Truncated Möbius inverse identity for the Riemann zeta function ($\zeta (s){\sum }_{m\le M}\mu (m)m^{-s}=1+{\sum }_{\min (M,T)<n\le MT}{a}_n{n}^{-s}+O(T^{-\sigma }{M}^{1-\sigma }\log M)$)

For $T/2\le \Im s\le T$ and $0<\Re s=\sigma \le 1$, 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 ${\sum }_{m\le M}{m}^{-\sigma }\ll M^{1-\sigma }\log M$.