The real part (red) and imaginary part (blue) of the critical line Re(s) = 1/2 of the Riemann zeta-function. You can see the first non-trivial zeros at Im(s) = ±14.135, ±21.022 and ±25.011.

A Value Graph of zeta, that is, Re(zeta) vs. Im(zeta), along the critical line s = it + 1/2, with t running from 0 to 34

The Riemann hypothesis (also called the Riemann zeta-hypothesis), first formulated by Bernhard Riemann in 1859, is one of the most famous and important unsolved problems in mathematics. It has been an open question for well over a century, despite attracting concentrated efforts from many outstanding mathematicians. Unlike some other celebrated problems, it is more attractive to professionals in the field than to amateurs. In number theory, a local zeta-function is a generating function Z(t) for the number of solutions of a set of equations defined over a finite field F, in extension fields Fk of F. The analogy with the Riemann zeta function comes via consideration of the logarithmic derivative . ... Image File history File links Download high resolution version (1276x784, 29 KB)Graph of real (red) and imaginary (blue) parts of the critical line Re(z)=1/2 of the Riemann zeta function. ... Image File history File links Download high resolution version (1276x784, 29 KB)Graph of real (red) and imaginary (blue) parts of the critical line Re(z)=1/2 of the Riemann zeta function. ... Image File history File links Zeta_polar. ... Image File history File links Zeta_polar. ... Bernhard Riemann. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...

The Riemann hypothesis (RH) is a conjecture about the distribution of the zeros of the Riemann zeta-function ζ(s). The Riemann zeta-function is defined for all complex numbers s ≠ 1. It has zeros at the negative even integers (i.e. at s = −2, s = −4, s = −6, ...). These are called the trivial zeros. The Riemann hypothesis is concerned with the non-trivial zeros, and states that: In mathematics, a conjecture is a mathematical statement which appears likely to be true, but has not been formally proven to be true under the rules of mathematical logic. ... In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ... In mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...

The real part of any non-trivial zero of the Riemann zeta function is ½.

Thus the non-trivial zeros should lie on the so-called critical line ½ + it with t a real number and i the imaginary unit. The Riemann zeta-function along the critical line is sometimes studied in terms of the Z-function, whose real zeros correspond to the zeros of the zeta-function on the critical line. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, the imaginary unit (or sometimes the Latin or the Greek iota, see below) allows the real number system to be extended to the complex number system . ... In mathematics, the Z-function is a function used for studying the Riemann zeta-function along the critical line where the real part of the argument is one-half. ...

The Riemann hypothesis is one of the most important open problems of contemporary mathematics, mainly because a large number of deep and important other results have been proven under the condition that it holds. Most mathematicians believe the Riemann hypothesis to be true. (J. E. Littlewood and Atle Selberg have been reported as skeptical. Selberg's skepticism, if any, waned from his young days. In a 1989 paper, he suggested that an analogue should hold for a much wider class of functions, the Selberg class). A $1,000,000 prize has been offered by the Clay Mathematics Institute for the first correct proof.^[1] John Edensor Littlewood (June 9, 1885 - September 6, 1977) was a British mathematician. ... Atle Selberg (born June 17, 1917) is a Norwegian mathematician known for his work in analytic number theory, and in the theory of automorphic forms, in particular bringing them into relation with spectral theory. ... In mathematics, the Selberg class S is an axiomatic definition of the class of L-functions. ... The Clay Mathematics Institute (CMI) is a private, non-profit foundation, based in Cambridge, Massachusetts. ...

The Clay Mathematics Institute (CMI) is a private, non-profit foundation, based in Cambridge, Massachusetts, and dedicated to increasing and disseminating mathematical knowledge. ... Diagram of complexity classes provided that P ≠ NP. The existence of problems outside both P and NP-complete in this case was established by Ladner. ... The Hodge conjecture is a major unsolved problem of algebraic geometry. ... In mathematics, the Poincaré conjecture (IPA: [])[1] is a conjecture about the characterization of the three-dimensional sphere amongst three-dimensional manifolds. ... The Clay Mathematics Institute has offered the prize of 1 million dollars for each of 7 great problems in mathematics. ... This article or section is in need of attention from an expert on the subject. ... In mathematics, the Birch and Swinnerton-Dyer conjecture relates the rank of the abelian group of points over a number field of an elliptic curve E to the order of zero of the associated L-function L(E, s) at s = 1. ...

History

Unsolved problems in mathematics: Does every non-trivial zero of the Riemann zeta function have real part ½?

