Equality of Dedekind sums modulo 72Z


According to the wide appearance of Dedekind sums in different applications on different subjects, a new approach for the equivalence of the essential and sufficient condition for 12s(a_1,b)-.12s(a_2,b) in.24Z and 12s(a_1,b)-12s(a_2,b) in 72Z where s(a,b)=∑_(k mod b)▒((ak/b))((k/b)) and the equality of .two .Dedekind sums with their connections is given. The conditions for 12s(a_1,b)- i12s(a_2,b) in 24Z which is equivalent to 12s(a_1,b)–12s(a_2,b) in 72Z were demonstrated with condition that of 9 does not divide b. Some applications for the important of Dedekind sums were given.