Evaluation of Weber’s functions at quadratic irrationalities

ABSTRACT. Let η(z) denote the Dedekind eta function. Let ax2+ bxy+ cy2 be a positive-definite, primitive, integral, binary quadratic form of discriminant d( = b2 − 4ac < 0). The value of |η((b + √d)/2a) | is determined for an arbitrary discriminant d. This result generalizes the corresponding result when d is fundamental, which was obtained by van der Poorten and Williams [16] in 1999. As a consequence of our evaluation of |η(z) | for z = ((b + √d)/2a), formulae are obtained for the moduli of Weber’s functions f(z), f1(z) and f2(z) [20: p. 114]. From these formulae the values of f( √−m), f1(√−m) and f2(√−m) are determined for an arbitrary positive integer m. 1. Introduction. The Dedekind eta function η(z) is defined for all z = x+ iy ∈ C with y> 0 by η(z) = epiiz/12 m=