Since the Smith–Volterra–Cantor set S has positive Lebesgue measure, this means that V ′ is discontinuous on a set of positive measure. By Lebesgue's criterion for Riemann integrability, V ′ is not Riemann integrable. If one were to repeat the construction of Volterra's function with the ordinary measure-0 Cantor set C in place of the "fat" (positive-measure) Cantor set S, one would obtain a function with many similar properties, but the derivative would then be discontinuous on the measure-0 set C instead of the positive-measure set S, and so the resulting function would have a Riemann integrable derivative.

The mathematics of almost all eigenvalue problems encountered in wave physics is essentially the same, but the richest source of such problems is quantum mechanics, where the eigenvalues are the energies of stationary states ("levels"), rather than frequencies as in acoustics or optics, and the operator is the hamiltonian.

If the Riemann hypothesis is true, Im tn = 0 for all n, and the function f(u), constructed from the primes, has a discrete spectrum; that is, the support of its Fourier transform is discrete. If the Riemann hypothesis is false, this is not the case. The frequencies tn are reminiscent of the decomposition of a musical sound into its constituent harmonics. Therefore there is a sense in which we can give a one-line nontechnical statement of the Riemann hypothesis: "The primes have music in them."