= Bilinear shift averaging for a logarithmic phase
{title2=$S=Z^{-2}\sum_{n\in I}\sum_{x,y\le Z}(n+xy)^{-it}+O(Z^2)$}
Translating an integer interval by $xy$ changes a sum of unit-modulus terms by at most $2xy$. Averaging gives the displayed formula. For $n\asymp N$, $Z=N^{2/5}$, expand $-t\log(1+xy/n)$ to degree $r=\lfloor5.01\log t/\log N\rfloor$. Its remainder is at most $tN^{-(r+1)/5}<t^{-1/500}$. This reduces the logarithmic <exponential sum> to bilinear polynomial sums while keeping explicit boundary and approximation errors.
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