The
Riemann hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in ...
is one of the most important
conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 19 ...
s in
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
. It is a statement about the zeros of the
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
. Various geometrical and arithmetical objects can be described by so-called global
''L''-functions, which are formally similar to the Riemann zeta-function. One can then ask the same question about the zeros of these ''L''-functions, yielding various generalizations of the Riemann hypothesis. Many mathematicians believe these generalizations of the Riemann hypothesis to be true. The only cases of these conjectures which have been proven occur in the
algebraic function field
In mathematics, an algebraic function field (often abbreviated as function field) of ''n'' variables over a field ''k'' is a finitely generated field extension ''K''/''k'' which has transcendence degree ''n'' over ''k''. Equivalently, an algebraic ...
case (not the number field case).
Global ''L''-functions can be associated to
elliptic curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
s,
number fields (in which case they are called
Dedekind zeta-functions),
Maass form
In mathematics, Maass forms or Maass wave forms are studied in the theory of automorphic forms. Maass forms are complex-valued smooth functions of the upper half plane, which transform in a similar way under the operation of a discrete subgroup \ ...
s, and
Dirichlet characters (in which case they are called
Dirichlet L-functions). When the Riemann hypothesis is formulated for Dedekind zeta-functions, it is known as the extended Riemann hypothesis (ERH) and when it is formulated for Dirichlet ''L''-functions, it is known as the generalized Riemann hypothesis or generalised Riemann hypothesis (see
spelling differences) (GRH). These two statements will be discussed in more detail below. (Many mathematicians use the label ''generalized Riemann hypothesis'' to cover the extension of the Riemann hypothesis to all global ''L''-functions,
not just the special case of Dirichlet ''L''-functions.)
Generalized Riemann hypothesis (GRH)
The generalized Riemann hypothesis (for Dirichlet ''L''-functions) was probably formulated for the first time by
Adolf Piltz
Adolf Piltz (8 December 1855 – 1940) was a German mathematician who contributed to number theory. Piltz was arguably the first to formulate a generalized Riemann hypothesis
The Riemann hypothesis is one of the most important conjectures in mat ...
in 1884. Like the original Riemann hypothesis, it has far reaching consequences about the distribution of
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s.
The formal statement of the hypothesis follows. A
Dirichlet character is a
completely multiplicative arithmetic function ''χ'' such that there exists a positive integer ''k'' with for all ''n'' and whenever . If such a character is given, we define the corresponding Dirichlet ''L''-function by
:
for every
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
''s'' such that . By
analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new ...
, this function can be extended to a
meromorphic function (only when
is primitive) defined on the whole complex plane. The generalized Riemann hypothesis asserts that, for every Dirichlet character ''χ'' and every complex number ''s'' with , if ''s'' is not a negative real number, then the real part of ''s'' is 1/2.
The case for all ''n'' yields the ordinary Riemann hypothesis.
Consequences of GRH
Dirichlet's theorem states that if ''a'' and ''d'' are
coprime natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
s, then the
arithmetic progression ''a'', , , , ... contains
infinitely many prime numbers. Let denote the number of prime numbers in this progression which are less than or equal to ''x''. If the generalized Riemann hypothesis is true, then for every coprime ''a'' and ''d'' and for every ,
:
where
is
Euler's totient function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ot ...
and
is the
Big O notation
Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Lan ...
. This is a considerable strengthening of the
prime number theorem
In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying ...
.
If GRH is true, then every proper subgroup of the multiplicative group
omits a number less than , as well as a number coprime to ''n'' less than . In other words,
is generated by a set of numbers less than . This is often used in proofs, and it has many consequences, for example (assuming GRH):
*The
Miller–Rabin primality test is guaranteed to run in polynomial time. (A polynomial-time primality test which does not require GRH, the
AKS primality test, was published in 2002.)
*The
Shanks–Tonelli algorithm is guaranteed to run in polynomial time.
*The Ivanyos–Karpinski–Saxena deterministic algorithm for factoring polynomials over finite fields with prime constant-smooth degrees is guaranteed to run in polynomial time.
If GRH is true, then for every prime ''p'' there exists a
primitive root mod ''p'' (a generator of the multiplicative group of integers modulo ''p'') that is less than
Goldbach's weak conjecture also follows from the generalized Riemann hypothesis. The yet to be verified proof of
Harald Helfgott
Harald Andrés Helfgott (born 25 November 1977) is a Peruvian mathematician working in number theory. Helfgott is a researcher ('' directeur de recherche'') at the CNRS at the Institut Mathématique de Jussieu, Paris.
Early life and education
...
of this conjecture verifies the GRH for several thousand small characters up to a certain imaginary part to obtain sufficient bounds that prove the conjecture for all integers above 10
29, integers below which have already been verified by calculation.
Assuming the truth of the GRH, the estimate of the character sum in the
Pólya–Vinogradov inequality can be improved to
, ''q'' being the modulus of the character.
Extended Riemann hypothesis (ERH)
Suppose ''K'' is a
number field (a finite-dimensional
field extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
of the
rationals ''Q'') with
ring of integers
In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often deno ...
O
''K'' (this ring is the
integral closure In commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over ''A'', a subring of ''B'', if there are ''n'' ≥ 1 and ''a'j'' in ''A'' such that
:b^n + a_ b^ + \cdots + a_1 b + a_0 = 0.
That is to say, ''b'' is ...
of the
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s ''Z'' in ''K''). If ''a'' is an
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
of O
''K'', other than the zero ideal, we denote its
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
by ''Na''. The
Dedekind zeta-function of ''K'' is then defined by
:
for every complex number ''s'' with real part > 1. The sum extends over all non-zero ideals ''a'' of O
''K''.
The Dedekind zeta-function satisfies a functional equation and can be extended by
analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new ...
to the whole complex plane. The resulting function encodes important information about the number field ''K''. The extended Riemann hypothesis asserts that for every number field ''K'' and every complex number ''s'' with ζ
''K''(''s'') = 0: if the real part of ''s'' is between 0 and 1, then it is in fact 1/2.
The ordinary Riemann hypothesis follows from the extended one if one takes the number field to be ''Q'', with ring of integers ''Z''.
The ERH implies an effective version
of the
Chebotarev density theorem
Chebotarev's density theorem in algebraic number theory describes statistically the splitting of primes in a given Galois extension ''K'' of the field \mathbb of rational numbers. Generally speaking, a prime integer will factor into several ideal ...
: if ''L''/''K'' is a finite Galois extension with Galois group ''G'', and ''C'' a union of conjugacy classes of ''G'', the number of
unramified primes of ''K'' of norm below ''x'' with Frobenius conjugacy class in ''C'' is
:
where the constant implied in the big-O notation is absolute, ''n'' is the degree of ''L'' over ''Q'', and Δ its discriminant.
See also
*
Artin's conjecture
*
Dirichlet L-function
*
Selberg class
In mathematics, the Selberg class is an axiomatic definition of a class of ''L''-functions. The members of the class are Dirichlet series which obey four axioms that seem to capture the essential properties satisfied by most functions that are co ...
*
Grand Riemann hypothesis In mathematics, the grand Riemann hypothesis is a generalisation of the Riemann hypothesis and generalized Riemann hypothesis. It states that the nontrivial zeros of all automorphic L-function, automorphic L-function, ''L''-functions lie on the crit ...
References
Further reading
*
{{L-functions-footer
Zeta and L-functions
Algebraic geometry
Conjectures
Unsolved problems in mathematics
Bernhard Riemann