<jats:title>Abstract</jats:title>
<jats:p>
We show that the sign constancy for the values of certain weighted summatory functions of the von Mangoldt function implies the Riemann Hypothesis or the Generalized Riemann Hypothesis for Dirichlet
<jats:italic>L</jats:italic>
-functions. While such sign constancy is challenging to establish individually, we prove that the summatory functions under study have constant signs on average.
</jats:p>
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