An arithmetic function arising from Carmichael’s conjecture
Let denote Euler’s totient function. A century-old conjecture of Carmichael asserts that for every , the equation has a solution . This suggests defining as the number of solutions to the equation . (So Carmichael’s conjecture asserts that always.) Results on are scattered throughout the literature. For example, Sierpiński conjectured, and Ford proved, that the range of contains every natural number . Also, the maximal order of has been investigated by Erdős and Pomerance. In...