}}\log n}=n^{(1/\log \log n)\log 2}=n^{\log 2/\log \log n}.$$

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$$d(n)\approx 2^{\pi (x)}=2^{\log n/\log \log n}=n^{\log 2/\log \log n}.$$