What does this equation mean?

This is the same trick used to derive Wien’s displacement law: because u\,g(u) rises from zero, passes through a single interior maximum, and falls back to zero, the integral of the absolute value of its derivative over all u is exactly twice that maximum, with no need to evaluate the integral directly. Finding the maximum of u\,g(u) = 24\,$u^2$/($e^{2\pi u}$-1) means solving $e^{2\pi u}(1-\pi u)$=1 , the same class of transcendental equation that fixes the peak of a Planck spectrum weighted by an extra power of frequency. Solved by Newton’s method to five-figure precision, the nontrivial root is $u^\ast$ $\approx$ 0.25363 , at which $u^\ast$ g($u^\ast$) $\approx$ 0.3937 . The leading-order sensitivity is therefore $D_{\rm det}(a,\, a+\delta a) \;\approx\; 0.394\, \frac{|\delta a|}{a}$. Two accelerated worldlines whose accelerations differ by one part in a thousand disagree, on this measure, by about four parts in ten thousand in normalized excitation content — a small number, appropriately, since a one-part-in-a-thousand physical difference should produce a comparably graded disagreement rather than a large one. This is the sense in which the coefficient 0.394 is an exact result: it is the exact leading term of a convergent expansion in $\delta$ a/a , obtained from an exact closed-form spectral shape by ordinary calculus, with the only numerical step being the solution of one transcendental equation to a stated precision. It is illustrative, not a report of anything measured, and it says nothing about accelerations that are not small perturbations of one another.