FALSEHOOD OF THE RIEMANN HYPOTHESIS

[Last updated: July 18, 2026 (Version 4)] Regarding the prime number classification process based on the evaluation formulas of P. Dusart and Ch. Axler and the P. Dusart-type complementary formula (shown in Figures 1 and 2), my analysis revealed that the relative left-right positioning of the value -1+(loglog n-C)/log n (satisfying π(x)-li(x)=0) and the value -1+(loglog n-2)/log n was reversed; I have therefore corrected this. Additionally, while I previously proved in Chapter 5 that lim n→∞[n-li(n{log n+loglog n-1+(loglog n-2)/log n})]=0, I have now also verified the validity of this result through numerical calculations using PARI/GP. Details regarding this are provided in Chapters 5 and 6 (specifically pages 36–41), so please take a look. Thank you for your attention. 【ABSTRACT】This paper examines the asymptotic formula for prime numbers Pn = n{log n+loglog n +O(1)} and the error term O(1). By analyzing Cippola ’s asymptotic expansion of prime numbers, Rosser, Schoenfeld ’s prime number evaluation formula, and P. Dusart, Ch. Axler ’s prime number evaluation formula, I realized that it is necessary to treat the error term O(1) as a numerical function. 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.