Jacob's ladders and new equivalents of the Fermat-Wiles theorem connected with some cross-bred of the formulae of Hardy-Littlewood-Ingham (1926) and of Ingham (1926)
Jan Moser
TL;DR
The paper investigates new connections between zeta-energy integrals on the critical line and in the critical strip by leveraging Jacob's ladders to create non-local interactions between energy blocks. It establishes a principal asymptotic for the ratio of partial energies on these two domains, then develops cross-breeding constructions that yield ζ-equivalents of Fermat-Wiles through functionals that equate to prescribed values (including cases tied to Fermat-type identities). By incorporating higher-moment formulas for $\zeta$, the work extends these Hamiltonian-like energy interactions to 4th-moment settings and culminates in a bilinear asymptotic formula capturing the coupled energies of the two zeta-oscillation families. The results provide a novel, non-linear framework linking classical zeta moments with Fermat-type diophantine analogues, via the geometry of Jacob's ladders and cross-bred energy functionals.
Abstract
The main result of this paper is new formula connecting certain $zeta$-integral on the critical line with a $ζ$-integral in the critical strip. Further, a kind of cross-breeding of the Hardy-Littlewood-Ingham formula and Ingham formula produces new $ζ$-equivalent of the Fermat-Wiles theorem.