Summary: We express the number of distinct primitive cuboids with given odd diagonal in terms of the twisted Euler function with alternating Dirichlet character of period four, and two counting formulas for binary sums of squares. Based on the asymptotic behaviour of the sums of these formulas, we derive an approximation formula for the cumulative number of primitive cuboids.

11D45, 11N37, 11A25, 11B34

arithmetic function, twisted Euler function, Dirichlet L-function, Dirichlet beta function, Catalan's constant, Lehmer's totient sum, Pythagorean quadruple