Pith. Sign in

REVIEW

Nontrivial Zeros of the Riemann Zeta Function and Its Derivative

PaperPDF from arXiv

data files

ReviewThe objections and verdict, as JSON Signed receiptProof this review has not been altered
X Bluesky LinkedIn Reddit Hacker News
Chuancun Yin, Xianghua Zhao, Zuolu Hao, Zhongyang Sun

Number Theory · Submitted 9 Nov 2026 · 21 references

Pith's verdict: This paper attempts to prove the Riemann Hypothesis but contains a fatal logical gap: the deduction that |ζ(s)| ≠ |ζ(1−s)| from |E(s)| ≠ 1 fails when ζ(1−s)=0, which is precisely the zero case; the argument is circular and does not hold up.

T0 · AI review

How this review was made: An AI referee (deepseek/deepseek-v4-flash) read the paper on November 10, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

  1. T0 AI review An AI read the paper and answered a fixed list of questions.
  2. T1 Audit Automatic checks that the references and files the paper relies on exist and match.
  3. T2 Compute The paper's code or calculations re-run with the same results.
  4. T3 Formal statement The claims written out in a formal math language a computer can read.
  5. T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 1 major objection · 4 minor · Ask about this paper · Read the full review →
Challenge this review Comment (0) Read on arXiv ↗
Cite ▾
arXiv ID2611.12438 (version 1)
Pith IDNOOAXFQU, Pith's permanent record of this paper and its review
CitationChuancun Yin, Xianghua Zhao, Zhongyang Sun, Zuolu Hao. "Nontrivial Zeros of the Riemann Zeta Function and Its Derivative." arXiv:2611.12438 (2026). https://pith.science/paper/NOOAXFQU
Cite this reviewBibTeX for Pith's review of this paper
Claim paper

Why Pith says this

Highlight any text to comment on it.

Referee report

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

A purported proof of the Riemann Hypothesis via a modulus inequality for the functional equation factor.

The reading

This paper attempts to prove the Riemann Hypothesis, which asserts that all nontrivial zeros of the Riemann zeta function ζ(s) lie on the critical line Re(s)=1/2. The authors' approach relies on the functional equation ζ(s)=E(s)ζ(1-s) with E(s)=2^s $π^{{s-1}}$ sin(πs/2) Γ(1-s). For any fixed a≠1/2 and sufficiently large imaginary part b, they argue that |E(a+ib)| tends to either 0 or ∞, so |E(a+ib)|≠1. From this they infer that |ζ(a+ib)|≠|ζ(1-a-ib)|, which they claim precludes a zero at s=a+ib because a zero would force equality of these absolute values. The paper then computes a bound b(a)≈6.2898 such that for all b>b(a) the inequality holds, and notes that the smallest known zero has imaginary part ≈14.13, concluding that all nontrivial zeros satisfy Re(s)=1/2.

What carries the argument

The key object is the function E(s)=2^s $π^{{s-1}}$ sin(πs/2) Γ(1-s) appearing in the functional equation ζ(s)=E(s)ζ(1-s). The authors analyze the modulus |E(a+ib)|, show it is strictly increasing in b for 0<a<1/2, and prove that for a≠1/2 the limit as b→∞ is either 0 or ∞. They then claim that |E(a+ib)|≠1 for b>b(a) implies |ζ(a+ib)|≠|ζ(1-a-ib)|, which they take as a sufficient condition for the absence of a zero at s=a+ib.

Extended reading notes

Core claim

The paper claims to have proven the Riemann Hypothesis: all nontrivial zeros of ζ(s) lie on the line Re(s)=1/2. The central result is stated as Theorem 1.1, and the proof is presented in Section 3 through Propositions 3.1 and 3.2.

Load-bearing premise

The proof assumes that the inequality |E(a+ib)|≠1 implies |ζ(a+ib)|≠|ζ(1-a-ib)|, but when ζ(1-a-ib)=0 both absolute values are zero regardless of |E|, so the inequality does not follow unless it is also shown that ζ(1-a-ib) cannot vanish for the relevant parameters.

Editorial extensions

If this is right

  • If the Riemann Hypothesis is proved, the error term in the Prime Number Theorem improves to O(√x log x), the best possible bound.
  • Over a thousand conditional theorems in number theory would become unconditional, including results in arithmetic geometry and computational complexity.
  • The proof would confirm that the zeros of ζ(s) are all simple and satisfy the Gaussian unitary ensemble spacing statistics, strengthening the link between number theory and quantum chaos.
  • The generalized Riemann Hypothesis for other L-functions might become more tractable through analogous functional-equation arguments.
  • The exact bound b(a)≈6.2898 would become a universal threshold for the imaginary part of any off-line zero.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The proof implicitly assumes that when ζ(1-s)=0, the modulus inequality still yields a contradiction, but the functional equation then gives ζ(s)=0, making the absolute values equal anyway—the paper does not address this case.
  • If the logical gap in Proposition 3.1 is closed, the method would provide the first complete proof of the Riemann Hypothesis; if not, the argument reduces to a rediscovery of known asymptotic bounds.
  • The threshold b(a)≈6.2898 is far below the first zero height ~14.13, so the proof essentially reduces the claim to a finite check that is already known numerically.
  • The paper's reliance on the strict monotonicity of |E(a+ib)| in b (Lemma 2.1) is sound, but its connection to the vanishing of ζ(s) requires a careful handling of the case where ζ(1-s)=0.
Original abstract

The nontrivial zeros of the Riemann zeta function lie at the heart of analytic number theory. Through the explicit formulas, they govern the distribution of prime numbers: the closer all nontrivial zeros lie to the critical line $\Re(s)=\frac12$, the more uniformly the primes are distributed. The Riemann Hypothesis-asserting that all nontrivial zeros satisfy $\Re(s)=\frac12$-is equivalent to the optimal error bound $O(\sqrt{x}\log x)$ in the Prime Number Theorem. Beyond number theory, the zeros of $\zeta(s)$ and its derivative $\zeta'(s)$ are deeply intertwined: the horizontal distribution of $\zeta'$-zeros near the critical line encodes information about the spacing of consecutive $\zeta$-zeros. This paper establishes some results concerning the nontrivial zeros of the Riemann zeta function.

References

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.