Pseudodifferential arithmetic, disproof of Riemann and proof of Lindelöf hypotheses

This paper employs pseudodifferential arithmetic to construct an explicit operator that allegedly disproves the Riemann hypothesis by showing the real parts of non-trivial zeros have a measure of at least 0.5, while simultaneously claiming to prove the Lindelöf hypothesis.

The Gist

The Big Mystery: The Riemann Hypothesis

Imagine the Riemann Zeta function as a giant, complex musical instrument. When you play it, it produces a series of notes (zeros). For over 160 years, mathematicians have believed that all the "interesting" notes (non-trivial zeros) are perfectly tuned to a single, specific frequency line called the Critical Line (where the real part of the number is exactly 1/2).

This belief is the Riemann Hypothesis. If true, it means the distribution of prime numbers (the building blocks of arithmetic) is incredibly orderly. If false, the music is chaotic.

The New Detective: Pseudodifferential Arithmetic

The author, André Unterberger, decides to investigate this mystery not with the usual tools of calculus (the standard way to analyze music), but with a new, strange tool he calls "Pseudodifferential Arithmetic."

Think of this new tool as a magic microscope that can look at numbers in two dimensions at once:

1. Position: Where the number is (like a location on a map).
2. Momentum: How the number is moving or changing (like its speed).

Usually, in math, you look at position or momentum. This paper forces them to look at both simultaneously, creating a "shadow" or a "ghost" of the numbers.

The Strategy: Building a Machine

Unterberger builds a mathematical machine (an operator) designed to act like a filter.

• He feeds this machine a special distribution of numbers (a "cloud" of points) that is constructed specifically to react to the zeros of the Zeta function.
• He then runs a test: He asks, "If the Riemann Hypothesis is true, what should this machine output?"

The Prediction: If the hypothesis is true, the machine's output should be very small and well-behaved (it should stay within strict limits).

The Twist: The Machine Breaks the Rules

Here is where the paper gets exciting. Unterberger runs the test using his new "Pseudodifferential Arithmetic" microscope.

1. The Disproof: He discovers that the machine cannot stay within those strict limits. The output is too wild. The math shows that if the Riemann Hypothesis were true, the machine would have to produce a "ghost" (a mathematical singularity) that simply cannot exist.

   • The Conclusion: The Riemann Hypothesis is false. The "notes" of the Zeta function are not all on the single line. Some of them are drifting off into the chaos.

2. The Proof of the Lindelöf Hypothesis: While breaking the first hypothesis, he accidentally solves a different puzzle called the Lindelöf Hypothesis.

   • The Analogy: Imagine the Zeta function is a balloon. The Lindelöf Hypothesis claims the balloon won't expand infinitely in certain directions.
   • Unterberger proves that even though the notes are chaotic (Riemann is false), the balloon does stay within a manageable size. He proves the function is "bounded" (it doesn't explode) in the critical area.

The "Shadow" of the Zeros

The most surprising part of the paper is a conclusion about the "drifting" notes.

• Unterberger proves that the "drifting" notes aren't just scattered randomly.
• He calculates that the measure (the total "width" or "density") of the real parts of these zeros is at least 1/2.
• In plain English: Imagine the critical line is a tightrope. The Riemann Hypothesis says all the acrobats are on the rope. Unterberger says, "No, the acrobats have fallen off, and they are spread out over a wide area that is at least as wide as the rope itself."

The "Magic Mirror" (Pseudodifferential Arithmetic)

How did he do it? He used a technique that treats numbers like waves.

• The Old Way: Looking at a number as a static dot.
• The New Way: Looking at a number as a wave that can be reflected, stretched, and compressed.
• He uses a "reflection" trick. He takes a function, flips it inside out (like turning a sock inside out), and sees how it interacts with the prime numbers. This interaction reveals a hidden algebraic structure that standard calculus misses.

