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Number theory Millennium Prize Problem Status: Open Status last verified

Riemann Hypothesis

All non-trivial zeros of the Riemann zeta function have real part 1/2. If true, it pins down how regularly the prime numbers are distributed.

Problem statement

The Riemann zeta function, defined by ζ(s) = Σ n^(−s) for Re(s) > 1 and extended to the complex plane by analytic continuation, has all its non-trivial zeros on the line Re(s) = 1/2. The official statement is Bombieri's problem description.

ζ(s) = Σ_{n≥1} n^(−s)    (Re s > 1, continued analytically)

Conjecture: ζ(ρ) = 0 and ρ not in {−2, −4, −6, …}  ⇒  Re(ρ) = 1/2

Source: Enrico Bombieri — Problems of the Millennium: the Riemann Hypothesis (official problem description)

Introduction

The prime numbers look irregular one by one, but on a large scale they thin out in a very predictable way. Riemann (1859) connected that pattern to the zeros of a single function, the zeta function ζ(s).

The zeros at −2, −4, −6, … are called trivial. The Riemann Hypothesis says every other zero lies on the vertical line Re(s) = 1/2, the critical line. Where the zeros lie controls how far the count of primes can drift from its average.

It is one of the Clay Millennium Prize Problems.

Definitions

Non-trivial zero
A zero of ζ(s) in the critical strip 0 < Re(s) < 1. All non-trivial zeros lie in that strip.
Critical line
The line Re(s) = 1/2.
Prime-counting function π(x)
The number of primes ≤ x.

What is known

Each statement says how certain it is: an established (proved) result, an experimental observation, an open conjecture, or an unverified claim.

  • Established result The prime number theorem (Hadamard and de la Vallée Poussin, 1896): π(x) is asymptotic to x / log x. Its proof rests on ζ(s) having no zeros on the line Re(s) = 1. [Enrico Bombieri]
  • Established result Infinitely many zeros lie on the critical line (Hardy, 1914), and a positive proportion of them do. [Enrico Bombieri]
  • Experimental observation Platt and Trudgian (2021) verified rigorously, with interval arithmetic, that every zero with imaginary part between 0 and 3·10^12 lies on the critical line. [Dave Platt, Tim Trudgian]
  • Open conjecture Every non-trivial zero lies on the critical line (the hypothesis itself; unproven). [Enrico Bombieri]

Important limitations

  • Computing zeros up to any height, however rigorously, cannot prove the hypothesis: there are infinitely many zeros.
  • Floating-point computation of ζ(s) is not rigorous on its own; serious verification uses interval arithmetic and error bounds.
  • Many statements are known to be equivalent to the hypothesis; proving one of them is as hard as the original.

Status history

  1. Status: Open Named one of the seven Millennium Prize Problems by the Clay Mathematics Institute.
  2. Status: Open Checked against the Clay Mathematics Institute page: still unsolved.

This status is maintained by the Zerethon IDE team from authoritative sources. Contributions, runs and AI answers never change it.