The Riemann Zeta Function

The Riemann zeta function is defined, for complex arguments $s$ with real part greater than 1, by an elegant infinite sum:

$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} = 1 + \frac{1}{2^s} + \frac{1}{3^s} + \frac{1}{4^s} + \cdots$

It looks modest. It is the most important object in analytic number theory and the home of the most famous unsolved problem in mathematics.

Why it connects to the primes

Leonhard Euler’s key insight, around 1737, was that the zeta function can also be written as an infinite product over the primes:

$\zeta(s) = \prod_{p \text{ prime}} \frac{1}{1 - p^{-s}}$

The proof uses only the fact that every positive integer has a unique factorization into primes. This Euler product is remarkable: it says the analytic behavior of $\zeta$ encodes information about the primes themselves.

Setting $s = 1$ gives the harmonic series $1 + 1/2 + 1/3 + \cdots$, which diverges. The right-hand product also diverges — which Euler took as a new proof that there are infinitely many primes.

Riemann’s breakthrough

In 1859, Bernhard Riemann published an eight-page paper, “On the Number of Prime Numbers less than a Given Magnitude.” In it he made a bold move: extend the definition of $\zeta(s)$ from real to complex $s$, using the technique of analytic continuation.

The extended zeta function is defined (almost) everywhere in the complex plane and has:

• Trivial zeros at $s = -2, -4, -6, \ldots$ (the negative even integers)
• Non-trivial zeros, which are the interesting ones. They all lie in the strip $0 < \text{Re}(s) < 1$

Riemann conjectured that every non-trivial zero has real part exactly $\tfrac{1}{2}$ — that is, they all lie on the “critical line” $\text{Re}(s) = 1/2$. This is the Riemann Hypothesis.

Why the hypothesis matters

The zeros of the zeta function are intimately linked to the distribution of primes. Riemann showed that the prime counting function $\pi(x)$ — the number of primes up to $x$ — can be written as an explicit formula involving these zeros. The Prime Number Theorem (proved in 1896 by Hadamard and de la Vallée Poussin) is equivalent to the weaker statement that $\zeta(s)$ has no zeros on the line $\text{Re}(s) = 1$.

If the Riemann Hypothesis is true, the error term in the Prime Number Theorem is as small as possible. Hundreds of other theorems in number theory are proven “conditionally on the Riemann Hypothesis” — meaning: assuming RH, we can show this. A proof would cascade through the whole field.

Today

As of this writing (2025), the first $10^{13}$ zeros have all been computed and verified to lie on the critical line. No counterexample has ever been found. The Riemann Hypothesis remains one of the seven Millennium Prize Problems, with a $1 million reward offered by the Clay Mathematics Institute for a proof (or disproof).

Deep connections have been discovered between the zeta function and:

• Random matrix theory — the spacings between zeta zeros match eigenvalue spacings of random Hermitian matrices
• Quantum chaos — the zeta function has been conjectured to encode the spectrum of a quantum system whose classical limit is chaotic
• Other L-functions — the zeta function is the prototype of a vast family of “L-functions,” each with its own Riemann Hypothesis

Hearing the primes through the zeta function has become a central theme of 21st-century mathematics. Whether a proof will come next year or next century, no one knows.

Frequently asked

Who first studied the zeta function?

Leonhard Euler studied the real-valued version in the 18th century, discovering its connection to the primes. Bernhard Riemann, in an 1859 paper, extended it to complex arguments and formulated the Riemann Hypothesis.

What is the Riemann Hypothesis?

The conjecture that every non-trivial zero of the zeta function has real part exactly 1/2. It is the most famous unsolved problem in mathematics and one of the seven Millennium Prize Problems, with a $1 million reward.

Why does the zeta function connect to primes?

Euler showed that ζ(s) = ∏ (1 − p⁻ˢ)⁻¹, where the product runs over all primes. This Euler product identity turns facts about the zeta function into facts about primes.

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Person

Bernhard Riemann

German mathematician of extraordinary depth. In a brief life he transformed analysis, number theory, and differential geometry, and posed the hypothesis that remains mathematics' most famous open problem.

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Leonhard Euler

Swiss mathematician of extraordinary range and productivity. Formulated Euler's identity, founded graph theory, and produced more mathematical output than any mathematician before or since.

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Riemann Hypothesis

The most famous unsolved problem in mathematics. A proof would unlock dozens of other theorems in number theory.

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