= Fractional-part continuation formula for the Riemann zeta function
{title2=$\zeta(s)=\sum_{n\le x}n^{-s}+x^{1-s}/(s-1)+\{x\}x^{-s}-s\int_x^\infty\{w\}w^{-s-1}\,dw$}
<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.
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