An effective Linear Independence conjecture for the zeros of the Riemann zeta function and applications

Abstract

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.