What is the Riemann Hypothesis? Who was Riemann? How is it connected to prime numbers? What other areas of mathematics does it relate to? Why is it important? Are there any proposed proofs circulating?

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Many consider it to be the most important unsolved problem in pure mathematics Bombieri It is of great interest in number theory because it implies results about the distribution of prime numbers. The Riemann hypothesis and some of its generalizations, along with Goldbach's conjecture and the twin prime conjecture , comprise Hilbert's eighth problem in David Hilbert 's list of 23 unsolved problems ; it is also one of the Clay Mathematics Institute's Millennium Prize Problems.

The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields. These are called its trivial zeros. However, the negative even integers are not the only values for which the zeta function is zero. The other ones are called non-trivial zeros. The Riemann hypothesis is concerned with the locations of these non-trivial zeros, and states that:. The books Edwards , Patterson , Borwein et al.

The Riemann zeta function is defined for complex s with real part greater than 1 by the absolutely convergent infinite series. Leonhard Euler already considered this series in the s for real values of s, in conjunction with his solution to the Basel problem. He also proved that it equals the Euler product. The Riemann hypothesis discusses zeros outside the region of convergence of this series and Euler product.

To make sense of the hypothesis, it is necessary to analytically continue the function to obtain a form that is valid for all complex s. This is permissible because the zeta function is meromorphic , so its analytic continuation is guaranteed to be unique and functional forms equivalent over their domains.

One begins by showing that the zeta function and the Dirichlet eta function satisfy the relation. But the series on the right converges not just when the real part of s is greater than one, but more generally whenever s has positive real part.

If s is a positive even integer this argument does not apply because the zeros of the sine function are cancelled by the poles of the gamma function as it takes negative integer arguments. The functional equation also implies that the zeta function has no zeros with negative real part other than the trivial zeros, so all non-trivial zeros lie in the critical strip where s has real part between 0 and 1.

Of course one would wish for a rigorous proof here; I have for the time being, after some fleeting vain attempts, provisionally put aside the search for this, as it appears dispensable for the immediate objective of my investigation.

His formula was given in terms of the related function. Riemann's formula is then. The function li occurring in the first term is the unoffset logarithmic integral function given by the Cauchy principal value of the divergent integral. This formula says that the zeros of the Riemann zeta function control the oscillations of primes around their "expected" positions. The practical uses of the Riemann hypothesis include many propositions known true under the Riemann hypothesis, and some that can be shown to be equivalent to the Riemann hypothesis.

Riemann's explicit formula for the number of primes less than a given number in terms of a sum over the zeros of the Riemann zeta function says that the magnitude of the oscillations of primes around their expected position is controlled by the real parts of the zeros of the zeta function. In particular the error term in the prime number theorem is closely related to the position of the zeros.

Von Koch proved that the Riemann hypothesis implies the "best possible" bound for the error of the prime number theorem. A precise version of Koch's result, due to Schoenfeld , says that the Riemann hypothesis implies. Schoenfeld also showed that the Riemann hypothesis implies. The Riemann hypothesis implies strong bounds on the growth of many other arithmetic functions, in addition to the primes counting function above.

The statement that the equation. From this we can also conclude that if the Mertens function is defined by. Littlewood , ; see for instance: paragraph For the meaning of these symbols, see Big O notation. The determinant of the order n Redheffer matrix is equal to M n , so the Riemann hypothesis can also be stated as a condition on the growth of these determinants. The Riemann hypothesis is equivalent to several statements showing that the terms of the Farey sequence are fairly regular.

The Riemann hypothesis also implies quite sharp bounds for the growth rate of the zeta function in other regions of the critical strip.

For example, it implies that. However, some gaps between primes may be much larger than the average. Many statements equivalent to the Riemann hypothesis have been found, though so far none of them have led to much progress in proving or disproving it. Some typical examples are as follows. The Riesz criterion was given by Riesz , to the effect that the bound.

Nyman proved that the Riemann hypothesis is true if and only if the space of functions of the form. Salem showed that the Riemann hypothesis is true if and only if the integral equation. Weil's criterion is the statement that the positivity of a certain function is equivalent to the Riemann hypothesis.

Related is Li's criterion , a statement that the positivity of a certain sequence of numbers is equivalent to the Riemann hypothesis. Several applications use the generalized Riemann hypothesis for Dirichlet L-series or zeta functions of number fields rather than just the Riemann hypothesis. Many basic properties of the Riemann zeta function can easily be generalized to all Dirichlet L-series, so it is plausible that a method that proves the Riemann hypothesis for the Riemann zeta function would also work for the generalized Riemann hypothesis for Dirichlet L-functions.

Several results first proved using the generalized Riemann hypothesis were later given unconditional proofs without using it, though these were usually much harder. Many of the consequences on the following list are taken from Conrad Some consequences of the RH are also consequences of its negation, and are thus theorems. The method of proof here is truly amazing. If the generalized Riemann hypothesis is true, then the theorem is true. If the generalized Riemann hypothesis is false, then the theorem is true.

