An Integral Related with the Riemann Hypothesis

  1. Huan Xiao

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Abstract

In this note we prove that the Riemann hypothesis is false. The proof is by contradiction based on a criterion of Hu.

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  1. PREreview

    This Zenodo record is a permanently preserved version of a PREreview. You can view the complete PREreview at https://prereview.org/reviews/15368713.

    This manuscript presents a thoughtful and original attempt to examine the Riemann Hypothesis using an integral criterion introduced by Hu. The author's approach is to assume the hypothesis is true and evaluate an associated integral expression, aiming to show a contradiction with the expected result. This method is structured and direct, breaking the integral into three components—S₁, S₂, and S₃—and calculating them step by step.

    The presentation is clear, and the motivation behind the work is evident. Taking on a question like the Riemann Hypothesis is never easy, and it's important to recognize the effort, creativity, and time required to pursue such a direction. Even if some of the technical details require revision, the approach itself is worthwhile, and the paper represents an earnest contribution to the ongoing conversation surrounding this important problem.

    Improvements to Consider

    1. Revisit the evaluation of S₁ in connection with Lemma 4. The integral used in Lemma 4 appears to omit part of the expression. Including the real component may change the value of S₁. Reviewing this step carefully can help ensure the conclusion is accurate.

    2. Clarify the exchange of summation and integration. The paper exchanges the order of summation and integration when computing S₁. To support this step, a short explanation or reference would help show that the operation is valid under the conditions used.

    3. Explain the use of regularization in evaluating S₃. The regularization technique involving the digamma function is an interesting choice. A bit of explanation about why this method applies in this context would be helpful for readers.

    4. Give a brief explanation of the numerical comparison. The contradiction is presented by comparing two values at the end. Including a few lines explaining how these values were obtained would make the result more transparent.

    5. Add references for similar approaches. Citing other work involving integral criteria or regularization in number theory could help place the paper in context and strengthen the background.

    6. Include a short discussion of limitations. Acknowledging areas that still need further analysis shows maturity and balance. Even a brief note on what could be explored next would improve the manuscript.

    Conclusion

    This work reflects an independent and determined effort to approach the Riemann Hypothesis in a fresh way. While certain calculations may need to be re-examined, the structure of the paper is solid, and the idea is clearly expressed. It's encouraging to see someone take on such a difficult and meaningful problem with care and originality.

    Progress in mathematics often comes through careful refinement and continued exploration. What matters most is the commitment to revisiting ideas, deepening understanding, and building on the work already done. The reviewer encourages the author to continue developing this line of inquiry, strengthen the key steps, and remain engaged with this important area of research.

    Competing interests

    The author declares that they have no competing interests.

  2. Version published to 10.20944/preprints202409.1294.v1