Volume 11 Issue 4 ( December 2024)

Riemann zeta function is particularly one of the famous types of ‘L- function’. Since it can be represented as prime’s product, it is very useful to study about properties and behaviour of this to have a deep understanding about the distribution of primes This paper aims to enhance our understanding of prime distribution by establishing precise upper and lower bounds for the zeta function and identifying a zero-free region. These findings contribute to a deeper comprehension of bounded gaps between primes, marking significant progress toward resolving two of mathematics most famous unsolved problems: the Riemann Hypothesis and the Twin Prime Conjecture. Furthermore, by leveraging the complexities of quantum chaos, this work establishes connections between the Riemann zeta function and climate system dynamics, offering innovative models for precise forecasting and maintenance of climatic behavior, bridging the gap between quantum theory and climate science.

Keywords: Distribution of primes; L- function; Riemann hypothesis; Riemann zeta function; Twin prime conjecture; Zero free region