 Fourier coefficients of a periodic logarithmic singularity

For the period-one continuation of $\log |x-1/2|$, symmetry and integration by parts give $c_m=-(-1{)}^m\mathrm{Si}(\pi m)/(\pi m)$ for $m\ne 0$. Thus the sine integral asymptotic gives $c_m=-(-1{)}^m/(2m)+1/({\pi }^2{m}^2)+O(m^{-4})$ as $m\to +\infty$. Translation of the logarithmic singularity causes the alternating phase; the endpoint derivative mismatch causes the next term.