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- W4205653604 abstract "Under the assumption that the Riemann hypothesis is true, von Koch deduced the improved asymptotic formula $theta(x) = x + O(sqrt{x} times log^{2} x)$, where $theta(x)$ is the Chebyshev function. A precise version of this was given by Schoenfeld: He found under the assumption that the Riemann hypothesis is true that $theta(x) < x + frac{1}{8 times pi} times sqrt{x} times log^{2} x$ for every $x geq 599$. On the contrary, we prove if there exists some real number $x geq 10^{8}$ such that $theta(x) > x + frac{1}{log log log x} times sqrt{x} times log^{2} x$, then the Riemann hypothesis should be false. In this way, we show that under the assumption that the Riemann hypothesis is true, then $theta(x) < x + frac{1}{log log log x} times sqrt{x} times log^{2} x$ for all $x geq 10^{8}$." @default.
- W4205653604 created "2022-01-25" @default.
- W4205653604 creator A5039521075 @default.
- W4205653604 date "2022-01-12" @default.
- W4205653604 modified "2023-09-28" @default.
- W4205653604 title "Possible Counterexample of the Riemann Hypothesis" @default.
- W4205653604 doi "https://doi.org/10.33774/coe-2022-kxpg8-v3" @default.
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