Riemann Hypothesis on Extremely Abundant Number

The Riemann Hypothesis (RH) is renowned for its profound connection to the distribution of prime numbers and remains one of the central unsolved problems in mathematics. A deep understanding of prime distribution is essential for developing efficient algorithms and advancing number theory. The RH asserts that all non-trivial zeros of the Riemann zeta function are complex numbers with real part $\frac{1}{2}$, and it is widely regarded as the most important open problem in pure mathematics. Several equivalent formulations of the RH exist. Gr\"onwall's function $G$ is defined for all natural numbers $n > 1$ by \[ G(n) = \frac{\sigma(n)}{n \cdot \log \log n}, \] where $\sigma(n)$ denotes the sum of the divisors of $n$ and $\log$ is the natural logarithm. This paper leverages the properties of extremely abundant numbers, which are the left-to-right maxima of the function $n \mapsto G(n)$. In 2014, Nazardonyavi and Yakubovich established that the RH holds if and only if there are infinitely many extremely abundant numbers. Using this criterion, we introduce a novel approach that yields a complete proof of the RH.