Can anyone, who has knowledge of the following, share some more details about it because not much information is available publicly regarding the same:
RH is equivalent to the assertion that for all $n\ge3$ $$| \log \operatorname{lcm}(1,2,\dots, n) - n | < \sqrt{n}\log^2(n)$$ where $\operatorname{lcm}$ denotes the least common multiple.
Moreover, I also want to confirm if this equivalent problem is correct and if yes, where we can find the proof of this equivalence?
The only article we can find publicly is the "Prime Number Races" written by Granville and Martin who mention this equivalence on Page Number 9 of their article as:
Riemann’s prediction has been made more precise over the years, and it can now be expressed very explicitly as:
$$| \ln \operatorname{lcm}(1,2,\dots, x) - x | ≤ 2\sqrt{x}\ln^2(x)$$ when x ≥ 100.
In fact, this inequality is equivalent to the celebrated Riemann Hypothesis, perhaps the most prominent open problem in mathematics.
Sources: [1]: http://oeis.org/A003418 [2]: https://en.wikipedia.org/wiki/Chebyshev_function [3]: https://dms.umontreal.ca/~andrew/PDF/PrimeRace.pdf [4]: https://mathoverflow.net/q/232413
I agree with you that it would've been nice if Granville and that other folk included a reference :)
As you know, $\ln\mathop{\rm lcm}(1,2,\dots,x)$ is the same as the Chebyshev function $\psi(x)$. The Riemann hypothesis Wikipedia page contains the assertion that RH implies $$ |\psi(x)-x| < \frac1{8\pi} \sqrt x \ln^2 x \quad\text{for all } x\ge 73.2 $$ (which is stronger than the inequality in the OP) and gives this attribution: L. Schoenfeld, Sharper bounds for the Chebyshev functions θ(x) and ψ(x). II, Mathematics of Computation 30 (1976), 337–360.
For the reverse implication, it is known that any inequality of the form $|\psi(x)-x| < Cx^\theta$ for $x$ sufficiently large implies that $\zeta(s)$ has no zeros with real part greater than $\theta$. (This is a consequence of "Landau's theorem", and a version can be found, for example, as Theorem 15.2 in Montgomery and Vaughan's Multiplicative Number Theory I. Classical Theory.) In particular, $|\psi(x)-x| < \frac1{8\pi} \sqrt x \ln^2 x$ implies $|\psi(x)-x| < Cx^{1/2+\varepsilon}$ for any $\varepsilon>0$ when $x$ is sufficiently large, and thus $\zeta(s)$ has no zeros with real part exceeding $\frac12+\varepsilon$ for every $\varepsilon>0$, which implies the Riemann hypothesis.