FALSEHOOD OF THE RIEMANN HYPOTHESIS

[Updated: June 3, 2026 (Version 3)] I apologize for the repeated corrections. Regarding Figure 2, the classfied process for prime numbers in both the evaluation formulas of P. Dusart and Ch. Axler, and the P. Dusart- type complement formula, was unclear, so I have corrected it. Additionally, the limit calculations on pages 29 to 31 of section 5 have been explained in a more easily understandable way. Please replace it with version 3. Thank you very much for your understanding. 【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.