Title:

On arithmetics in Cayley's algebra and multiplicative functions

The aim of this thesis is to discuss fully the characterisation and basic properties of the arithmetics of Cayley's algebra C and to define certain multiplicative functions by summing homogeneous polynomials in four (or eight) indeterminates over the components of all elements of constant norm m of an arithmetic of C. The algebra C is introduced by reviewing some of the relevant work by Cayley, Dickson, Artin, Zorn and others. A generalisation of Dickson's condensed law of multiplication is then used to obtain certain automorphisms of C. The maximal and non maximal arithmetics of C are characterized. Isomorphisms and ether relations between certain of the arithmetics are obtained. Treatments of the arithmetics by Coxeter, Dickson and Kirmse are reviewed. Results on congruence modulo a rational integer in any maximal arithmetic of C are proved. For example, it is shown that Any element zeta of odd norm of a maximal arithmetic of Jw of C is congruent modulo 2 in JW to an element, unique apart from sign, of norm 1 of JW. This result is used, under certain conditions, to characterize and count the factors of an element of an arithmetic of C. For example, it is proved that Any element z of maximal arithmetic JW of C for which N z = mn, where m, n are positive rational integers for which (m,n) = 1, has precisely 240 factorizations zetaeta in JW such that N zeta = m and Neta = n. Results on factorization in the arithmetics of C, needed for the construction of multiplicative functions are thus established. The section on ideals in C contains some improvements on Mahler's results on the same subject. For example, we prove that the basis of any ideal in C is a rational integer. Finally, a systematic account of identities and multiplicative functions defined as above by using the arithmetics of C not previously used by Rankin for this purpose is given. While the identities and. multiplicative functions defined by using Hurwitz maximal quaternion arithmetic are easily related to those constructed by Rankin, the remaining arithmetics not considered by Rankin appear to give new identities and functions.
