Harmonic series summation lemma and Vasyunin's formulas

Introducting a rather simple map of the Nyman–Beurling approach, we arrive at Vasyunin style contagent formulas for the relevant inner products. This is accomplished by means of a summation formula for cetain series derived from arithmetical progression subseries of the harmonic series. The main attractiveness of the resulting contagent sums for the inner products is the fact that the coefficients of the contagents are only $\pm1$, which gives perhaps some hope to lighten up the calculation of the Gramians.