Generalized Riemann hypothesis
Encyclopedia
The Riemann hypothesis
is one of the most important conjecture
s in mathematics
. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global L-function
s, 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 function field case (not the number field case).
Global L-functions can be associated to elliptic curve
s, number fields (in which case they are called Dedekind zeta-functions), Maass forms, and Dirichlet character
s (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 (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.)
Like the original Riemann hypothesis, it has far reaching consequences about the distribution of prime number
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 χ(n + k) = χ(n) for all n and χ(n) = 0 whenever gcd(n, k) > 1. If such a character is given, we define the corresponding Dirichlet L-function by
for every complex number
s with real part > 1. By analytic continuation
, this function can be extended to a meromorphic function
defined on the whole complex plane. The generalized Riemann hypothesis asserts that for every Dirichlet character χ and every complex number s with L(χ,s) = 0: if the real part of s is between 0 and 1, then it is actually 1/2.
The case χ(n) = 1 for all n yields the ordinary Riemann hypothesis.
states that if a and d are coprime
natural number
s, then the arithmetic progression
a, a+d, a+2d, a+3d, … contains infinitely many prime numbers. Let π(x,a,d) 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 ε > 0
where φ(d) is Euler's totient function
and O is the Big O notation
. This is a considerable strengthening of the prime number theorem
.
If GRH is true, then every proper subgroup of the multiplicative group
omits a number 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):
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) which is less than
Goldbach's weak conjecture
also follows from the generalized Riemann hypothesis.
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.
of the rationals
Q) with ring of integers
OK (this ring is the integral closure of the integer
s Z in K). If a is an ideal of OK, other than the zero ideal we denote its norm 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 OK.
The Dedekind zeta-function satisfies a functional equation and can be extended by analytic continuation
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 ζK(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.
Riemann hypothesis
In mathematics, the Riemann hypothesis, proposed by , is a conjecture about the location of the zeros of the Riemann zeta function which states that all non-trivial zeros have real part 1/2...
is one of the most important conjecture
Conjecture
A conjecture is a proposition that is unproven but is thought to be true and has not been disproven. Karl Popper pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is contrasted by hypothesis , which is a testable statement based on accepted grounds...
s in mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global L-function
L-function
The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary analytic number theory. In it, broad generalisations of the Riemann zeta function and the L-series for a Dirichlet character are constructed, and their general properties, in most cases...
s, 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 function field case (not the number field case).
Global L-functions can be associated to elliptic curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
s, number fields (in which case they are called Dedekind zeta-functions), Maass forms, and Dirichlet character
Dirichlet character
In number theory, Dirichlet characters are certain arithmetic functions which arise from completely multiplicative characters on the units of \mathbb Z / k \mathbb Z...
s (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 (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 Piltz in 1884.Like the original Riemann hypothesis, it has far reaching consequences about the distribution of prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s.
The formal statement of the hypothesis follows. A Dirichlet character
Dirichlet character
In number theory, Dirichlet characters are certain arithmetic functions which arise from completely multiplicative characters on the units of \mathbb Z / k \mathbb Z...
is a completely multiplicative
Multiplicative function
In number theory, a multiplicative function is an arithmetic function f of the positive integer n with the property that f = 1 and whenevera and b are coprime, then...
arithmetic function
Arithmetic function
In number theory, an arithmetic function is a real or complex valued function ƒ defined on the set of natural numbers In number theory, an arithmetic (or arithmetical) function is a real or complex valued function ƒ(n) defined on the set of natural numbers In number theory, an arithmetic (or...
χ such that there exists a positive integer k with χ(n + k) = χ(n) for all n and χ(n) = 0 whenever gcd(n, k) > 1. If such a character is given, we define the corresponding Dirichlet L-function by
for every complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s with real part > 1. By analytic continuation
Analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...
