标题： 一个关于黎曼zeta函数零点的有效线性无关性猜想及其应用 显示英文标题

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

摘要： 使用导致对数中线性型理论中Lang-Waldschmidt猜想的相同启发式论证，我们提出了黎曼zeta函数非平凡零点纵坐标线性独立性猜想的有效版本。 然后假设该猜想，我们得到了素数定理中误差项的Omega结果，这些结果被Montgomery认为是可能的最佳结果。 这被声称出现在Montgomery和Vaughan在其经典著作中提到的Monach的论文中，但该结果或其证明在该论文或文献中均无法找到。 此外，如果除了这个有效的线性独立性猜想外，我们还假设Gonek和Hejhal关于黎曼zeta函数导数的负矩的猜想，我们可以证明莫比乌斯函数的求和函数$M(x)$是$\Omega_{\pm}\left(\sqrt{x}(\log\log\log x)^{5/4}\right)$。 这在条件上解决了Gonek的一个猜想的一部分。 显示英文摘要

摘要： 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.