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- W4206541330 abstract "For every prime number $p_{n}$, we define the sequence $X_{n} = prod_{q leq p_{n}} frac{q}{q-1} - e^{gamma} times log theta(p_{n})$, where $theta(x)$ is the Chebyshev function and $gamma approx 0.57721$ is the Euler-Mascheroni constant. The Nicolas theorem states that the Riemann hypothesis is true if and only if the $X_{n} > 0$ holds for all prime $p_{n} > 2$. For every prime number $p_{k}$, $X_{k} > 0$ is called the Nicolas inequality. We show if the sequence $X_{n}$ is strictly decreasing for $n$ big enough, then the Riemann hypothesis must be true. For every prime number $p_{n} > 2$, we define the sequence $Y_{n} = frac{e^{frac{1}{2 times log(p_{n})}}}{(1 - frac{1}{log(p_{n})})}$ and show that $Y_{n}$ is strictly decreasing for $p_{n} > 2$. For all prime $p_{n} > 286$, we demonstrate that the inequality $X_{n} < e^{gamma} times log Y_{n}$ is always satisfied. We prove that $lim_{{nto infty }}X_{n}=lim_{{nto infty }}(log Y_{n})=0$." @default.
- W4206541330 created "2022-01-26" @default.
- W4206541330 creator A5039521075 @default.
- W4206541330 date "2021-07-29" @default.
- W4206541330 modified "2023-09-25" @default.
- W4206541330 title "The Nicolas criterion for the Riemann Hypothesis" @default.
- W4206541330 doi "https://doi.org/10.33774/coe-2021-wfzw6-v3" @default.
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