ON THE MULTIPLICATIVE ORDER OF a MODULO n

Let n be a positive integer and αn be the arithmetic function which assigns the multiplicative order of an modulo n to every integer a coprime to n and vanishes elsewhere. Similarly, let βn assign the projective multiplicative order of a n modulo n to every integer a coprime to n and vanish elsewhere. In this paper, we present a study of these two arithmetic functions. In particular, we prove that for every positive integers n1 and n2 with the same square-free part, there exists an exact relationship between the functions αn1 and αn2 and between the functions βn1 and βn2 . This allows us to reduce the determination of αn and βn to the case where n is square-free. These arithmetic functions recently appeared in the context of an old problem of Molluzzo, and more precisely in the study of which arithmetic progressions yield a balanced Steinhaus triangle in Z/nZ for n odd.