THE MOMENTS OF MINKOWSKI ?(x) FUNCTION: DYADIC PERIOD FUNCTIONS
نویسنده
چکیده
We examine the generating function of moments of the Minkowski question mark function ?(x), which describes the distribution of rationals according to their continued fraction expansion. It appears that the generating function possesses certain modular properties and is defined in C \ R≥1. The exponential generating function satisfies the integral equation, with kernel being the Bessel function of the first kind. Finally, the solution of this integral equation leads to the definition of dyadic period functions of weight 2 and index λ. Such a form is defined and is holomorphic in the domain C \ R≥1, it satisfies the semi-modular functional equation, and it exists for every λ, which is the eigen-value of the properly defined Hilbert-Schmidt integral operator. Mathematical subject classification: 11A55, 11B37, 11F11, 26A30, 33C10.
منابع مشابه
The Moments of Minkowski Question Mark Function: the Dyadic Period Function
The Minkowski question mark function ?(x) arises as a real distribution of rationals in the Farey tree. We examine the generating function of moments of ?(x). It appears that the generating function is a direct dyadic analogue of period functions for Maass wave forms and it is defined in the cut planeC\(0,∞). The exponential generating function satisfies the integral equation with kernel being ...
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