  Using the same heuristic argument leading to the Lang-Waldschmidt Conjecture
in the theory of linear forms in logarithms, we formulate an effective version
of the Linear Independence conjecture for the ordinates of the non-trivial
zeros of the Riemann zeta function. Then assuming this conjecture, we obtain
Omega results for the error term in the prime number theorem, which are
conjectured to be best possible by Montgomery. This was claimed to appear in
Monach's thesis by Montgomery and Vaughan in their classical book, but such a
result or its proof is nowhere to be found in this thesis or in the
litterature. Moreover, if in addition to this effective Linear Independence
conjecture we assume a conjecture of Gonek and Hejhal on the negative moments
of the derivative of the Riemann zeta function, we can show that the summatory
function of the M\"obius function $M(x)$ is
$\Omega_{\pm}\left(\sqrt{x}(\log\log\log x)^{5/4}\right)$. This conditionally
resolves a part of a conjecture of Gonek.
