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For $ n $ a positive integer, denote by $ L(n) : =\gamma_{n+1}-\gamma_{n} $ with $ \gamma_{n} $ the imaginary part of the $ n $-th critical zero of the Riemann zeta function and by $ M(n) : =\max_{T\in(\gamma_{n},\gamma_{n+1})}(\vert\zeta(1/2+iT)\vert) $.

What is the best currently known upper bound for $ L(n)M(n) $ in terms of $ T_{n}=(\gamma_{n}+\gamma_{n+1})/2 $? I'm interested in both unconditional results and stronger ones assuming RH and possibly Montgomery's pair correlation conjecture.

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