Jacob's ladders, new equivalent of the Fermat-Wiles theorem generated by certain cross-breed of Ingham and Heath-Brown formula (1979) and some chains of equivalents
Jan Moser
TL;DR
The paper addresses the problem of obtaining new Fermat-Wiles-type equivalents via zeta-function functionals. It builds a cross-breed of Ingham-type and Heath-Brown-type formulas together with Jacobman-style ladder constructions to define a $\zeta$-functional that yields a limit equal to $x$ for all $x>0$ and $\sigma\ge \tfrac{1}{2}+\epsilon$, including a Fermat-Wiles equivalent in a special rational substitution. The main contributions are (i) a new $\zeta$-functional and its Fermat-Wiles-type equivalence, (ii) a detailed structure of almost linear increments of the Hardy-Littlewood integral expressed through nonlinear, nonlocal compositions of zeta-integrals, and (iii) the existence of a continuum of finite chains of equivalences parameterized by $x$. The results provide a novel framework linking high-degree zeta-integrals with Fermat-type relations, revealing a rich family of asymptotic equivalences and chain structures with potential implications for analytic number theory and the study of zeta-values.
Abstract
In this paper we obtain a new equivalent of the Fermat-Wiles theorem based on a kind of cross-bred of Ingham and D. R. Heath-Brown formula. Further, we prove the existence of infinite set of finite chains of a kind of equivalent expressions of mathematical analysis.