theory deals with qualitative properties rather than quantitative properties of numbers. Discard the ones that are not in lowest terms; keep only those that are in lowest terms: Then bring in trigonometry: The value of the sum is −1, because 42 has an odd number of prime factors and none of them is repeated: 42 = 2 × 3 × 7. (If there had been an even'' number of non-repeated factors then the sum would have been 1; if there had been any repeated prime factors (e.g., 60 = 2 × 2 × 3 × 5) then the sum would have been 0; the sum is the Möbius function evaluated at 42.) This hints at the possibility of applying Fourier analysis to number theory. Solving non-trigonometric equations Various types of equations can be solved using trigonometry. For example, a linear difference equation or linear differential equation with constant coefficients has solutions expressed in terms of the eigenvalues of its characteristic equation; if some of the eigenvalues are complex, the complex terms can be replaced by trigonometric functions of real terms, showing that the dynamic variable exhibits oscillations. Similarly, cubic equations with three real solutions have an algebraic solution that is unhelpful in that it contains cube roots of complex numbers; again an alternative solution exists in terms of trigonometric functions of real terms. References Trigonometry