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Riemann hypothesis

Conjecture that all nontrivial zeros of the Riemann zeta function have real part 1/2; a central open problem in number theory with deep consequences for the distribution of prime numbers.

Author: Leandro Alegsa Creation date: November 13, 2022 Last updated: June 10, 2026

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The Riemann hypothesis is a conjecture in analytic number theory that concerns the locations of the complex zeros of the Riemann zeta function. It is widely regarded as one of the most important unsolved problems in pure mathematics: the question is simple to state but has resisted proof for more than a century. The problem is usually presented as a yes/no question about whether every nontrivial zero of the zeta function lies on a specific vertical line in the complex plane. Many survey accounts describe the conjecture as a touchstone for progress in understanding prime numbers and related structures in number theory. For a formal statement see this description.

Statement and definitions

The object at the center of the conjecture is the Riemann zeta function, an analytic function of a complex variable often written ζ(s). The function has a meromorphic continuation to the whole complex plane and satisfies a functional equation relating ζ(s) to ζ(1−s); for background see Riemann zeta function. Zeros of ζ(s) fall into two classes: "trivial" zeros at certain negative even integers, and the nontrivial zeros whose real parts lie between 0 and 1. The Riemann hypothesis asserts that every nontrivial zero has real part equal to 1/2. This vertical line Re(s)=1/2 is called the critical line; a proof of the conjecture would amount to establishing that all nontrivial zeros lie on that line. The search for a rigorous proof occupies researchers across several mathematical disciplines.

History and development

The problem is named after Bernhard Riemann, who in 1859 published an influential paper connecting the zeta function to the distribution of primes; see Bernhard Riemann. Riemann observed properties of ζ(s) and wrote the basic conjecture about its zeros, but gave no proof. Subsequent work established many related facts: analytic continuation, the functional equation, and links to the prime-counting function. Hardy proved that infinitely many zeros lie on the critical line, and later results pushed that further, but a complete proof remains elusive. Computational verification has checked that many of the low-lying zeros do lie on the critical line, and extensive numerical work supports the conjecture though it does not substitute for a proof.

Why it matters

The Riemann hypothesis has broad implications for how primes are distributed. Precise error estimates in the prime number theorem and in counting primes in short intervals would follow from a proof. Equivalently, several statements in number theory can be shown to be true if the hypothesis holds; conversely, a counterexample would have wide-reaching consequences. Because of these connections, mathematicians working in analytic number theory, algebraic number theory and related areas pay close attention to partial results and to methods that might lead toward a resolution. For the community’s perspective, see material aimed at mathematicians.

There are many statements known to be equivalent to the Riemann hypothesis, including certain bounds on error terms in counting functions and criteria such as Li's criterion.
The Generalized Riemann Hypothesis (GRH) extends the assertion to L-functions associated with number fields and characters; GRH has further applications in algebraic number theory and computational problems.
Connections to random matrix theory and the statistical distribution of zeros suggest deep links to physical models and to conjectures about correlations between zeros.

Current status and significance

Despite intense effort, the Riemann hypothesis remains open. Numerical checks have confirmed that many nontrivial zeros lie on the critical line, and partial theorems establish weaker but important constraints. Because of its foundational role, the Clay Mathematics Institute included the conjecture among its Millennium Prize Problems, offering a substantial award for a correct proof. Proof or disproof would reshape aspects of analytic number theory and our understanding of prime numbers. For accessible introductions and further reading consult general expositions and encyclopedic treatments such as those aimed at students and researchers: see a basic entry on the conjecture (Riemann hypothesis), background on pure mathematics (pure mathematics), and overviews of the zeta function (ζ(s)).

Because of its mix of elementary statement and deep consequences, the Riemann hypothesis continues to inspire new techniques and cross-disciplinary links. Whether its resolution will come from refinements of classical analysis, from algebraic or geometric methods, or from unexpected connections to other fields remains one of the great open questions in modern mathematics.

Introduction

Prime numbers

→ Main article: Prime number

At the centre of number theory, that branch of mathematics which deals with the properties of the natural numbers 1, 2, 3, 4 ... , are the prime numbers 2, 3, 5, 7, 11 ... . These are distinguished by the property of having exactly two divisors, namely 1 and themselves. The 1 is not a prime number. Already Euclid could show that there are infinitely many prime numbers, which is why the list 2, 3, 5, 7, 11 ... will never end. His result is called Euclid's theorem.

