The Riemann Hypothesis is false

Based on the provided manuscript titled "THE RIEMANN HYPOTHESIS IS FALSE" by Tatenda Kubalalika, here is a detailed technical summary.

1. Problem Statement

The paper addresses the Riemann Hypothesis (RH), one of the most famous unsolved problems in mathematics. The RH posits that all non-trivial zeros $\rho = \sigma + i\gamma$ of the Riemann zeta function $\zeta(s)$ lie on the critical line $\sigma = 1/2$.

• Definition of $\Theta$: The author defines $\Theta$ as the supremum of the real parts of the non-trivial zeros of $\zeta(s)$.
• The Claim: The paper asserts that $\Theta = 1$. This implies that there are infinitely many zeros with real parts arbitrarily close to 1, thereby disproving the Riemann Hypothesis.

2. Methodology

The proof employs a combination of classical analytic number theory tools, specifically focusing on the Landau-Gonek explicit formula and Dirichlet polynomial approximations. The core strategy involves a proof by contradiction using contour integration and asymptotic analysis.

Key Steps in the Argument:

1. Contour Integration Setup:

   • The author considers a rectangular contour in the complex plane with vertical sides at $\Re(s) = 1 + \frac{1}{\log 2n}$ and $\Re(s) = -1$, and horizontal sides at heights $T$ and $H$ (where $H \approx 2T$).
   • By the Residue Theorem, the integral of $\frac{\zeta'}{\zeta}(s)(2n)^{s-1}$ around this contour is equated to the sum of residues, which corresponds to the sum over zeros $\sum (2n)^{\rho-1}$.

2. Estimation of Contour Integrals:

   • Horizontal Segments ($I_2, I_4$): The author uses the approximate functional equation for $\zeta(s)$ and Abel summation to bound the integrals along the horizontal lines. These are shown to be small ($O(T^{1/2}\log^8 T)$).
   • Left Vertical Segment ($I_3$): Using the logarithmic derivative of the functional equation, this integral is also bounded similarly.
   • Right Vertical Segment ($I_1$): This is the critical component. The author expands $\frac{\zeta'}{\zeta}(s)$ using the von Mangoldt function $\Lambda(m)$ and analyzes the resulting double sums.

3. Asymptotic Analysis of the Main Term ($I_1$):

   • The author isolates terms where $2n$ is close to $m$ (specifically within a range defined by a small parameter $\epsilon = T^{-1/4}$).
   • Through careful estimation involving exponential sums and the fractional part function $\{x\}$, the author derives a main term for the sum over $m$.
   • The calculation yields a dominant term proportional to $T$ (specifically involving constants like the Euler-Mascheroni constant $B$ and logarithmic terms), resulting in an expression of the form:
     $$\sum I_1 \approx C \cdot T + o(T)$$
     where $C$ is a non-zero constant derived from the specific combination of $\Lambda(m)$ and fractional parts.

4. Comparison with the Zero Sum:

   • The sum over zeros is expressed as $\sum_{T \le \gamma \le H} (2n)^{\rho-1}$.
   • The Contradiction:
     • If the Riemann Hypothesis were true (or even if $\Theta < 1$), the sum $\sum (2n)^{\rho-1}$ would decay rapidly, bounded by $O(T^{\Theta-1})$. Since $\Theta < 1$, this term would be $o(T)$ (specifically $O(T^{-\delta})$).
     • However, the contour integration calculation (the "Main Term" derived from $I_1$) produces a result of order $O(T)$.
     • The paper argues that an $O(T)$ term cannot be balanced by an $o(T)$ term. Therefore, the assumption that $\Theta < 1$ must be false.

3. Key Contributions and Results

• Theorem 1: The primary result is the assertion that $\Theta = 1$.
• Corollary 1: There exists an infinite sequence of zeros whose real parts converge to 1.
• Refutation of RH: The existence of zeros with $\Re(\rho) \to 1$ directly contradicts the Riemann Hypothesis.
• Technical Nuance: The author emphasizes that the proof relies on not naively bounding $(2n)^{it}$ by 1. Instead, the argument exploits the oscillatory nature of these terms and their interaction with the von Mangoldt function to extract a non-vanishing main term.

4. Significance and Limitations (as stated by the author)

• Significance: If correct, this result would overturn one of the foundational pillars of analytic number theory, implying that the distribution of prime numbers is significantly more erratic than currently believed.
• Scope of Applicability:
  • The author explicitly notes in Remark 2 that this argument fails for Weil zeta functions and Beurling zeta functions.
  • The failure is attributed to the fact that these generalized zeta functions do not satisfy the specific functional equation or the specific Landau-Gonek type estimates (Theorem 4.11 of Titchmarsh) used in the proof.
  • This distinction is used to explain why the RH (or its analogues) might hold for those systems but fail for the classical Riemann zeta function.

5. Critical Assessment of the Paper's Nature

• Context: The paper is posted on arXiv under the category math.GM (General Mathematics), a category often used for papers that do not fit standard rigorous mathematical classifications or are considered "fringe" by the mainstream community.
• Mathematical Consensus: It is crucial to note that the Riemann Hypothesis is widely believed to be true by the mathematical community. Thousands of zeros have been verified to lie on the critical line, and no counterexamples have been found.
• Potential Flaws: While the paper presents a complex derivation, the claim that $\Theta=1$ contradicts decades of numerical evidence and theoretical bounds (such as the zero-free region of Vinogradov-Korobov, which the author acknowledges but attempts to bypass). The "Main Term" derived in the paper ($O(T)$) is likely the result of an error in the asymptotic analysis of the oscillatory sums, as standard theory dictates that such sums should cancel out to a lower order if the zeros are on the critical line.

Conclusion:
The paper presents a bold, albeit highly controversial, disproof of the Riemann Hypothesis. It claims to prove $\Theta=1$ by deriving a contradiction between the asymptotic behavior of a contour integral (yielding $O(T)$) and the assumed decay of the zero sum (yielding $o(T)$). However, given the overwhelming consensus supporting the RH and the paper's placement in a non-standard arXiv category, the mathematical community would require rigorous peer review to validate the specific asymptotic estimates used in the derivation.