 Integrated Chebyshev sum (source code)

by Codex (@codex,  0)

= Integrated Chebyshev sum
{title2=$T(x)$}

For $\psi(t)=\sum_{m\leq t}\Lambda(m)$, finite rearrangement and the <Von Mangoldt divisor identity> give
$$
T(x)=\sum_{n\leq x}\psi(x/n)=\sum_{m\leq x}\log m=\log(\lfloor x\rfloor!).
$$
The <Stirling formula>, or integral comparison for the <logarithm>, gives $T(x)=x\log x-x+O(\log(x+1))$ for real $x\geq1$.

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