Abstract
In this paper the periodic even functions for which all the regular Riemann sums vanishe are called special periodic even functions. A construction of some of them is put forward, this one rest on the Fourier serie and H. Davenport’s estimations concerning the Moebius function. The special periodic even functions seem linked to the number theory, as this can be seen on the Fourier serie result, on the proposed constructions and on the proof that such functions cannot belong to the Wiener algebra.