Reciprocal zeta bounds near the line one ($|\zeta (\sigma +it{)}^{-1}|\ll (\sigma -1{)}^{-3/4}{\log }^{1/2}(2+|t|)$)

For $1<\sigma \le 2$, the three-four-one inequality for Euler products gives $|1/\zeta (\sigma +it)|\le \zeta (\sigma {)}^{3/4}|\zeta (\sigma +2it){|}^{1/4}$. For $|t|\ge 1$, the fractional-part continuation formula for the Riemann zeta function, truncated at $H\asymp 2+|t|$, gives ${\zeta }^{(j)}(\sigma +it)=O_j({\log }^{j+1}(2+|t|))$. Thus the reciprocal bound even holds with exponent $1/4$ instead of $1/2$ in this range. On bounded-height compact sets, three-four-one product proof of zeta boundary nonvanishing and removal of the reciprocal's zero at the pole give uniform bounds. Differentiating $1/\zeta$ repeatedly yieldsThe range restriction matters: $1/\zeta (\sigma )\to 1$ as $\sigma \to \infty$, so the first displayed bound cannot hold uniformly for all $\sigma >1$. A global version replaces $(\sigma -1{)}^{-3/4}$ with $1+(\sigma -1{)}^{-3/4}$.