Prime Numbers and L-Functions

This paper examines the intrinsic relationship between complex analysis and the distribution of prime numbers through the analytical framework of L-functions, with particular emphasis on the Riemann zeta function. Beginning from the Euler product representation, the study explores how L-functions encode prime number properties within the complex plane and how analytic continuation extends their domain to reveal deeper structural symmetries. Central to this investigation is the Riemann Hypothesis, which asserts that all non-trivial zeros of the zeta function lie on the critical line Re(s) = ½. By deriving the logarithmic derivative of the zeta function and employing the Chebyshev explicit formula, the analysis elucidates how non-trivial zeros govern prime density and distribution. Numerical evaluation using the first ten non-trivial zeros demonstrates an enhanced approximation of the prime-counting function ψ(x) compared to simpler asymptotic estimates, thereby illustrating the increasing precision obtained with higher-order zero inclusion. The results affirm that the zeta function serves as a unifying construct linking prime distribution with complex analytic behavior, reinforcing the Riemann Hypothesis as a cornerstone conjecture in analytic number theory with profound implications for both theoretical mathematics and computational applications.