= Solution
For a locally integrable periodic <function>, its <Fourier series> coefficients are $c_m=\int_0^1 f(x)e^{-2\pi imx}dx$. Window the function to one period by $w(x)=H(x)-H(x-1)$. With the stated <Fourier transform> convention,
$$
\boxed{c_m=\widehat{fw}(2\pi m).}
$$
There is no extra factor of $2\pi$: the Fourier coefficient already integrates over a period of length one.
For a completely arbitrary periodic <distribution>, multiplication by a discontinuous window need not be defined, for example at a derivative of a delta supported on a period boundary. The sharp-window assertion therefore presupposes a windowing convention or enough regularity at the endpoints. The <smooth-window Fourier coefficients of a periodic distribution> use a smooth compactly supported partition-of-unity window $\chi$ with $\sum_{j\in\mathbb Z}\chi(x+j)=1$; then $c_m=\widehat{\chi f}(2\pi m)$ in the <distribution> sense. The logarithmic example below is locally integrable, so the printed sharp window is legitimate and no ambiguity affects it.
For $f(x)=\log|x-1/2|$, put $u=x-1/2$. Symmetry gives, for integer $m\ne0$,
$$
c_m=2(-1)^m\int_0^{1/2}\log u\cos(2\pi m u)du.
$$
An <integration by parts> has no endpoint term: $u\log u\to0$ at zero and $\sin(\pi m)=0$ at the upper endpoint. It follows that
$$
\boxed{c_m=-\frac{(-1)^m}{\pi m}\operatorname{Si}(\pi m),}
$$
where the <sine integral> is $\operatorname{Si}(z)=\int_0^z\sin t/t\,dt$. Its limiting value $\pi/2$ follows from the Abel-regularized calculation in part (ii); <integration by parts> of its tail gives $\operatorname{Si}(\pi m)=\pi/2-(-1)^m/(\pi m)+O(m^{-3})$ for positive integer $m$. Hence the <Fourier coefficients of a periodic logarithmic singularity> have
$$
\boxed{c_m=-\frac{(-1)^m}{2m}+\frac1{\pi^2m^2}+O(m^{-4}),\qquad m\to+\infty.}
$$
The first term is the requested leading behaviour. Its alternating sign records the singularity at half a period, and its $m^{-1}$ decay records the logarithmic cusp. The $m^{-2}$ correction comes from the derivative mismatch across the periodically joined endpoints. Also $c_0=-1-\log2$ and $c_{-m}=c_m$, consistent with the real symmetric profile.
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