Zeros and the Riemann Hypothesis

The location of zeros of the zeta function controls the distribution of prime numbers. Understanding these zeros is one of mathematics' deepest mysteries.

DefinitionTrivial and Non-Trivial Zeros

The zeros of $\zeta(s)$ divide into two types:

Trivial zeros: $s = -2, -4, -6, \ldots$ (arising from the functional equation)
Non-trivial zeros: Located in the critical strip $0 < \Re(s) < 1$

All non-trivial zeros are symmetric about $\Re(s) = 1/2$ (by the functional equation) and about the real axis (by reality of $\zeta$ on the real line).

TheoremRiemann Hypothesis (RH)

Conjecture: All non-trivial zeros of $\zeta(s)$ lie on the critical line $\Re(s) = 1/2$ .

Equivalently: if $\zeta(\rho) = 0$ and $0 < \Re(\rho) < 1$ , then $\rho = 1/2 + it$ for some $t \in \mathbb{R}$ .

ExampleConsequences of RH

If the Riemann Hypothesis is true:

$\pi(x) = \text{Li}(x) + O(\sqrt{x} \log x)$ (optimal error in Prime Number Theorem)
$\psi(x) - x = O(\sqrt{x} \log^2 x)$ (Chebyshev function)
Many explicit inequalities on prime gaps and prime counting functions
Improved bounds in the Lindelöf hypothesis: $\zeta(1/2 + it) = O(t^{\epsilon})$

DefinitionZero Counting Functions

Let $N(T)$ denote the number of zeros $\rho = \beta + i\gamma$ with $0 < \gamma \leq T$ . The Riemann-von Mangoldt formula gives:

N(T) = \frac{T}{2\pi} \log\frac{T}{2\pi e} + O(\log T)

This shows there are infinitely many non-trivial zeros, distributed roughly with density $\frac{1}{2\pi} \log T$ per unit length.

Remark

As of 2024, billions of zeros have been computed numerically, and all satisfy RH. The first 10 trillion non-trivial zeros lie on the critical line. However, no proof exists that even one zero must lie on the critical line!

ExampleExplicit Formula

The zeros $\rho$ appear in the explicit formula for the prime counting function:

\psi(x) = x - \sum_{\rho} \frac{x^{\rho}}{\rho} - \log(2\pi) - \frac{1}{2}\log(1-x^{-2})

Each zero contributes an oscillatory term $x^{\rho}/\rho$ . If $\Re(\rho) = 1/2$ , these terms are $O(\sqrt{x})$ ; if any zero has $\Re(\rho) > 1/2$ , the error grows faster.

The Riemann Hypothesis remains one of the seven Millennium Prize Problems. Its resolution would revolutionize our understanding of prime distribution and have profound implications across mathematics and cryptography.