Correlations of multiplicative functions with their partial sums

Abstract

Let $ζ(.)$ denote the Riemann zeta function and let $a(.)$ and $A(.)$ respectively denote a multiplicative function and its corresponding summatory function. We consider the correlation $$ \langle a(n)A(n-1) \rangle (T) = \frac{1}{ζ(1+δ(T))}\sum_{n\leq T^{1-c}}\frac{a(n)A(n-1)}{n^{1+δ(T)}} $$ where $0<c<1$ is arbitrary and $0<δ(T)=O\left(T^{c-1}\right)$ is suitably chosen. Let $μ(.)$ and $λ(.)$ denote the Möbius 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 $ρ=1/2+ i γ$ of $ζ(s)$ we show that $$ \langle μ(n)M(n-1) \rangle (T)= -\frac{3}{π^{2}}\left(1-T^{(c-1)δ(T)}\right)+\sum_{0<γ<T}\frac{1}{\left|ρζ&#39;(ρ)\right|^{2}} $$ and $$ \langle λ(n)L(n-1) \rangle (T)=\frac{1}{2}\left(\frac{1}{ζ^{2}(1/2)}-1+T^{(c-1)δ(T)}\right)+\sum_{0<γ<T}\left|\frac{ζ(2ρ)}{ρζ&#39;(ρ)}\right|^{2} $$ as $T\rightarrow \infty$ where $0\leq T^{(c-1)δ(T)}<1$. These results combined with numerical observations suggest that there is anticorrelation between $μ(n)$ and $M(n-1)$ as well as between $λ(n)$ and $L(n-1)$, where the correlation is computed using a logarithmic average. This would imply effective upper bounds on $\left|1/ζ&#39;(ρ)\right|$.