An identity for the Dedekind eta-function involving two independent complex variables

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...

Quotients 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 ...

Values of the Dedekind Eta Function at Quadratic Irrationalities

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 ) .

An identity involving the function ep(n)

Let p be any fixed prime, n be any positive integer, ep(n) denotes the largest exponent of power p in n. That is, ep(n) = m, if p | n and p n. In problem 68 of [1], Professor F.Smarandache asked us to study the properties of the sequence {ep(n)}. About the elementary properties of this function, many scholars have studied it (see reference [2]-[7]), and got some useful results. For examples, Li...

On Humbert matrix function of two complex variables under differential operator