On some Dirichlet series related to the Riemann zeta function. I (Q1568647)

Assuming the Riemann Hypothesis, the author constructs modified von Mangoldt functions \(\widetilde{\Lambda}(n)\) having slightly better mean-value properties than \(\Lambda(n)\). Essentially, the author proves that for every \(\delta\in (0,\frac 12)\) there is a \(\widetilde{\Lambda} (n)\) such that \[ \widetilde{\Lambda} (n)= \Lambda(n) (1+o(1)), \tag{i} \] \[ \sum_{n\leq x} \widetilde{\Lambda} (n) (1-\tfrac{n}{x})= \tfrac{x}{2}+ O(x^{1/4+\delta}). \tag{ii} \] {}.