Sharp bounds for the Bernoulli numbers

We determine the best possible real constants a\alpha and b\beta such that the inequalities (2(2n)!)/((2p)²ⁿ)] 1/(1-2^a-2n)] \leqq |B_2n| \leqq (2(2n)!)/((2p)²ⁿ)] 1/(1-2^b-2n)]{2(2n)! \over(2\pi)^{2n}} {1 \over 1-2^{\alpha -2n}} \leqq |B_{2n}| \leqq {2(2n)! \over (2\pi )^{2n}}\, {1 \over 1-2^{\beta -2n}}hold for all integers n\geqq 1n\geqq 1. Here, B₂, B₄, B₆,... are Bernoulli numbers.