The prime numbers are, in a sense, the atoms of the integers, since every positive integer can be uniquely decomposed into such multiplicatively. For example, 21 = 3 - 7 and 110 = 2 - 5 - 11. Despite this elementary property, after several millennia of mathematical history, there is still no known simple pattern to which the prime numbers conform in their sequence. Their nature is one of the most important open questions in mathematics.

The prime number theorem

→ Main article: Prime number set

Even if the detailed understanding of the sequence 2, 3, 5, 7, 11 ... of prime numbers is considered to be unattainably distant, one can look for patterns if one broadens one's view. This is helped, for example, by the idea that statistical methods can often be used to describe the behaviour of very large numbers of people (for example, in terms of consumption and voting behaviour) with surprising precision, even though a single person is extremely complex. Roughly speaking, this has to do with the fact that larger and larger relevant data sets provide increasingly reliable information. In the case of prime numbers, such an expansion leads, among other things, to the question of how many primes there are below a fixed chosen number.

For example, only 4 prime numbers, namely 2, 3, 5 and 7, are smaller than the number 10. In the case of 50, there are already 15 smaller prime numbers, namely

$2,3,5,7,11,13,17,19,23,29,31,37,41,43,47.$

One question in number theory is whether there is a universal and simple principle for at least estimating how many prime numbers there are under a given bound. Such a principle was first recognized in 1792/93 by the then 15-year-old Carl Friedrich Gauss, after he had studied logarithm tables. Gauss conjectured that the number of all primes from 2 to a large number x is approximately equal to the area between the x-axis and the function $t\mapsto {\tfrac {1}{\log(t)}}$ in the interval from 2 to x. Here $\log(t)$ the natural logarithm. Thus the approximation

Number of primes to x $\approx \int _{2}^{x}{\frac {1}{\log(t)}}\mathrm {d} t.$

The integral to the right cannot be calculated elementary closed, because the reciprocal logarithm has no elementary root function. It thus defines an "independent" function, also known as the integral logarithm. Gauss did not present a mathematical proof of his conjecture, and it was over 100 years before such a proof was provided - independently by Jacques Hadamard and Charles-Jean de La Vallée Poussin - in 1895. Here, proof does not mean that all conceivable values have been tried through, which is impossible with infinitely many numbers, but that a logical argument based on the mathematical axioms proves the facts in full generality. The theorem thus demonstrated is still called the prime number theorem.

The approximation given in the prime number theorem provides quite good values. For example, below the number 73 893 there are exactly 7293 prime numbers, and the following applies

$\int _{2}^{73\,893}{\frac {1}{\log(t)}}\mathrm {d} t\approx 7331{,}35.$

The prime number theorem captures the average behavior of prime number distances. One interpretation of its statement is that a random number between 2 and a very large n with approximate probability ${\tfrac {1}{\log(n)}}$ is a prime number. However, it does not give arbitrarily detailed information about the prime sequence.

Riemann's ideas

Original work from 1859

In 1859, in gratitude for his admission to the Berlin Academy of Sciences, Bernhard Riemann wrote a 9-page paper that was later to lay the foundations for modern analytic number theory. His work aimed at proving and further deepening Gauss's conjecture on the prime number theorem. However, since the essay was extremely sketchy and many of the statements made in it were not rigorously proved, it was to be some time before mathematicians accepted the assertions made there. To this day, all of Riemann's statements in his paper, with the exception of the Riemann conjecture formulated there in a subordinate clause, are considered proven.

The Riemann zeta function

→ Main article: Riemann zeta function

One possible tool for proving this formula is the Riemannzeta function. Here it is exploited that it expresses the law of unique prime factorization in the language of analysis. So the properties of the prime numbers are stored hidden in this function. The decisive characteristics, which allow conclusions about the prime numbers, are the zeros of the zeta function, i.e. all points, at which it takes the value 0. These generate a correction term of the above formula, which converts it into an exact expression. So this resulting exact formula knows the distribution of the prime numbers to the last detail. However, this does not mean that the questions about prime numbers have been solved: the computational effort increases very strongly with increasing values and thus practical calculations with this formula are not effective. In contrast, modern prime number tests are more suitable for numerical research. However, the exact formula is of theoretical interest: namely, it conceals the error gap between the simple prediction and the actual prime number distribution. It is assumed that this error is as small as possible (within the spectrum of all possibilities). Within the exact formula, which is to output the number of primes under the number , terms with $x^{\rho }$ summed up, where ρ $\rho$ the zeros. Now, if the real part of ρ becomes $\rho$ larger, this also increases the size of $x^{\rho },$ which would have the consequence that the distance between the estimate of the prime set and the actual distribution would also be larger. It can be shown that the real part of infinitely many values of ρ equal to $\rho$ , ${\tfrac {1}{2}}$ so the error will in any case have a minimum magnitude of . $x^{{\frac {1}{2}}+\varepsilon }$ However, the Riemann conjecture now states that there are no other zeros that behave differently from those previously known in the critical strip.

