$$\prod _{k=2}^n\left(\log_2k\right)$$
Can somebody help me with bounds of this expressions. I see only the rude measure:
$$\log_2n\le \prod _{k=2}^n\left(\log_2k\right)\le \left(\log_2n\right)^n$$
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1It will suffice to find bounds for the log of your product, $\sum_{k=2}^n \log_2 \log_2 k$, which is done at https://math.stackexchange.com/questions/1084665/getting-tight-asymptotic-upper-and-lower-bounds-of-product-logs?rq=1 – Michael Lugo Mar 30 '16 at 16:40
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Yes, but I have a multiply and there is a sum, isn't it different for the both situations ? – Ilya.K. Mar 30 '16 at 17:37