Friday, 3 January 2014

summation - Finding an upperbound for sumlimitsni=2bigg(prodlimitsik=2dfracpk2pkbigg)




I was wondering whether there exists a known upperbound for:



f(n)=ni=2(ik=2pk2pk)



For example:



f(4)=13+1335+135357+135935711



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 nk=2pk2pk<nk=2pk1pk=2nk=1pk1pk2eγlnn



This means that there ε>0, constant such that nk=2pk2pk<(1+ε)2eγlnn


always.




For more details see Mertens' theorems. Or section 22.8 of this book.


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