Thus, the theorem is true!! Care should be taken to understand what is meant by saying the generalized Riemann hypothesis is false: one should specify exactly which class of Dirichlet series has a counterexample. This concerns the sign of the error in the prime number theorem.

Skewes' number is an estimate of the value of x corresponding to the first sign change. Littlewood's proof is divided into two cases: the RH is assumed false about half a page of Ingham , Chapt.

V , and the RH is assumed true about a dozen pages. This is the conjecture first stated in article of Gauss's Disquisitiones Arithmeticae that there are only a finite number of imaginary quadratic fields with a given class number. Theorem Hecke; Assume the generalized Riemann hypothesis for L -functions of all imaginary quadratic Dirichlet characters.

Then there is an absolute constant C such that. Theorem Deuring; Theorem Heilbronn; In the work of Hecke and Heilbronn, the only L -functions that occur are those attached to imaginary quadratic characters, and it is only for those L -functions that GRH is true or GRH is false is intended; a failure of GRH for the L -function of a cubic Dirichlet character would, strictly speaking, mean GRH is false, but that was not the kind of failure of GRH that Heilbronn had in mind, so his assumption was more restricted than simply GRH is false.

In J. Nicolas proved Ribenboim , p. The method of proof is interesting, in that the inequality is shown first under the assumption that the Riemann hypothesis is true, secondly under the contrary assumption.

The Riemann hypothesis can be generalized by replacing the Riemann zeta function by the formally similar, but much more general, global L-functions. It is these conjectures, rather than the classical Riemann hypothesis only for the single Riemann zeta function, which account for the true importance of the Riemann hypothesis in mathematics. The generalized Riemann hypothesis extends the Riemann hypothesis to all Dirichlet L-functions.

The extended Riemann hypothesis extends the Riemann hypothesis to all Dedekind zeta functions of algebraic number fields. The extended Riemann hypothesis for abelian extension of the rationals is equivalent to the generalized Riemann hypothesis. The Riemann hypothesis can also be extended to the L -functions of Hecke characters of number fields. The grand Riemann hypothesis extends it to all automorphic zeta functions , such as Mellin transforms of Hecke eigenforms.

Artin introduced global zeta functions of quadratic function fields and conjectured an analogue of the Riemann hypothesis for them, which has been proved by Hasse in the genus 1 case and by Weil in general. Arithmetic zeta functions generalise the Riemann and Dedekind zeta functions as well as the zeta functions of varieties over finite fields to every arithmetic scheme or a scheme of finite type over integers.

Selberg introduced the Selberg zeta function of a Riemann surface. These are similar to the Riemann zeta function: they have a functional equation, and an infinite product similar to the Euler product but taken over closed geodesics rather than primes. The Selberg trace formula is the analogue for these functions of the explicit formulas in prime number theory. Selberg proved that the Selberg zeta functions satisfy the analogue of the Riemann hypothesis, with the imaginary parts of their zeros related to the eigenvalues of the Laplacian operator of the Riemann surface.

The Ihara zeta function of a finite graph is an analogue of the Selberg zeta function , which was first introduced by Yasutaka Ihara in the context of discrete subgroups of the two-by-two p-adic special linear group. A regular finite graph is a Ramanujan graph , a mathematical model of efficient communication networks, if and only if its Ihara zeta function satisfies the analogue of the Riemann hypothesis as was pointed out by T.

Montgomery suggested the pair correlation conjecture that the correlation functions of the suitably normalized zeros of the zeta function should be the same as those of the eigenvalues of a random hermitian matrix. Odlyzko showed that this is supported by large-scale numerical calculations of these correlation functions.

Dedekind zeta functions of algebraic number fields, which generalize the Riemann zeta function, often do have multiple complex zeros Radziejewski This is because the Dedekind zeta functions factorize as a product of powers of Artin L-functions , so zeros of Artin L-functions sometimes give rise to multiple zeros of Dedekind zeta functions. Other examples of zeta functions with multiple zeros are the L-functions of some elliptic curves : these can have multiple zeros at the real point of their critical line; the Birch-Swinnerton-Dyer conjecture predicts that the multiplicity of this zero is the rank of the elliptic curve.

There are many other examples of zeta functions with analogues of the Riemann hypothesis, some of which have been proved. Goss zeta functions of function fields have a Riemann hypothesis, proved by Sheats Several mathematicians have addressed the Riemann hypothesis, but none of their attempts have yet been accepted as a correct solution. Watkins lists some incorrect solutions, and more are frequently announced.

Odlyzko showed that the distribution of the zeros of the Riemann zeta function shares some statistical properties with the eigenvalues of random matrices drawn from the Gaussian unitary ensemble. In a connection with this quantum mechanical problem Berry and Connes had proposed that the inverse of the potential of the Hamiltonian is connected to the half-derivative of the function. Connes This yields a Hamiltonian whose eigenvalues are the square of the imaginary part of the Riemann zeros, and also that the functional determinant of this Hamiltonian operator is just the Riemann Xi function.


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