Values of the Dedekind Eta Function at Quadratic Irrationalities
نویسندگان
چکیده
Let d be the discriminant of an imaginary quadratic field. Let a, b, c be integers such that b − 4ac = d, a > 0, gcd(a, b, c) = 1. The value of |η ( (b + √ d)/2a ) | is determined explicitly, where η(z) is Dedekind’s eta function η(z) = e ∞ ∏ m=1 (1− e) ( im(z) > 0 ) .
منابع مشابه
Evaluation of the Dedekind Eta Function
We extend the methods of Van der Poorten and Chapman [7] for explicitly evaluating the Dedekind eta function at quadratic irrationalities. Via Hecke L-series we obtain evaluations in some new cases. Specifically we provide further evaluations at points in imaginary quadratic number fields with class numbers up to four. We also describe techniques, which make use of modular equations, which prov...
متن کاملClass Invariants from a New Kind of Weber-like Modular Equation
A new technique is described for explicitly evaluating quotients of the Dedekind eta function at quadratic integers. These evaluations do not make use of complex approximations but are found by an entirely ‘algebraic’ method. They are obtained by means of specialising certain modular equations related to Weber’s modular equations of ‘irrational type’. The technique works for a large class of et...
متن کاملIntegrals of Eisenstein Series and Derivatives of L-functions
In his lost notebook, Ramanujan recorded a formula relating a “character analogue” of the Dedekind eta-function, the integral of a quotient of eta-functions, and the value of a Dirichlet Lfunction at s = 2. Here we derive an infinite family of formulas which includes Ramanujan’s original formula as a special case. Our results depend on a representation of values of the derivatives of Dirichlet ...
متن کاملQuotients of Values of the Dedekind Eta Function
Inspired by Riemann’s work on certain quotients of the Dedekind Eta function, in this paper we investigate the value distribution of quotients of values of the Dedekind Eta function in the complex plane, using the form η(Ajz) η(Aj−1z) , where Aj−1 and Aj are matrices whose rows are the coordinates of consecutive visible lattice points in a dilation XΩ of a fixed region Ω in R, and z is a fixed ...
متن کاملLinking Numbers and Modular Cocycles
It is known that the 3-manifold SL(2,Z)\ SL(2,R) is diffeomorhic to the complement of the trefoil knot in S3. As is shown by E. Ghys the linking number of the trefoil with a modular knot associated to a hyperbolic conjugacy is related to the classical Dedekind symbol. These symbols arose historically in the transformation property of the logarithm of Dedekind’s eta function. In this paper we st...
متن کامل