I was wondering whether there exists a known upperbound for:
f(n)=n∑i=2(i∏k=2pk−2pk)
For example:
f(4)=13+1⋅33⋅5+1⋅3⋅53⋅5⋅7+1⋅3⋅5⋅93⋅5⋅7⋅11
I've searched around for a bit, but since english is not my native language, I've been unable to phrase this question in a way that google understands.
I'm really hoping for something in terms of log(n) or better.
Any kind of help is really appreciated.
Answer
Well n∏k=2pk−2pk<n∏k=2pk−1pk=2n∏k=1pk−1pk∼2e−γlnn
This means that there ∃ε>0, constant such that n∏k=2pk−2pk<(1+ε)2e−γlnn
always.
For more details see Mertens' theorems. Or section 22.8 of this book.
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