The Final Verdict

This paper is a shock to the mathematical world because:

1. It claims to disprove the most famous unsolved problem in math (Riemann).
2. It claims to prove a related, difficult problem (Lindelöf).
3. It does so by inventing a new language ("Pseudodifferential Arithmetic") that mixes the rigid rules of number theory (primes) with the fluid rules of physics (waves and operators).

The Bottom Line: The author argues that the universe of prime numbers is more chaotic than we thought (Riemann is false), but it is still "well-behaved" enough to be predictable in a specific way (Lindelöf is true). He used a new mathematical lens to see a pattern that everyone else missed.

(Note: In the world of mathematics, such bold claims require intense peer review. While the paper presents a complete logical argument, the mathematical community would need to verify every step of this "magic microscope" before accepting the disproof of the Riemann Hypothesis.)

Technical Summary

Technical Summary: Pseudodifferential Arithmetic, Disproof of Riemann and Proof of Lindelöf Hypotheses

Author: André Unterberger
Source: arXiv:2208.12937v43 [math.NT]

1. Problem Statement

The paper addresses two of the most famous unsolved problems in analytic number theory:

1. The Riemann Hypothesis (RH): The conjecture that all non-trivial zeros of the Riemann zeta function $\zeta(s)$ lie on the critical line $\text{Re}(s) = 1/2$.
2. The Lindelöf Hypothesis: The conjecture that $\zeta(1/2 + it) = O(t^\epsilon)$ for every $\epsilon > 0$, implying that $\zeta(s)$ is bounded on vertical lines within the critical strip.

The author argues that a purely analytic approach is insufficient and that arithmetic properties, specifically those related to prime numbers, must be integrated into the analysis. The paper aims to disprove the Riemann Hypothesis and prove the Lindelöf Hypothesis using a novel framework called Pseudodifferential Arithmetic.

2. Methodology: Pseudodifferential Arithmetic

The core methodology combines Weyl symbolic calculus (pseudodifferential operators) with automorphic distribution theory and congruence algebra.

• Weyl Calculus ($\Psi$): The author utilizes the Weyl quantization rule, which maps a distribution $S(x, \xi)$ on the phase plane $\mathbb{R}^2$ to a linear operator $\Psi(S)$ acting on functions in $L^2(\mathbb{R})$. The "Planck constant" is set to 2 for arithmetic applications.
• Key Distributions:
  • $T_\infty(x, \xi)$: A distribution defined as a sum over integer pairs $(j, k)$ weighted by an arithmetic function $a((j,k)) = \prod_{p|(j,k)} (1-p)$. This distribution decomposes over the zeros of the zeta function.
  • Eisenstein Distributions ($E_{-\nu}$): Homogeneous distributions invariant under $SL(2, \mathbb{Z})$. They serve as the spectral building blocks for $T_\infty$.
  • $T_N$ and $T_{\infty, 2}$: Finite approximations and variants of $T_\infty$ used to bridge the gap between continuous analysis and discrete arithmetic.
• The Hermitian Form: The central object of study is the hermitian form $\langle v | \Psi(Q^{2i\pi E} T_\infty) u \rangle$, where $E$ is the Euler operator and $Q$ is a squarefree integer. The behavior of this form as $Q \to \infty$ encodes information about the zeros of $\zeta(s)$.
• Arithmetic Reduction: A crucial step involves transferring the problem from the space of Schwartz functions $\mathcal{S}(\mathbb{R})$ to the finite ring $\mathbb{Z}/(2N^2)\mathbb{Z}$ via a linear map $\theta_N$. This allows the hermitian form to be expressed as a finite-dimensional sum involving Möbius functions and congruences, effectively "discretizing" the operator.

3. Key Contributions and Derivations

A. The Criterion for RH

The paper establishes a criterion where the Riemann Hypothesis is equivalent to specific growth estimates of the hermitian form. Specifically, if RH holds, the form should satisfy:
$$\langle v | \Psi(Q^{2i\pi E} T_\infty) v \rangle = O(Q^{1/2 + \epsilon})$$
The author constructs a generating series $F_\epsilon(s)$ based on these forms. If RH were true, this series would possess specific analytic properties (entirety or specific pole structures).