Deciphering the error is not relevant to numerics. Rather, pure mathematics strives to learn the hitherto hidden reason why the error (if true) turns out to be as small as possible. Mathematicians hope for a fundamental insight into the nature of numbers behind the formal justification of this regularity.

The Riemann zeta function

→ Main article: Riemann zeta function

The Riemann zeta function is a complex-valued function which, for a complex number with a real part $\operatorname {Re} (s)>1$ by the infinite sum

$\zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}=1+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+{\frac {1}{4^{s}}}+\dotsb$

is defined.

One of the most important properties of the Riemann zeta function is its connection with prime numbers. It establishes a relationship between complex analysis and number theory (see analytic number theory) and forms the starting point of the Riemann conjecture. The following expression, which goes back to Leonhard Euler (1748), represents the connection formulaically as

$\zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}=\prod _{p\ {\text{prim}}}{\frac {1}{1-{\frac {1}{p^{s}}}}}={\frac {1}{\left(1-{\frac {1}{2^{s}}}\right)\left(1-{\frac {1}{3^{s}}}\right)\left(1-{\frac {1}{5^{s}}}\right)\dotsm }},$

where $\Pi _{p}$ is an infinite product over all primes . The expression follows immediately from the theorem on the uniqueness of prime decomposition and the summation formula for the geometric series.

The function can be uniquely analytically continued beyond the original convergence region of Euler's sum or product formula to the entire complex plane - except for . s=1 One obtains a meromorphic function

$\zeta (s)={\frac {1}{\Gamma (s)}}\left({\frac {1}{s-1}}-{\frac {1}{2s}}+\sum \limits _{n=2}^{\infty }{\frac {B_{n}}{n!}}{\frac {1}{s+n-1}}+\int \limits _{1}^{\infty }{\frac {x^{s-1}}{\mathrm {e} ^{x}-1}}\,\mathrm {d} x\right),$

where Γ $\Gamma$ is the gamma function and $B_{n}$ are the Bernoulli numbers. At the point s=1 it has a simple pole. The other singularities of this representation are all liftable because the whole function has a simple zero $1/\Gamma$ at each of these points . $0,-1,-2,-3,\dotsc$

Questions and Answers

Q: What is the Riemann hypothesis?

A: The Riemann hypothesis is a mathematical question (conjecture) that asks a question about a special thing called the Riemann zeta function.

Q: What type of mathematics does the Riemann hypothesis relate to?

A: The Riemann hypothesis relates to pure mathematics, which is a type of mathematics that is about thinking about mathematics, rather than trying to put it into the real world.

Q: Who was Bernhard Riemann?

A: Bernhard Riemann was a man who lived in the 1800s and whose name has been given to this conjecture.

Q: What would be the outcome if someone could prove the Riemann hypothesis?

A: If someone could prove the Riemann hypothesis, mathematicians would be able to know more about prime numbers and how to find them.

Q: How much money has been offered for proof of this conjecture?

A: The Clay Mathematics Institute has offered $1,000,000 for proof of this conjecture.

Q: Is there only one answer for this conjecture?

A: Yes, there are only two possible answers for this conjecture - "yes" or "no".

Riemann hypothesis

Statement and definitions

History and development

Why it matters

Equivalent formulations and related conjectures

Current status and significance

Introduction

Prime numbers

The prime number theorem

Riemann's ideas

Original work from 1859

The Riemann zeta function

The Riemann zeta function

Questions and Answers

Q: What is the Riemann hypothesis?

Q: What type of mathematics does the Riemann hypothesis relate to?

Q: Who was Bernhard Riemann?

Q: What would be the outcome if someone could prove the Riemann hypothesis?

Q: How much money has been offered for proof of this conjecture?

Q: Is there only one answer for this conjecture?