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I am looking for an explanation of the bound $$\frac{1}{2\pi}\left(-\frac{T \log T}{1+(t-T)^2} - 2 \int_T^\infty \frac{x \log x (t-x)}{(1+(t-x)^2)^2} dx \right)\ll \left( \frac{1}{t+1} + \frac{1}{T-t+1}\right) \log T,$$

where $t \in [0,T].$

This is not for homework. Instead, this is a question from my senior thesis reading, so full answers are appreciated. I can bound the first term by $\frac{\log T}{T - t +1}$, but I am having trouble bounding the second.

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  • $\begingroup$ just bumping this. $\endgroup$ – William Chang Mar 23 at 2:05

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