Fractional-part continuation formula for the Riemann zeta function ($\zeta (s)={\sum }_{n\le x}{n}^{-s}+x^{1-s}/(s-1)+\{x\}{x}^{-s}-s{\int }_x^{\infty }\{w\}{w}^{-s-1}dw$)

Abel summation applied to the counting function $\lfloor w\rfloor$ gives the displayed identity for $\Re s>1$ and every $x>0$. Since $0\le \{w\}<1$, the integral is locally uniformly convergent and holomorphic for $\Re s>0$. This supplies a meromorphic continuation of the Riemann zeta function to that half-plane, with its sole pole at one and residue one. Keeping the fractional endpoint term makes the formula valid at noninteger cutoffs.