Distribution of values of a real function means,moments, and symmetry

For a distribution functionD we define itsabsolute andsigned moments of orderk∈R, which generalise in a natural way the Hamburger moments of orders an even and an odd natural number. Similarly, for a real functionh we define itsabsolute andsigned asymptotic means of orderk∈R. We show that if the means exist on an infinite and bounded set of values ofk, then they exist on an intervalI and coincide onI ^o with the moments ofD=D _h, the distribution function of the values ofh, which is shown to exist (in the sense of Wintner). We also give a sufficient condition forD _h to be symmetric. These results apply to a class of functionsh that contain in particular error terms related to the Euler phi function and to the sigma divisor function. A further application on a certain class of converging trigonometrical series implies in particular classical results of A. Wintner establishing the existence for such functions of a distribution function as well as Hamburger moments of arbitrarily large orders. The remainder term of the prime number theorem belongs to this class provided the Riemann hypothesis holds, and the distribution function of its values is shown to be “almost” symmetric.