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For questions related to 'elementary' proofs in a technical sense, which has nothing to do with the difficulty of the argument or result. A typical example would be 'elementary' proofs of the Prime Number Theorem, which avoid complex analysis. The tag is however not limited to this particular notion of 'elementary.'
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Implicit constant in Tenenbaum's result
You can also find a very clean explicit version of Tenenbaum's inequality as Proposition 2.10 in www.dms.umontreal.ca/~andrew/PDF/Pretentious010611.pdf . The proof is short enough that you should be a …