RAMANUJANS CLASS INVARIANTS, KRONECKERS LIMIT FORMULA, AND MODULAR EQUATIONS

In his notebooks, Ramanujan gave the values of over 100 class invarian ts which he had calculated. Many had been previously calculated by Hei nrich Weber, but approximately half of them had not been heretofore: d etermined. G. N. Watson wrote several papers devoted to the calculatio n of class invariants, but his methods were not entirely rigorous. Up until the past few years, eighteen of Ramanujan's class invariants rem ained to be verified. Five were verified by the authors in a recent pa per. For the remaining class invariants, in each case, the associated imaginary quadratic field has class number 8, and moreover there are t wo classes per genus. The authors devised three methods to calculate t hese thirteen class invariants. The first depends upon Kronecker's lim it formula, the second employs modular equations, and the third uses c lass field theory to make Watson's ''empirical method'' rigorous.