Let $s=\sigma+it$, $0\leq \sigma\leq 1$, $|t|\geq 1$, say. Using Euler-Maclaurin, one can easily show that, for $x\geq |t|$, $$\zeta(s) = \sum_{n\leq x} \frac{1}{n^s} + \frac{x^{1-s}}{s-1} + O\left(\frac{1}{x^\sigma}\right).$$ (In fact, the implied constant is at most $5/6$.)

What about $1/\zeta(s)$? Can one also express $$\frac{1}{\zeta(s)} - \sum_{n\leq x} \frac{\mu(n)}{n^s}$$ as the sum of a leading term plus a much smaller error term, and, if so, what is the best known bound?

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