Jacob's ladders and the equivalent of the Fermat-Wiles theorem generated by the Hardy-Littlewood formula (1921)

1. Problem Statement

The paper addresses two primary objectives:

1. Generating New Equivalents of the Fermat-Wiles Theorem: The author seeks to demonstrate that a specific, 105-year-old lemma (Lemma 18) from a 1921 paper by G.H. Hardy and J.E. Littlewood can be utilized to generate a continuum set of new functional equivalents to the Fermat-Wiles theorem (which states that $x^n + y^n = z^n$ has no non-trivial integer solutions for $n \geq 3$).
2. Critique of Trigonometric Sum Methods: The paper critically analyzes the limitations of the method of trigonometric sums (Weyl's method) in estimating the Riemann zeta-function, specifically in relation to the zeros of its derivative, $Z'(t)$. It contrasts the author's previous results with those of A.A. Karatsuba, arguing that Karatsuba's claims of improvement are based on incomplete applications of the author's earlier work and that the method of trigonometric sums faces "extremely distanced" limits compared to the Riemann Hypothesis.

2. Methodology

The paper employs a combination of analytic number theory techniques, asymptotic analysis, and the theory of the Riemann zeta-function.

• Hardy-Littlewood Lemma 18: The core mathematical engine is Lemma 18 from Hardy and Littlewood's 1921 work, which provides an estimate for the integral of the square of the function $J(t, H)$ (an integral of the Hardy-Littlewood function $X(t)$).
  • The paper utilizes the asymptotic relationship between the Hardy-Littlewood function $X(t)$ and Riemann's $Z(t)$ function: $X(t) \sim (\frac{\pi}{2})^{1/4} Z(t)$.
• Jacob's Ladders: The author introduces a specific geometric structure called "Jacob's ladders," defined by a sequence of reverse iterations $\phi^{-r}_1(T)$. These iterations create a partition of the real line into segments $[ \frac{r-1}{T}, \frac{r}{T} ]$ that behave asymptotically like an expanding universe. This structure allows for the decomposition of integrals involving $|\zeta(1/2 + it)|^2$ into asymptotically equal parts.
• Functional Construction: By combining the Hardy-Littlewood integral estimates with the Jacob's ladder partitions and specific substitutions (e.g., $T = \tau^{1/a}$, $\tau = x\rho$), the author constructs a complex functional limit involving integrals of $Z^2(t)$ and $X(u)$.
• Asymptotic Analysis of $Z'(t)$: In the second half, the paper analyzes the distribution of zeros of the derivative $Z'(t)$ using elementary trigonometric sums $S(a, b)$. It compares the author's 1981 asymptotic formulas with Karatsuba's later work.

3. Key Contributions

A. New $\zeta$-Equivalents of the Fermat-Wiles Theorem

The author proves that the Hardy-Littlewood Lemma 18 can generate a continuum set of new conditions equivalent to the Fermat-Wiles theorem.

• The Functional: A specific limit functional is derived (Theorem 1) involving parameters $x, a, \alpha$:
  $$\lim_{\rho \to \infty} \frac{1}{\rho} \left\{ \int_{[x\rho]_1}^{[x\rho + C]_1} Z^2(t) dt \right\}^{-\alpha} \times \int \left[ \int X(u) du \right]^2 dt = x$$
• The Equivalence: When the variable $x$ is replaced by a Fermat rational ($x = \frac{x^n + y^n}{z^n}$), the theorem states that if the Fermat-Wiles theorem is true (i.e., no integer solutions exist for $n \geq 3$), then the limit of this functional cannot equal 1.
  • Result: The condition $\text{Limit} \neq 1$ on the set of all Fermat rationals is a new $\zeta$-equivalent of the Fermat-Wiles theorem. Since the parameters $a$ and $\alpha$ range over a continuum, this generates a continuum set of such equivalents.

B. Critique of Karatsuba and Limits of Trigonometric Sums

The author provides a rigorous critique of A.A. Karatsuba's paper (dedicated to I.M. Vinogradov's 90th anniversary):

• Priority and Generality: The author argues that Karatsuba's results on the zeros of $Z^{(k)}(t)$ are merely specific instances or generalizations of the author's earlier, more general theorems (published in 1981) regarding $Z'(t)$. The author claims Karatsuba ignored the general structure of the author's work, focusing only on specific corollaries (like Kolesnik's exponent) while omitting the necessary assumptions.
• Vinogradov's Scepticism: The paper invokes I.M. Vinogradov's skepticism regarding the method of trigonometric sums. It argues that while trigonometric sums can improve error terms slightly (e.g., from $1/6$ to $35/216$), they are "extremely distanced" from the limits imposed by the Riemann Hypothesis.
• The "Alice in Wonderland" Limit: The author demonstrates that under the Riemann Hypothesis, the interval containing zeros of $Z'(t)$ shrinks to $O(1/\ln \ln T)$, whereas trigonometric sum methods yield intervals of size $O(T^\epsilon)$. The gap is so vast that improving trigonometric sums to reach the Riemann Hypothesis limit is deemed practically impossible without new theoretical breakthroughs in zeta-function theory.

4. Key Results

1. Theorem 1 & 2: Establishment of a functional limit involving $Z(t)$ and $X(t)$ that equals a constant $x$. When $x$ represents a Fermat rational, the limit $\neq 1$ constitutes a new equivalent of the Fermat-Wiles theorem.
2. Corollary 1 (Fermat Rationals): Explicitly formulates the $\zeta$-condition for Fermat rationals, proving the existence of a continuum of such equivalents.
3. Theorem 3 (Riemann Hypothesis Implications): Under the Riemann Hypothesis, within an infinitesimal interval $(T, T + \frac{1}{\ln \ln \ln T})$, there are packed zeros of odd order for functions $Z^{(k)}(t)$ for $k$ up to a specific logarithmic bound. This contrasts sharply with results derived solely from trigonometric sums.
4. Refutation of Karatsuba's Claims: The paper asserts that Karatsuba's Lemma 5 is essentially a generalization of the author's Formula 1 (1981) and that Karatsuba failed to prove the necessary bounds on trigonometric sums to validate his claims of "improvement" over the author's work.

5. Significance

• Reinterpretation of Classical Lemmas: The paper highlights the enduring power of Hardy and Littlewood's Lemma 18, showing it can be repurposed decades later to generate deep connections between the Riemann zeta-function and Fermat's Last Theorem.
• New Analytic Equivalents: It provides a novel class of analytic conditions (involving integrals of $Z^2(t)$) that are mathematically equivalent to the Fermat-Wiles theorem, offering a different perspective on the problem through the lens of the zeta-function's zero distribution.
• Methodological Warning: The paper serves as a significant commentary on the limitations of current analytic number theory tools (specifically trigonometric sums). It reinforces the view that significant progress in the distribution of zeta zeros likely requires new theoretical frameworks beyond incremental improvements of Weyl's method.
• Historical Context: The author clarifies the historical priority of his results regarding $Z'(t)$ and the distribution of its zeros, addressing perceived omissions in the work of prominent mathematician A.A. Karatsuba.

In summary, Jan Moser's paper bridges classical analysis with modern number theory, using "Jacob's ladders" to construct new functional equivalents of Fermat's Last Theorem while simultaneously critiquing the current boundaries of trigonometric sum methods in the study of the Riemann zeta-function.