FALSEHOOD OF THE RIEMANN HYPHOTHESIS

I have published a revised and updated version of my paper (DOI:[10.13140/RG.2.2.18216.02565]), originally published on ResearchGate in December 2025, on Zenodo on May 5, 2026 (DOI:[10.5281/zenodo.19985210]) After publication, I needed to correct an error, so I further revised the paper and created version 2. I would like to thank all the readers who have read my paper, and I sincerely apologize for publishing a revised version again. 【Updated: May 22, 2026 (Version 2)】 The arguments and key points have been thoroughly updated and strengthened to correct errors pointed out in the previous version. The fundamental conclusions remain unchanged. The correction concerns the view that the formulas of P. Dusart, Ch. Axler and Rosser, Schoenfeld were independent and that prime numbers were included in one of them, but this was incorrect. This is because the P. Dusart, Ch. Axler formula is derived from the Rosser, Schoenfeld formula and has strict restrictions; therefore, the P. Dusart, Ch. Axler formula must be included in the Rosser, Schoenfeld formula. Thus, these two formulas are not independent. In this new version of the paper, we have shown that there exists a "P. Dusart-type complementary formula" that is included in the Rosser, Schoenfeld formula but does not satisfy the P. Dusart, Ch. Axler formula, i.e., it is in the form of a complement. Furthermore, by showing that prime numbers oscillate between these two formulas—the P. Dusart, Ch. Axler formula and "the P. Dusart-type complementary formula"—we conclude that the Riemann hypothesis is false. We are confident that this new version is even more convincing than the previous one. Please take a look. 【ABSTRACT】This paper examines the asymptotic formula for prime numbers Pn = n{logn+loglogn+O(1)} and the error term O(1). Considering Littlewood’s theorem, the error term O(1) must be composed of two formulas: P. Dusart, Ch. Axler’s formula and the P. Dusart-type complementary formula to which prime numbers contained in Rosser, Schoenfeld ’s formula but outside the range specified by P. Dusart, Ch. Axler ’s formula belong. In other words, Rosser, Schoenfeld ’s formula is formed by combining these two formulas. However, these two formulas cannot be used simultaneously, and prime numbers must be included in one of these formulas. This is what had made solving the Riemann hypothesis so difficult. And, we will consider the theorem by H.von Koch and show the falsehood of the Riemann hypothesis.