B. The Disproof of the Riemann Hypothesis

Through a detailed analysis of the generating series $F_\epsilon(s)$ using contour integration and residue calculus:

1. Decomposition: The series is decomposed into parts involving Eisenstein distributions and homogeneous components of the test functions.
2. Residue Analysis: The author analyzes the singularities of the integral representation. A critical singularity arises at $\mu = -1/2$ (related to the pole of the Eisenstein series).
3. Contradiction: The author proves that if there were a zero $\rho$ of $\zeta(s)$ with $\text{Re}(\rho) > 1/2$, it would force a singularity in the analytically continued function $F_\epsilon(s)$ that contradicts the established entire nature of the function derived from pseudodifferential arithmetic.
4. Result: The paper concludes that the Riemann Hypothesis is false. It further proves that the set of real parts of the non-trivial zeros, denoted $Z$, has a Lebesgue measure of at least $1/2$. Specifically, it is shown that$\sigma_0 = \sup { \text{Re}(\rho) : \zeta(\rho)=0 } \ge 4/7$(and later refined to$\sigma_0 = 1$).

C. Proof of the Lindelöf Hypothesis

Using a similar framework but with a modified distribution $S_\infty$ (involving coefficients $1+p$instead of$1-p$), the author proves the Lindelöf Hypothesis.

• The proof demonstrates that $\zeta(s)$ is bounded on vertical lines $\text{Re}(s) = \alpha$ for $1/2 < \alpha < 1$.
• The argument relies on the fact that if $\sigma_0 < 1$, the boundedness would be impossible due to the functional equation and the location of the zeros. Since the Lindelöf bound is proven, and the RH is disproven, the result is consistent with $\sigma_0 = 1$.

4. Key Results

1. Disproof of RH: The Riemann Hypothesis is false. The set of real parts of the non-trivial zeros of $\zeta(s)$ is not a single point ($1/2$) but a set with measure$\ge 1/2$.
2. Accumulation on Re(s)=1: The zeros of $\zeta(s)$ accumulate on the line $\text{Re}(s) = 1$.
3. Proof of Lindelöf: The Lindelöf Hypothesis is true. $\zeta(s)$ is bounded on lines $\text{Re}(s) = \alpha$ for $1/2 < \alpha < 1$.
4. Generalization: The results extend to Dirichlet $L$-functions, proving that for any primitive character $\chi$, the supremum of the real parts of the zeros of $L(s, \chi)$ is 1.
5. Measure of Zeros: The closure of the set of real parts of the zeros has a Lebesgue measure $\ge 1/2$.

5. Significance and Implications

• Paradigm Shift: The paper proposes a radical shift in how the Riemann Hypothesis is approached, moving away from pure complex analysis toward a synthesis of operator theory, harmonic analysis, and number theory (specifically congruence algebra).
• New Tools: It introduces "Pseudodifferential Arithmetic" as a rigorous tool for translating analytic problems into finite algebraic structures, allowing for the explicit computation of spectral properties.
• Controversial Claims: The claim that RH is false is a monumental and highly controversial assertion in mathematics. If correct, it would overturn centuries of number theory that assumes RH. The paper asserts that the "measure of the closure of the set of real parts" being $\ge 1/2$ implies a dense distribution of zeros off the critical line.
• Lindelöf Validity: The proof of the Lindelöf Hypothesis is significant as it is a major open problem. The author notes that while Lindelöf is often considered a consequence of RH, this paper proves it independently, even in the context where RH is false.

Conclusion:
André Unterberger's paper presents a comprehensive, albeit highly technical and unconventional, argument that the Riemann Hypothesis is false and the Lindelöf Hypothesis is true. By constructing a specific operator via Weyl calculus and analyzing its arithmetic properties through Eisenstein distributions, the author derives a contradiction to the existence of zeros strictly on the critical line, concluding that zeros must exist with real parts extending up to 1. The work relies heavily on the interplay between continuous spectral theory and discrete modular arithmetic.