As we know that RH is the most popular unsolved problem in all of maths. If this hypothesis is true, then will it give us the power to predict the exact number of primes less then a given number? And how it will effect the e-commerce system?
2025-01-13 09:04:36.1736759076
Will Riemann Hypothesis, if true, give us the exact value of number of primes less than a given number?
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The Prime Number Theorem states that $\pi(x)$, the number of primes up to a number $x$, is $$\pi\left(x\right)=\textrm{Li}\left(x\right)+O\left(x\exp\left(-C\sqrt{\log\left(x\right)}\right)\right)$$ where $C>0$ and $\textrm{Li}\left(x\right)$ is the logarithmic integral of $x$. Now if the Riemann Hypothesis is true, then you can take $$\pi\left(x\right)=\textrm{Li}\left(x\right)+O\left(\sqrt{x}\log\left(x\right)\right)$$ and this is, essentially, the best possible. So you haven't the exact number but a better approximation.