= Solution
The factor called $N$ in the formula must be the stellar <number density> $n$, not a dimensionless count. Indeed $\sigma^3/(G^2m^2)$ has dimensions time per volume, so a density in the denominator is essential. If $N_*$ is the population in a representative volume $V$, then $n=N_*/V$.
For a weak encounter between equal-mass <stars>, let $u$ be the asymptotic relative <speed> and $b$ the <impact parameter>. In the straight-line impulse approximation, one star's transverse kick is
$$
\Delta v_\perp=\int_{-\infty}^{\infty}\frac{Gmb\,dt}{(b^2+u^2t^2)^{3/2}}=\frac{2Gm}{bu}.
$$
The other receives the opposite kick, so the relative-velocity change is $\Delta u_\perp=4Gm/(bu)$. Encounters in $(b,b+db)$ occur at rate $2\pi b\,db\,nu$. Independent random kicks add in squared magnitude, giving
$$
D_v:=\frac{d\langle|\Delta\mathbf v|^2\rangle}{dt}=\frac{8\pi nG^2m^2}{u}\int_{b_{\min}}^{b_{\max}}\frac{db}{b}=\frac{8\pi nG^2m^2\ln\Lambda}{u},\qquad D_u=4D_v.
$$
This is the <random-walk derivation of stellar relaxation>. The <Coulomb logarithm in stellar dynamics> is $\ln\Lambda=\ln(b_{\max}/b_{\min})$. The upper cutoff is the size or inhomogeneity scale beyond which independent local encounters are inappropriate. The lower cutoff is of order the strong-deflection scale $G(2m)/u^2$, or a larger stellar-size/softening scale if applicable. A self-gravitating system commonly has $\Lambda$ of order its total number of stars; that population count is distinct from $n$ in the diffusion rate.
To state the numerical convention explicitly, use representative encounter speed $u=\sqrt2\sigma$ and define the relaxation estimate from the relative-velocity diffusion rate with reference squared velocity $3\sigma^2$. This gives
$$
\boxed{T_R:=\frac{3\sigma^2}{D_u}=\frac{3}{16\pi\sqrt2}\frac{\sigma^3}{nG^2m^2\ln\Lambda}.}
$$
This recovers the printed coefficient under a specified convention. There is no convention-independent exact numerical prefactor determined by a representative one-dimensional velocity alone. Using the single-star diffusion rate with the same $3\sigma^2$ reference instead gives four times this time; averaging over a full velocity distribution also changes the prefactor. The <relaxation time conventions in stellar dynamics> must therefore distinguish single-star and relative kicks. \b[The physical scaling is $T_R\propto\sigma^3/(nG^2m^2\ln\Lambda)$, and the printed $N$ must be interpreted as number density.]
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