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Riemann Hypothesis — Definition, Formula & Examples

The Riemann Hypothesis is an unproven conjecture stating that every non-trivial zero of the Riemann zeta function has a real part equal to 12\frac{1}{2}. It is one of the most famous open problems in mathematics and carries a $1 million Millennium Prize.

Let ζ(s)=∑n=1∞n−s\zeta(s) = \sum_{n=1}^{\infty} n^{-s} for Re(s)>1\text{Re}(s) > 1, extended to a meromorphic function on C\mathbb{C}. The zeta function has "trivial" zeros at s=−2,−4,−6,…s = -2, -4, -6, \ldots The Riemann Hypothesis asserts that all remaining (non-trivial) zeros satisfy Re(s)=12\text{Re}(s) = \tfrac{1}{2}, meaning they lie on the critical line in the complex plane.

Key Formula

ζ(s)=∑n=1∞1ns,Re(s)>1\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}, \quad \text{Re}(s) > 1
Where:
  • ss = A complex number $s = \sigma + it$ where $\sigma$ and $t$ are real
  • nn = Positive integer index of summation
  • ζ(s)\zeta(s) = The Riemann zeta function, extended by analytic continuation to all $s \neq 1$

How It Works

The Riemann zeta function encodes deep information about the distribution of prime numbers. Euler showed that ζ(s)=∏p prime11−p−s\zeta(s) = \prod_{p\,\text{prime}} \frac{1}{1 - p^{-s}} for Re(s)>1\text{Re}(s) > 1, linking the zeta function directly to primes. The location of the non-trivial zeros controls how regularly primes are distributed among the integers. If the Riemann Hypothesis is true, the prime counting function π(x)\pi(x) deviates from its approximation Li(x)=∫2xdtln⁡t\text{Li}(x) = \int_2^x \frac{dt}{\ln t} by at most O(xln⁡x)O(\sqrt{x}\ln x), the best possible error bound. Billions of non-trivial zeros have been computed and all lie on the critical line, but no general proof exists.

Example

Problem: Verify that ζ(2)=π26\zeta(2) = \frac{\pi^2}{6} by computing the first few partial sums and observing convergence.
Step 1: Write out the series at s=2s = 2.
ζ(2)=112+122+132+142+⋯\zeta(2) = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots
Step 2: Compute partial sums to see convergence.
S4=1+0.25+0.1111+0.0625=1.4236S_4 = 1 + 0.25 + 0.1111 + 0.0625 = 1.4236
Step 3: Compare with the exact value (the Basel problem, solved by Euler).
π26≈1.6449\frac{\pi^2}{6} \approx 1.6449
Answer: The partial sums converge toward π26≈1.6449\frac{\pi^2}{6} \approx 1.6449. This is a known value of ζ(s)\zeta(s) on the real line — the Riemann Hypothesis concerns its zeros in the complex plane.

Why It Matters

A proof (or disproof) would transform analytic number theory, settling the tightest possible bounds on prime gaps and the error in the Prime Number Theorem. Cryptographic systems like RSA rely on the difficulty of factoring large numbers, and our understanding of prime distribution underpins the security assumptions behind them. The hypothesis also connects to random matrix theory in physics and appears in courses on complex analysis and algebraic number theory.

Common Mistakes

Mistake: Confusing the trivial zeros (s=−2,−4,−6,…s = -2, -4, -6, \ldots) with the non-trivial zeros the hypothesis addresses.
Correction: The trivial zeros are well-understood and lie on the negative real axis. The Riemann Hypothesis concerns only the non-trivial zeros, which all have 0<Re(s)<10 < \text{Re}(s) < 1, and conjectures they satisfy Re(s)=12\text{Re}(s) = \frac{1}{2}.