On algebraic values of function exp (2ni x + log log y)

Abstract

It is proved that for all but a finite set of the square-free integers $d$ the value of transcendental function $\exp~(2\pi i ~x+\log\log y)$ is an algebraic number for the algebraic arguments $x$ and $y$ lying in a real quadratic field of discriminant $d$. Such a value generates the Hilbert class field of the imaginary quadratic field of discriminant $-d$.