Multiplicative function anticorrelation and bounds on $1/ζ'(ρ)$

Abstract: Let $\zeta(s)$ denote the zeta function and let $\mu(.)$ and $M(.)$ denote the M\"obius function and the summatory M\"obius function respectively. Similarly, let $\lambda(.)$ and $L(.)$ denote the Liouville function and the summatory Liouville function respectively. Finding upper bounds on $1/\left|\zeta'(\rho)\right|$ is a longstanding open problem. Under the Riemann hypothesis and simplicity of the nontrivial zeros $\rho=1/2+ i \gamma$ of $\zeta(s)$ we show that numerical evidence for the result $$ \sum_{n\leq N}\frac{\mu(n)M(n-1)}{n}<0 $$ as $N\rightarrow \infty$ serves as numerical evidence for the bound $$ \frac{1}{\zeta'(\rho)}=o\left(\left|\rho\right|\right) $$ as $\left|\gamma\right|\rightarrow \infty$ and similarly, numerical evidence for $$ \sum_{n\leq N}\frac{\lambda(n)L(n-1)}{n}<0 $$ as $N\rightarrow \infty$ serves as numerical evidence for the bound $$ \frac{1}{\zeta'(\rho)}=o\left(\left|\rho\right|\log \log \left|\gamma\right| \right) $$ as $\left|\gamma\right|\rightarrow \infty$. We thus describe a new form of numerical evidence for effective upper bounds on $1/\left|\zeta'(\rho)\right|$ that involves demonstrating anticorrelation between multiplicative functions and their corresponding summatory functions, where the correlation is computed using a logarithmic average. Numerical results strongly indicate this anticorrelation, i.e., the negativity of the above sums.