= Hardy-Littlewood approximation to the Riemann zeta function
{c}
{title2=$\zeta(s)=\sum_{n\le x}n^{-s}+x^{1-s}/(s-1)+O(x^{-\sigma})$}
For $s=\sigma+it$, $\sigma>0$, and $x\ge|t|/\pi$, the displayed approximation holds away from the <pole>. Apply the <Van der Corput sum-integral lemma> to $-t\log w/(2\pi)$ on $w\ge x$, where its derivative has modulus at most one half. <Abel summation> with $w^{-\sigma}$ converts the bounded unweighted discrepancy to $O(x^{-\sigma})$. <Locally uniform convergence> of this weighted discrepancy extends the identity from $\sigma>1$ to $\sigma>0$. It turns estimates for finite <exponential sums> into estimates for the <Riemann zeta function>.
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