Jacob's ladders, next equivalents of the Fermat-Wiles theorem and new infinite sets of the equivalents generated by the Dirichlet's series

TL;DR

The paper addresses the problem of generating new Fermat-Wiles equivalents by exploiting deep connections between the Riemann zeta-function, Dirichlet-series, and Jacob's ladders. Its approach combines the Hardy-Littlewood mean-value formula, Selberg's results, and almost-linear zeta-integral behavior within the Jacobian ladder framework to construct three main equivalence schemes: products of three simplest $\zeta$-integrals, a linear-combination of almost-linear $\zeta$-integrals, and Dirichlet-series–based expressions. The key contributions are (i) a new $\zeta$-equivalent derived from a product of three integrals, (ii) a linear-combination $\zeta$-equivalent, (iii) an infinite family of $\mathfrak{D}$-equivalents generated by Dirichlet-series without ladders, and (iv) a unifying asymptotic framework linking $\zeta$, Dirichlet sums $D(x)$, and Jacob's ladders. These results broaden the landscape of Fermat-Wiles–type equivalences and provide a versatile set of tools for exploring mean-value structures in analytic number theory, with potential implications for further generalizations and connections to Dirichlet-series analysis.

Abstract

In this paper we obtain new sets of equivalents of the Fermat-Wiles theorem. Simultaneously, we obtain also asymptotic connections between the set of Dirichlet's series, certain segments of the Dirichlet's sum $\mfrak{D}(x)$, Riemann zeta-function and Jacob's ladders.

Theorems & Definitions (8)

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