Correlations of the Möbius and Liouville functions with their partial sums

Let [Formula: see text] denote the Riemann zeta function; let [Formula: see text] and [Formula: see text] denote the Möbius and Liouville functions, respectively, while [Formula: see text] and [Formula: see text] respectively, denote their corresponding summatory functions. We consider the correlations [Formula: see text] and [Formula: see text] where [Formula: see text] is arbitrary and [Formula: see text] is suitably chosen. Under the Riemann hypothesis and simplicity of the nontrivial zeros [Formula: see text] of [Formula: see text] we show that [Formula: see text] and [Formula: see text] as [Formula: see text] where [Formula: see text]. These results combined with numerical observations suggest that there is anticorrelation between [Formula: see text] and [Formula: see text] as well as between [Formula: see text] and [Formula: see text], where the correlation is computed using a logarithmic average. This would imply effective upper bounds on [Formula: see text].