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How to estimate the following integral:

$$\int_e^x \log{\log{t}} dt$$ so that the error term is within $$O\left(\frac{x}{\log^2{x}}\right)$$. Assume $$x>e$$

Any hint?

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integration by part and bound logarithmic integral. –  Soarer Sep 22 '11 at 7:28

1 Answer 1

up vote 6 down vote accepted

If you integrate by parts, you end up with

$$x\log\log\,x-\int_e^x\frac{\mathrm dt}{\log\,t}$$

The last bit evaluates to $\mathrm{li}(x)-\mathrm{Ei}(1)$, where $\mathrm{Ei}(x)$ is the exponential integral, and $\mathrm{li}(x)=\mathrm{Ei}(\log\,x)$ is the logarithmic integral.

Using this asymptotic series for the exponential integral, we obtain

$$x\log\log\,x-\frac{x}{\log\,x}\left(1+\frac1{\log\,x}+\frac2{(\log\,x)^2}+\cdots\right)$$

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