Jacob's ladders, new classes of sum variants of almost linear formula and corresponding $\zeta$-functionals and $\zeta$-equivalents of the Fermat-Wiles theorem

Technical Summary: Jacob's Ladders, New Classes of Sum Variants of Almost Linear Formula and Corresponding $\zeta$-Functionals and $\zeta$-Equivalents of the Fermat-Wiles Theorem

1. Problem Statement
The paper addresses the structural analysis of the Riemann zeta-function, specifically focusing on the Hardy-Littlewood integral of the squared modulus of the Riemann-Siegel function, $Z(t)$, defined as $Z(t) = e^{i\vartheta(t)}\zeta(\frac{1}{2} + it)$. The author seeks to derive new "sum variants" of an "almost linear formula" that describes the asymptotic behavior of this integral over specific intervals. Furthermore, the paper aims to construct new classes of $\zeta$-functionals and $\zeta$-equivalents of the Fermat-Wiles theorem (Fermat's Last Theorem) based on these integral properties. A central distinction in this work is the shift from using the sequence of zeros of $Z(t)$ (denoted $\{\gamma_k\}$) to using the Gramm's sequence (denoted $\{t_\nu\}$), defined by the roots of the equation $\vartheta(t) = \nu\pi$.

2. Methodology
The methodology relies on the theory of "Jacob's ladders," a framework developed in the author's previous works ([2]–[7]). Key methodological components include:

• Jacob's Ladders and Iterations: The paper utilizes the function $\varphi_1(T)$ and its iterations (direct and reverse) to define intervals $[T, \frac{1}{T}(T)]$, where $\frac{1}{T}(T) = \varphi_1^{-1}(T)$. These intervals expand as $T \to \infty$, behaving analogously to a one-dimensional Friedmann-Hubble expanding universe.
• Gramm's Sequence: Instead of the zeros of $Z(t)$, the construction is based on the sequence $\{t_\nu\}$ where $\vartheta(t_\nu) = \nu\pi$. The spacing between these points is governed by the Titchmarsh formula: $t_{\nu+1} - t_\nu \sim \frac{2\pi}{\ln t_\nu}$.
• Rectangular $\zeta$-Signals: The author defines rectangular signals $A_2(n)$ over intervals $[t_n, t_{n+1}]$ within the Jacob's ladder interval. The area of these rectangles is defined as $|A_2(n)| = Z_2(\tilde{t}(n))(t_{n+1} - t_n)$, where $Z_2(t) = |Z(t)|^2$ and $\tilde{t}(n)$ is a mean value point.
• Partitioning and Summation: The integral of $Z_2(t)$ over the interval $[T, \frac{1}{T}(T)]$ is decomposed into a sum of these rectangular areas. The paper further partitions these rectangles into smaller elementary rectangles $A_{r,p}^2(n)$ by equidistantly dividing the time and amplitude dimensions.
• Asymptotic Analysis: The paper employs asymptotic estimates ($T \to \infty$) to derive limits for sums of these areas and to construct functional limits involving a scaling parameter $\tau$.

3. Key Contributions and Results

• Fourth Sum Variant of the Almost Linear Formula:
  The paper establishes a new asymptotic formula for the sum of the areas of the rectangular signals generated by the Gramm's sequence:
  $$\sum_{n=1}^{N(T)} |A_2(n)| \sim (1-c)T, \quad T \to \infty$$
  This is identified as the "fourth sum variant" of the author's almost linear formula, distinct from previous variants based on the zeros of $Z(t)$.

• Canonical Formula for the Arithmetic Mean:
  A significant result is the derivation of the asymptotic behavior of the arithmetic mean of the areas $|A_2(n)|$. The paper proves:
  $$\frac{1}{N} \sum_{n=1}^{N(T)} |A_2(n)| \sim 2\pi, \quad T \to \infty$$
  This result is derived independently of any unproved hypotheses. The author notes that $2\pi$ corresponds to the area of a circle with radius $\sqrt{2}$ or a specific rectangle, suggesting an oscillation of the signal areas around this geometric constant.

• New $\zeta$-Functionals and $\zeta$-Equivalents of Fermat-Wiles:
  By applying a substitution $T = \frac{x}{1-c}\tau$, the paper derives new functionals. For a fixed $x > 0$:
  $$\lim_{\tau \to \infty} \frac{1}{\tau} \sum_{\tilde{t}(n) > \frac{x}{1-c}\tau} |A_2(n)| = x$$
  The paper extends this to Fermat's rationals ($x = \frac{x^m + y^m}{z^m}$ for $m \ge 3$). It posits a condition where the limit of the scaled sum does not equal 1 (or a specific fraction $l/sq$ for partitioned sums) on the set of Fermat's rationals. This non-equality is presented as a new "$\zeta$-equivalent" of the Fermat-Wiles theorem.

• Second Generation of Formulas:
  The paper introduces "second generation" classes of these formulas by partitioning the rectangular signals into $l$ sub-rectangles. It derives limits for sums of these sub-areas, showing they converge to $\frac{l}{sq}x$ (where $s, q$ are partition parameters and $l$ is the number of selected sub-rectangles).

4. Significance and Claims
The paper claims that these results represent new insights into the structure of the classical Hardy-Littlewood integral. Specifically:

• Independence from Unproved Hypotheses: A primary claim of significance is that the result regarding the arithmetic mean ($2\pi$) for the Gramm's sequence holds true independently of any unproved hypotheses (such as the Riemann Hypothesis or the Mertens Hypothesis).
• Contrast with Previous Work: The author explicitly contrasts this with results derived from the sequence of zeros $\{\gamma_k\}$. The analogous result for the zero-based sequence (Theorem 10) is shown to be conditional on the very strong Mertens hypothesis ($M(x) = O(\sqrt{x})$), which implies the Riemann Hypothesis and the simplicity of the zeros.
• Geometric Interpretation: The paper suggests a geometric interpretation of the analytic results, linking the oscillation of the zeta-function's energy areas to the constant $2\pi$ and the geometry of Jacob's ladders.

The author concludes that these findings provide a new class of increments for the Hardy-Littlewood integral and offer a novel perspective on the Fermat-Wiles theorem through the lens of $\zeta$-functionals generated by Gramm's sequence.