Statements in this class include the statement that P = NP, the Riemann hypothesis, and many other unsolved mathematical problems.

Among his major accomplishments were the 1940 proof, of the Riemann hypothesis for zeta-functions of curves over finite fields, and his subsequent laying of proper foundations for algebraic geometry to support that result ( from 1942 to 1946, most intensively ).

* The Riemann hypothesis

For example, the Riemann hypothesis is a conjecture from number theory that ( amongst other things ) makes predictions about the distribution of prime numbers.

Few number theorists doubt that the Riemann hypothesis is true ( it is said that Atle Selberg was once a sceptic, and J. E. Littlewood always was ).

Likewise, the Riemann hypothesis has a large number of consequences already proven.

The hypothesis that ƒ obey the Cauchy – Riemann equations throughout the domain Ω is essential.

Of the initial twenty-three Hilbert problems, most of which have been solved, only the Riemann hypothesis ( formulated in 1859 ) is included in the seven Millennium Prize Problems.

* The Riemann hypothesis

This result can sometimes substitute for the still-unproved generalized Riemann hypothesis.

# REDIRECT Riemann hypothesis

( Indeed, the behavior of this difference is very complicated and related to the Riemann hypothesis.

Because of the connection between the Riemann zeta function and π ( x ), the Riemann hypothesis has considerable importance in number theory: if established, it would yield a far better estimate of the error involved in the prime number theorem than is available today.

More specifically, Helge von Koch showed in 1901 that, if and only if the Riemann hypothesis is true, the error term in the above relation can be improved to

The constant involved in the big O notation was estimated in 1976 by Lowell Schoenfeld: assuming the Riemann hypothesis,

Under the assumption of the generalized Riemann hypothesis, it can be shown to take time

Under the Riemann hypothesis, the error term can be reduced:

The Riemann hypothesis has been verified for the first 10 trillion zeroes of the zeta function.

Other famous conjectures include the Collatz conjecture and the Riemann hypothesis.

* Bernhard Riemann formulates the Riemann hypothesis, one of the most important open problems of contemporary mathematics.

In 1956, he applied the same thinking to the Riemann – Roch theorem, which had already recently been generalized to any dimension by Hirzebruch.

There are Cauchy – Riemann equations, appropriately generalized, in the theory of several complex variables.

He claims to have proved several important conjectures in mathematics, including the generalized Riemann hypothesis ( GRH ).

In 1923, Hardy and Littlewood showed that, assuming the generalized Riemann hypothesis, the odd Goldbach conjecture is true for all sufficiently large odd numbers.

In 1997, Deshouillers, Effinger, te Riele and Zinoviev showed that the generalized Riemann hypothesis implies Goldbach's weak conjecture for all numbers.

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 ).

( Many mathematicians use the label generalized Riemann hypothesis to cover the extension of the Riemann hypothesis to all global L-functions,

The generalized Riemann hypothesis ( for Dirichlet L-functions ) was probably formulated for the first time by Piltz in 1884.

The generalized Riemann hypothesis asserts that for every Dirichlet character χ and every complex number s with L ( χ, s )

If the generalized Riemann hypothesis is true, then for every coprime a and d and for every ε > 0

Goldbach's weak conjecture also follows from the generalized Riemann hypothesis.

Dirichlet L-functions are generalizations of the Riemann zeta-function and appear prominently in the generalized Riemann hypothesis.

Similarly, under the generalized Riemann hypothesis, the Miller – Rabin test can be turned into a deterministic version ( called Miller's test ) with runtime Õ (( log n )< sup > 4 </ sup >).

" On generalized Riemann matrices ," Ann.

Its original version, due to Gary L. Miller, is deterministic, but the determinism relies on the unproven generalized Riemann hypothesis ; Michael O. Rabin modified it to obtain an unconditional probabilistic algorithm.

If the tested number n is composite, the strong liars a coprime to n are contained in a proper subgroup of the group, which means that if we test all a from a set which generates, one of them must be a witness for the compositeness of n. Assuming the truth of the generalized Riemann hypothesis ( GRH ), it is known that the group is generated by its elements smaller than O (( log n )< sup > 2 </ sup >), which was already noted by Miller.

The full power of the generalized Riemann hypothesis is not needed to ensure the correctness of the test: as we deal with subgroups of even index, it suffices to assume the validity of GRH for quadratic Dirichlet characters.

** In the category of Riemann surfaces, an automorphism is a bijective biholomorphic map ( also called a conformal map ), from a surface to itself.

For example, the automorphisms of the Riemann sphere are Möbius transformations.

His ' matrix divisor ' ( vector bundle avant la lettre ) Riemann – Roch theorem from 1938 was a very early anticipation of later ideas such as moduli spaces of bundles.

His first ( pre-IHÉS ) breakthrough in algebraic geometry was the Grothendieck – Hirzebruch – Riemann – Roch theorem, a far-reaching generalisation of the Hirzebruch – Riemann – Roch theorem proved algebraically ; in this context he also introduced K-theory.

The Grothendieck – Riemann – Roch theorem was announced by Grothendieck at the initial Mathematische Arbeitstagung in Bonn, in 1957.

The Bernoulli numbers appear in the Taylor series expansions of the tangent and hyperbolic tangent functions, in formulas for the sum of powers of the first positive integers, in the Euler – Maclaurin formula, and in expressions for certain values of the Riemann zeta function.

However, several of his students became influential mathematicians, among them Richard Dedekind, Bernhard Riemann, and Friedrich Bessel.

In mathematics, the Cauchy – Riemann differential equations in complex analysis, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which must be satisfied if we know that a complex function is complex differentiable.

Then f = u + iv is complex-differentiable at that point if and only if the partial derivatives of u and v satisfy the Cauchy – Riemann equations ( 1a ) and ( 1b ) at that point.

The sole existence of partial derivatives satisfying the Cauchy – Riemann equations is not enough to ensure complex differentiability at that point.

The reason why Euler and some other authors relate the Cauchy – Riemann equations with analyticity is that a major theorem in complex analysis says that holomorphic functions are analytic and viceversa.

Consequently, a function satisfying the Cauchy – Riemann equations, with a nonzero derivative, preserves the angle between curves in the plane.

That is, the Cauchy – Riemann equations are the conditions for a function to be conformal.

Conversely, if ƒ: C → C is a function which is differentiable when regarded as a function on R < sup > 2 </ sup >, then ƒ is complex differentiable if and only if the Cauchy – Riemann equations hold.

In other words, if u and v are real-differentiable functions of two real variables, obviously u + iv is a ( complex-valued ) real-differentiable function, but u + iv is complex-differentiable if and only if the Cauchy – Riemann equations hold.

Indeed, following, suppose ƒ is a complex function defined in an open set Ω ⊂ C. Then, writing for every z ∈ Ω, one can also regard Ω as an open subset of R < sup > 2 </ sup >, and ƒ as a function of two real variables x and y, which maps Ω ⊂ R < sup > 2 </ sup > to C. We consider the Cauchy – Riemann equations at z = 0 assuming ƒ ( z ) = 0, just for notational simplicity – the proof is identical in general case.