= Short-interval Perron bound for the second Chebyshev function
{title2=$\psi(x+y)-\psi(x)\ll y\int_{-T}^T|\zeta'(c+it)/\zeta(c+it)|\,dt+x\log^2x/T$}
For $c=1+1/\log x$, $1\le y\le x$ and $2\le T\le x$, apply the <truncated Perron formula> with <Von Mangoldt function> coefficients at both endpoints on the same vertical line. The difference kernel is $((x+y)^s-x^s)/s=\int_x^{x+y}u^{s-1}\,du$, of modulus $O(y)$. The usual near-integer error bound contributes $O(x\log^2x/T)$. The <logarithmic derivative> on this line is finite because of the <Euler product>.
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