---
tag: CXES7915R7
kind: statement
role: theorem
name: Non-vanishing of the zeta function on the line Re s = 1
msc: [11M06, 11N05]
uses: [1XWBY4N0VT, 0C3PN4075R]
about: [1XWBY4N0VT]
sources:
  - url: https://en.wikipedia.org/wiki/Prime_number_theorem
---

## Statement

Let $\zeta$ be the [[Riemann zeta function|1XWBY4N0VT]].
For every complex number $s \ne 1$ with $\operatorname{Re} s \ge 1$, one has $\zeta(s) \ne 0$.
In particular $\zeta(1 + it) \ne 0$ for every real $t \ne 0$.

## Remarks

For $\operatorname{Re} s > 1$ this follows from the [[Euler product formula|0C3PN4075R]], since a convergent product of nonzero factors is nonzero.
For the line $\operatorname{Re} s = 1$, the classical argument uses the inequality $3 + 4\cos\theta + \cos 2\theta = 2(1 + \cos\theta)^2 \ge 0$.
With the Euler product it gives $\abs{\zeta(\sigma)^3 \, \zeta(\sigma + it)^4 \, \zeta(\sigma + 2it)} \ge 1$ for $\sigma > 1$.
If $\zeta(1 + it) = 0$ with $t \ne 0$, then as $\sigma \to 1^+$ the left side behaves like $(\sigma - 1)^{-3} (\sigma - 1)^{4} \to 0$, because $\zeta$ has a simple pole at $1$ and $\zeta(1 + 2it)$ is finite; this is a contradiction.

## Explanations

### L1 human

The zeros of the zeta function control how regularly the primes are spread.
This theorem says that no zero lies on the line $\operatorname{Re} s = 1$, the edge of the region where the series converges.
It is the key analytic input in the proof of the prime number theorem.

### L2 human

The [[prime number theorem|0EHBY5Y84N]] $\pi(x) \sim x / \log x$ follows from this non-vanishing by a Tauberian theorem, such as the Wiener–Ikehara theorem, applied to $-\zeta'(s)/\zeta(s) = \sum_n \Lambda(n) n^{-s}$.
Hadamard and de la Vallée Poussin proved the prime number theorem this way in 1896.
Quantitative zero-free regions, such as $\sigma > 1 - c/\log(\abs{t} + 2)$, give the prime number theorem with an error term.
The same method shows $L(1 + it, \chi) \ne 0$ for [[Dirichlet L-functions|9D2Z543EJQ]].