FALSEHOOD OF THE RIEMAN HYPHOTHESIS

Note on this version: This paper is a revised and updated version of my paper published on ResearchGate in December 2025 (DOI: [10.13140/RG.2.2.18216.02565]). This version includes important corrections regarding the Riemann Hypothesis.

 My previous paper published on ResearchGate contained a significant error, and for a long time I mistakenly believed that limn→∞logPn/logn=1 was incorrect.I have since proven that this limit is actually correct (similarly for Pn ~ n log n). I deeply apologize for my oversight. I sincerely apologize for any inconvenience caused to my readers.

 In this revised version, I accept that limn→∞logPn/logn=1 (also for Pn ~ n log n) holds, and after carefully examining the distribution of prime numbers, I have concluded that the Riemann Hypothesis is false. This is a very meticulously analyzed paper, and I am confident that its content is compelling. 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: Rosser, Schoenfeld’s formula and P. Dusart, Ch.Axler’s formula. Both formulas cannot be used simultaneously; prime numbers must be contained within one of these formulas. This is what has 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.