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Let $\mu_k(n)$ denote the nth coefficient in the Dirichlet series for $\zeta^{-k}(s)$. It can be shown that if there exists a $u=u(x,s)$ such that $$|\sum_{n\leq x}\frac{\mu_k(n)}{n^s} |^{1/k}\leq u$$ then there is a $c=c(\sigma)$ such that $u$ is $\Omega(\log^c(x))$ when $\sigma$ is strictly less than unity.

Since the limit on left hand is $1/\zeta(s)$ if it converges, I am wondering if much is known about the modes of convergence of the series under the root on the left on various hypotheses such as Riemann etc?

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