The Riemann Zeta Function and Prime Numbers

Abstract: This thesis is the result of a literature study regarding the relationship between the Riemann zeta function and prime numbers. We introduce the $\zeta$ function, discussing its properties. Then, starting from Riemann's original series, we derive the Euler product formula and functional equation for $\zeta$. We then discuss finite-order integral functions and Hadamard products, as well as the Riemann $\xi$ function in order to determine some properties of the nontrivial zeros of $\zeta$. Then, we derive an integral function of the second Chebyshev function $\psi$, and use residue calculus in order to represent it as a sum over the zeros of $\zeta$. Lastly, we rewrite $\psi$ in order to get an estimate of $\pi$, proving the prime number theorem with a certain error term.