Jacob's ladders and new $\zeta$-functionals and corresponding $\zeta$-equivalents of the Fermat-Wiles theory based on sums of elementary $\zeta$-pulses

Technical Summary: Jacob's Ladders and New $\zeta$-Functionals

Problem Statement
The paper addresses the construction of "sum-variants" of previously established $\zeta$-functionals related to the Hardy-Littlewood integral. The primary goal is to transform continuous integral representations of the Riemann zeta-function's magnitude squared, $|\zeta(1/2 + it)|^2$, into discrete sum forms based on elementary signals (pulses). A secondary objective is to derive new "$\zeta$-equivalents" of the Fermat-Wiles theorem (Fermat's Last Theorem) based on these sum-variants, specifically utilizing isolated rectangle-shaped signals derived under the assumption of the Riemann Hypothesis (RH).

Methodology
The author employs the theory of "Jacob's ladders," specifically the reverse iterations of the function $\varphi_1(T)$, denoted as $\frac{r}{T}(T) = \varphi_1^{-r}(T)$. These iterations define a sequence of points that partition the integration domain $[T, \frac{1}{T}(T)]$ into "almost linear" increments.

The methodology proceeds through the following steps:

1. Partitioning: The interval $(T, \frac{1}{T}(T))$ is partitioned by the zeros $\{\gamma_n\}$ of the Riemann-Siegel function $Z(t)$. The integral of $Z^2(t)$ over this interval is decomposed into a sum of integrals over sub-intervals $[\gamma_n, \gamma_{n+1}]$.
2. Discretization (Sum-Variants):
   • First Variant: The integral over each sub-interval $[\gamma_n, \gamma_{n+1}]$ is treated as the area of a curvilinear signal $A_0(n)$.
   • Second Variant: Using the Mean Value Theorem, the area $A_0(n)$ is equated to the area of a rectangle $A_1(n)$ with height $Z^2(\bar{t}(n))$ and width $(\gamma_{n+1} - \gamma_n)$.
   • Third Variant (RH Dependent): Assuming the Riemann Hypothesis, the function $Z^2(t)$ is shown to have a single maximum $t_0(n)$ within each interval $(\gamma_n, \gamma_{n+1})$. Using the Bonnet Mean Value Theorem, the integral is transformed into the area of a rectangle $B(n)$ defined by the height $Z^2(t_0(n))$ and a specific width $[\xi_2(n) - \xi_1(n)]$, where $\xi_1$ and $\xi_2$ are Bonnet points.
3. Functional Construction: A limit process is applied where the upper bound of the integration (or summation) scales with a parameter $\tau$. The substitution $T = \frac{x}{1-c}\tau$ (where $c$ is Euler's constant) is used to normalize the sums.

Key Contributions and Results

1. New $\zeta$-Functionals:
   The paper establishes three distinct limit formulas that equate a normalized sum of signal areas to a fixed real number $x > 0$:

   • Sum of Curvilinear Signals:
     $$\lim_{\tau\to\infty} \frac{1}{\tau} \sum_{\gamma_n < [\frac{x}{1-c}\tau]_1} \int_{\gamma_n}^{\gamma_{n+1}} Z^2(t) dt = x$$
   • Sum of Rectangular Signals (Mean Value):
     $$\lim_{\tau\to\infty} \frac{1}{\tau} \sum_{\gamma_n < [\frac{x}{1-c}\tau]_1} Z^2(\bar{t}(n))(\gamma_{n+1} - \gamma_n) = x$$
   • Sum of Rectangular Signals (RH Maximum):
     Assuming RH, the paper derives:
     $$\lim_{\tau\to\infty} \frac{1}{\tau} \sum_{t_0(n) < [\frac{x}{1-c}\tau]_1} Z^2(t_0(n))[\xi_2(n) - \xi_1(n)] = x$$

2. $\zeta$-Equivalents of the Fermat-Wiles Theorem:
   By substituting Fermat's rationals $x \to \frac{x^m + y^m}{z^m}$ (for $m \geq 3$) into the functional limits, the author derives conditions that express the Fermat-Wiles theorem. Specifically, the paper claims that for all Fermat's rationals, the limit of the normalized sum is not equal to 1:
   $$\lim_{\tau\to\infty} \frac{1}{\tau} \sum (\dots) \neq 1$$
   This inequality is presented as a new "$\zeta$-equivalent" of the theorem. The paper provides three such equivalents corresponding to the three sum-variants described above.

3. Geometric Interpretation:
   The paper introduces a geometric transformation where the set of curvilinear signals $A_0(n)$ (bounded by $Z^2(t)$) is replaced by a disconnected set of isolated rectangular pulses $B(n)$ under the Riemann Hypothesis. The paper asserts that this transformation preserves the measure (area) of the signals.

Significance and Claims
The paper claims to provide a "more stable (or coded) form" of the Hardy-Littlewood integral by converting it into a sum of elementary pulses. The author posits that these sum-variants offer a new analytical perspective on the relationship between the Riemann zeta-function and the Fermat-Wiles theorem.

The significance is framed as follows:

• New Points of Contact: The work establishes further "points of contact" between the almost linear increments of the Hardy-Littlewood integral and Fermat's Last Theorem, extending previous work cited in the introduction.
• Role of RH: The paper demonstrates that the Riemann Hypothesis allows for a specific structural simplification of the $\zeta$-functional, replacing continuous curvilinear signals with discrete, isolated rectangular pulses while maintaining the functional limit.
• Conservation Laws: The author identifies a "global conservation law" (Property 6) on the Riemann Hypothesis that relates the maximum value of $Z^2(t)$, the mean value, and the lengths of the corresponding intervals.

The author explicitly states that the results are derived based on "almost linear increments" proven in prior works and that the construction of the $\zeta$-equivalent for Fermat's rationals relies on the consequences of the Riemann Hypothesis used in previous cosmological models by the same author. The paper does not propose experimental applications or future implications beyond the mathematical derivation of these equivalents.