Correlations of multiplicative functions with their partial sums
Let $\zeta(.)$ denote the Riemann zeta function and let $a(.)$ and $A(.)$ respectively denote a multiplicative function and its corresponding summatory function. We consider the correlation $$ \langle a(n)A(n-1) \rangle (T) = \frac{1}{\zeta(1+\delta(T))}\sum_{n\leq T^{1-c}}\frac{a(n)A(n-1)}{n^{1+\delta(T)}} $$ where $0<c<1$ is arbitrary and $0<\delta(T)=O\left(T^{c-1}\right)$ is suitably chosen. Let $\mu(.)$ and $\lambda(.)$ denote the M\"obius function and the Liouville function respectively while $M(.)$ and $L(.)$ denote their corresponding summatory functions. Under the Riemann hypothesis and simplicity of the nontrivial zeros $\rho=1/2+ i \gamma$ of $\zeta(s)$ we show that $$ \langle \mu(n)M(n-1) \rangle (T)= -\frac{3}{\pi^{2}}\left(1-T^{(c-1)\delta(T)}\right)+\sum_{0<\gamma<T}\frac{1}{\left|\rho\zeta'(\rho)\right|^{2}} $$ and $$ \langle \lambda(n)L(n-1) \rangle (T)=\frac{1}{2}\left(\frac{1}{\zeta^{2}(1/2)}-1+T^{(c-1)\delta(T)}\right)+\sum_{0<\gamma<T}\left|\frac{\zeta(2\rho)}{\rho\zeta'(\rho)}\right|^{2} $$ as $T\rightarrow \infty$ where $0\leq T^{(c-1)\delta(T)}<1$. These results combined with numerical observations suggest that there is anticorrelation between $\mu(n)$ and $M(n-1)$ as well as between $\lambda(n)$ and $L(n-1)$, where the correlation is computed using a logarithmic average. This would imply effective upper bounds on $\left|1/\zeta'(\rho)\right|$.