, this function can be extended to a meromorphic function
Meromorphic function
In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function...
defined on the whole complex plane. The generalized Riemann hypothesis asserts that for every Dirichlet character χ and every complex number s with L(χ,s) = 0: if the real part of s is between 0 and 1, then it is actually 1/2.
The case χ(n) = 1 for all n yields the ordinary Riemann hypothesis.
Consequences of GRH
Dirichlet's theoremDirichlet's theorem on arithmetic progressions
In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n ≥ 0. In other words, there are infinitely many primes which are...
states that if a and d are coprime
Coprime
In number theory, a branch of mathematics, two integers a and b are said to be coprime or relatively prime if the only positive integer that evenly divides both of them is 1. This is the same thing as their greatest common divisor being 1...
natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s, then the arithmetic progression
Arithmetic progression
In mathematics, an arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant...
a, a+d, a+2d, a+3d, … contains infinitely many prime numbers. Let π(x,a,d) 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 ε > 0
where φ(d) is Euler's totient function
Euler's totient function
In number theory, the totient \varphi of a positive integer n is defined to be the number of positive integers less than or equal to n that are coprime to n In number theory, the totient \varphi(n) of a positive integer n is defined to be the number of positive integers less than or equal to n that...
and O is the Big O notation
Big O notation
In mathematics, big O notation is used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions. It is a member of a larger family of notations that is called Landau notation, Bachmann-Landau notation, or...
. This is a considerable strengthening of the prime number theorem
Prime number theorem
In number theory, the prime number theorem describes the asymptotic distribution of the prime numbers. The prime number theorem gives a general description of how the primes are distributed amongst the positive integers....
.
If GRH is true, then every proper subgroup of the multiplicative group
omits a number 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 testAKS primality testThe AKS primality test is a deterministic primality-proving algorithm created and published by three Indian Institute of Technology Kanpur computer scientists, Manindra Agrawal, Neeraj Kayal, and Nitin Saxena, on August 6, 2002, in a paper titled "PRIMES is in P"...
, was published in 2002.) In practice the elliptic curve primality provingElliptic curve primality provingElliptic Curve Primality Proving is a method based on elliptic curves to prove the primality of a number . It is a general-purpose algorithm, meaning it does not depend on the number being of a special form...
algorithm is faster. - The Shanks–Tonelli algorithm is guaranteed to run in polynomial time.
If GRH is true, then for every prime p there exists a primitive root mod p
Primitive root modulo n
In modular arithmetic, a branch of number theory, a primitive root modulo n is any number g with the property that any number coprime to n is congruent to a power of g modulo n. In other words, g is a generator of the multiplicative group of integers modulo n...
(a generator of the multiplicative group of integers modulo p) which is less than
Goldbach's weak conjecture
Goldbach's weak conjecture
In number theory, Goldbach's weak conjecture, also known as the odd Goldbach conjecture, the ternary Goldbach problem, or the 3-primes problem, states that:...
also follows from the generalized Riemann hypothesis.
Assuming the truth of the GRH, the estimate of the character sum in the Pólya–Vinogradov inequality
Character sum
In mathematics, a character sum is a sum\Sigma \chi\,of values of a Dirichlet character χ modulo N, taken over a given range of values of n. Such sums are basic in a number of questions, for example in the distribution of quadratic residues, and in particular in the classical question of finding an...
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 extensionField extension
In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties...
of the rationals
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
Q) with ring of integers
Algebraic number
In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients. Numbers such as π that are not algebraic are said to be transcendental; almost all real numbers are transcendental...
OK (this ring is the integral closure of the integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s Z in K). If a is an ideal of OK, other than the zero ideal we denote its norm 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 OK.
The Dedekind zeta-function satisfies a functional equation and can be extended by analytic continuation
Analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...
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 ζK(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.
See also
- Artin's conjecture
- Dirichlet L-function
- Selberg classSelberg classIn mathematics, the Selberg class S 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 commonly called L-functions or zeta functions...