The Prime Counting Function and the Zeros of ζ(s)

Introduction

The distribution of prime numbers is among the oldest and deepest problems in mathematics. While primes appear irregular at the surface, they are governed — with remarkable precision — by the analytic properties of the Riemann zeta function. This post develops the connection between the prime counting function $\pi(x)$ and the non-trivial zeros of $\zeta(s)$ , culminating in the explicit formula and what the Riemann Hypothesis asserts about the error term.

The Zeta Function and Its Analytic Continuation

For $\Re(s) > 1$ , the Riemann zeta function is defined by the absolutely convergent Dirichlet series:

\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}

Via the Euler product — a consequence of unique factorization in $\mathbb{Z}$ — this equals:

\zeta(s) = \prod_{p ,\text{prime}} \frac{1}{1 - p^{-s}}, \quad \Re(s) > 1

The function admits a meromorphic continuation to all of $\mathbb{C}$ , with a simple pole of residue 1 at $s = 1$ and no other singularities. The completed zeta function:

\xi(s) = \frac{1}{2} s(s-1)\pi^{-s/2} \Gamma!\left(\tfrac{s}{2}\right)\zeta(s)

is entire and satisfies the functional equation $\xi(s) = \xi(1-s)$ .

Trivial and Non-Trivial Zeros

The functional equation forces $\zeta(s)$ to vanish at $s = -2, -4, -6, \ldots$ — the trivial zeros, arising from poles of $\Gamma(s/2)$ . All remaining zeros, the non-trivial zeros, lie in the critical strip $0 < \Re(s) < 1$ .

By the functional equation, if $\rho = \sigma + it$ is a non-trivial zero, so are $\bar{\rho}$ , $1 - \rho$ , and $1 - \bar{\rho}$ . They are symmetric about both the real axis and the critical line $\Re(s) = \frac{1}{2}$ .

The Riemann Hypothesis asserts:

Every non-trivial zero $\rho$ of $\zeta(s)$ satisfies $\Re(\rho) = \frac{1}{2}$ .

That is, all non-trivial zeros lie on the line $s = \frac{1}{2} + it$ , $t \in \mathbb{R}$ .

End