Jacob's ladders, point of contact of the remainder in the prime-number law with the Fermat-Wiles theorem and multiplicative puzzles on some sets of integrals

In this paper we prove, on the Riemann hypothesis, the existence of such increments of the Ingham integral (1932) that generate new functionals together with corresponding new $P\zeta$-equivalents of the Fermat-Wiles theorem. We obtain also new results in this direction.