We know that for $x$ being a half-integer $$\sum_{n\leq x}\Lambda(n)=x-\sum_{\rho}\frac{x^\rho}{\rho}+O(1).$$ Is there a similar formula for $\sum_{n\leq x}\mu(n)^2$ in the literature? The underlying Dirichlet series is $\frac{\zeta(s)}{\zeta(2s)}$ rather than $-\frac{\zeta'(s)}{\zeta(s)}$.
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5$\begingroup$ Yes, see this paper by Bartz from 1992, specifically Theorem 3: digizeitschriften.de/dms/img/… . The formula is more complicated than in the von Mangoldt case, and involves not only the zeros of the zeta function but the multiplicities as well. $\endgroup$– Ofir GorodetskyCommented Mar 16, 2017 at 16:12
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