= Truncated Möbius inverse identity for the Riemann zeta function
{title2=$\zeta(s)\sum_{m\le M}\mu(m)m^{-s}=1+\sum_{\min(M,T)<n\le MT}a_nn^{-s}+O(T^{-\sigma}M^{1-\sigma}\log M)$}
For $T/2\le\Im s\le T$ and $0<\Re s=\sigma\le1$, the <Hardy-Littlewood approximation to the Riemann zeta function> truncates zeta at $T$ with error $O(T^{-\sigma})$. Multiply the finite sums and use $a_n=\sum_{m\mid n,\ m\le M,\ n/m\le T}\mu(m)$. All coefficients with $1<n\le\min(M,T)$ vanish by the <Möbius divisor-sum identity>. The total error is bounded using $\sum_{m\le M}m^{-\sigma}\ll M^{1-\sigma}\log M$.
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