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The Riemann hypothesis is one of the most important conjectures 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-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. None of these conjectures have been proven or disproven, but many mathematicians believe them to be true. RH directs here. ...
In mathematics, a conjecture is a mathematical statement which has been proposed as a true statement, but which no one has yet been able to prove or disprove. ...
Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ...
In mathematics, the Riemann zeta function is a function which is of paramount importance in number theory, because of its relation to the distribution of prime numbers. ...
The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary number theory. ...
Global L-functions can be associated to elliptic curves, number fields (in which case they are called Dedekind zeta functions), Maass waveforms, 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 and when it is formulated for Dirichlet L-functions, it is known as the generalized Riemann hypothesis. These two statements will be discussed in more detail below. In mathematics, elliptic curves are defined by certain cubic (the superscript exponent is three, a. ...
In mathematics, an algebraic number field (or simply number field) is a finite field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and these days...
In mathematics, the Dedekind zeta function is a Dirichlet series defined for any algebraic number field K, and denoted ζK(s) where s is a complex variable. ...
In number theory, a Dirichlet character is a function χ from the positive integers to the complex numbers which has the following properties: There exists a positive integer k such that χ(n) = χ(n + k) for all n. ...
In number theory, a Dirichlet character is a function χ from the positive integers to the complex numbers which has the following properties: There exists a positive integer k such that χ(n) = χ(n + k) for all n. ...
Generalized Riemann hypothesis (GRH)
The generalized Riemann hypothesis 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 numbers. 1884 is a leap year starting on Tuesday (click on link to calendar). ...
In mathematics, a prime number, or prime for short, is a natural number greater than one and whose only distinct positive divisors are one and itself. ...
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 In number theory, a Dirichlet character is a function χ from the positive integers to the complex numbers which has the following properties: There exists a positive integer k such that χ(n) = χ(n + k) for all n. ...
In number theory, a multiplicative function is an arithmetic function f(n) of the positive integer n with the property that f(1) = 1 and whenever a and b are coprime, then f(ab) = f(a) f(b). ...
In number theory, an arithmetic function (or number-theoretic function) f(n) is a function defined for all positive integers and having values in the complex numbers. ...
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 complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. ...
A meromorphic function is a function that is holomorphic on an open subset of the complex number plane C (or on some other connected Riemann surface) except at points in a set of isolated poles, which are certain well-behaved singularities. ...
The case χ(n) = 1 for all n yields the ordinary Riemann hypothesis.
Consequences of GRH An arithmetic progression in the natural numbers is a set of numbers of the form a, a+d, a+2d, a+3d, ... where a and d are natural numbers and d is non-zero. Dirichlet's theorem states that if a and d are coprime, then such an arithmetic progression 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 Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...
In number theory, Dirichlets 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, or in other words: there are infinitely many primes which are congruent to a modulo d. ...
In mathematics, the integers a and b are said to be coprime or relatively prime if and only if they have no common factor other than 1 and −1, or equivalently, if their greatest common divisor is 1. ...
Infinity has discrete meanings in mathematics, philosophy, theology and everyday life. ...
where φ(d) denotes Euler's phi function and O is the Landau symbol. This is a considerable strengthening of the prime number theorem. In number theory, the totient φ(n) of a positive integer n is defined to be the number of positive integers less than or equal to n and coprime to n. ...
In complexity theory, computer science, and mathematics the Big O notation is a mathematical notation used to describe the asymptotic behavior of functions. ...
In number theory, the prime number theorem (PNT) describes the approximate, asymptotic distribution of the prime numbers. ...
If GRH is true, then for every prime p there exists a primitive root modulo p (a generator of the multiplicative group of integers modulo p) which is less than 70 (ln(p))2; this is often used in proofs. A primitive root modulo n is a concept from modular arithmetic in number theory. ...
Goldbach's weak conjecture also follows from the generalized Riemann hypothesis. In number theory, Goldbachs weak conjecture, also known as the odd Goldbach conjecture or the 3-primes problem, states that: Every odd number greater than 7 can be expressed as the sum of three odd primes. ...
If GRH is true, then the Miller-Rabin primality test is guaranteed to run in polynomial time. (A polynomial-time primality test which doesn't require GRH has recently been published; see prime number.) The Miller-Rabin primality test is a primality test: an algorithm which determines whether a given number is prime, similar to the Fermat primality test and the Solovay-Strassen primality test. ...
In mathematics, a prime number, or prime for short, is a natural number greater than one and whose only distinct positive divisors are one and itself. ...
Extended Riemann hypothesis (ERH) Suppose K is a number field (a finite-dimensional field extension of the rationals Q) with ring of integers OK (this ring is the integral closure of the integers 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 In mathematics, an algebraic number field (or simply number field) is a finite field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and these days...
In abstract algebra, an extension of a field K is a field L which contains K as a subfield. ...
In mathematics, a rational number (or informally fraction) is a ratio of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...
In mathematics, an algebraic number relative to a field F is any element x of a given field K containing F such that x is a solution of a polynomial equation of the form anxn + an−1xn−1 + ··· + a1x + a0 = 0 where n is a positive integer called the degree...
In abstract algebra, the concept of integral closure is a generalization of the set of all algebraic integers. ...
The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ...
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ...
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. In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. ...
The ordinary Riemann hypothesis follows from the extended one if one takes the number field to be Q, with ring of integers Z.
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