Riemann mentioned the conjecture that became known as the Riemann hypothesis in his 1859 paper On the Number of Primes Less Than a Given Magnitude, but as it was not essential to his central purpose in that paper, he did not attempt a proof. Riemann knew that the non-trivial zeros of the zeta-function were symmetrically distributed about the line s = ½ + it, and he knew that all of its non-trivial zeros must lie in the range 0 ≤ Re(s) ≤ 1. Image File history File links No higher resolution available. ... This article lists some unsolved problems in mathematics. ... In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ... In mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ... In mathematics, the real part of a complex number , is the first element of the ordered pair of real numbers representing , i. ... On the Number of Primes Less Than a Given Magnitude (or Über die Anzahl der Primzahlen unter einer gegebenen Grösse) is a seminal 8-page paper by Bernhard Riemann published in the November 1859 edition of the Monthly Reports of the Berlin Academy. ...

In 1896, Hadamard and de la Vallée-Poussin independently proved that no zeros could lie on the line Re(s) = 1. Together with the other properties of non-trivial zeros proved by Riemann, this showed that all non-trivial zeros must lie in the interior of the critical strip 0 < Re(s) < 1. This was a key step in the first complete proofs of the prime number theorem. This page is a candidate for speedy deletion. ... Charles-Jean de la Vallée-Poussin (August 14, 1866 - March 2, 1962) was a Belgian mathematician. ... In number theory, the prime number theorem (PNT) describes the approximate, asymptotic distribution of the prime numbers. ...

In 1900, Hilbert included the Riemann hypothesis in his famous list of 23 unsolved problems — it is part of Problem 8 in Hilbert's list, along with the Goldbach conjecture. When asked what he would do if awakened after having slept for five hundred years, Hilbert famously said his first question would be whether the Riemann hypothesis had been proven (Derbyshire 2003:197; Sabbagh 2003:69; Bollobas 1986:16). The Riemann Hypothesis is the only one of Hilbert's problems on the Clay Mathematics Institute Millennium Prize Problems. David Hilbert (January 23, 1862, Königsberg, East Prussia – February 14, 1943, Göttingen, Germany) was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. ... Hilberts problems are a list of twenty-three problems in mathematics put forth by German mathematician David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900. ... In mathematics, Goldbachs conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. ... The Clay Mathematics Institute (CMI) is a private, non-profit foundation, based in Cambridge, Massachusetts. ... The Clay Mathematics Institute (CMI) is a private, non-profit foundation, based in Cambridge, Massachusetts, and dedicated to increasing and disseminating mathematical knowledge. ...

In 1914, Hardy proved that an infinite number of zeros lie on the critical line Re(s) = ½. However, it was still possible that an infinite number (and possibly the majority) of non-trivial zeros could lie elsewhere in the critical strip. Later work by Hardy and Littlewood in 1921 and by Selberg in 1942 gave estimates for the average density of zeros on the critical line. G. H. Hardy Professor Godfrey Harold Hardy FRS (February 7, 1877 – December 1, 1947) was a prominent English mathematician, known for his achievements in number theory and mathematical analysis. ... In mathematics, the critical line theorem tells us that a positive percentage of the nontrivial zeros of the Riemann zeta function lie on the critical line. ... John Edensor Littlewood (June 9, 1885 - September 6, 1977) was a British mathematician. ...

Recent work has focused on the explicit calculation of the locations of large numbers of zeros (in the hope of finding a counterexample) and placing upper bounds on the proportion of zeros that can lie away from the critical line (in the hope of reducing this to zero).

The Riemann hypothesis and primes

The traditional formulation of the Riemann hypothesis obscures somewhat the true importance of the conjecture. The zeta-function has a deep connection to the distribution of prime numbers. Helge von Koch proved in 1901 that the Riemann hypothesis is equivalent to the following considerable strengthening of the prime number theorem: for every ε > 0, we have In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. ... Niels Fabian Helge von Koch (January 25, 1870 - March 11, 1924) was a Swedish mathematician, who gave his name to the famous fractal known as the Koch curve, which was one of the earliest fractal curves to have been described. ... In number theory, the prime number theorem (PNT) describes the approximate, asymptotic distribution of the prime numbers. ...

where π(x) is the prime-counting function, ln(x) is the natural logarithm of x, and the Landau notation is used on the right-hand side.^[2] A non-asymptotic version, due to Lowell Schoenfeld, says that the Riemann hypothesis is equivalent to In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number x. ... The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is an irrational constant approximately equal to 2. ... It has been suggested that this article or section be merged into Big O notation. ... Lowell Schoenfeld (1920-2002) was an American mathematician known for his work in analytic number theory. ...

The zeros of the Riemann zeta-function and the prime numbers satisfy a certain duality property, known as the explicit formulae, which shows that in the language of Fourier analysis the zeros of the Riemann zeta-function can be regarded as the harmonic frequencies in the distribution of primes. In mathematics, the explicit formulae for L-functions are a class of summation formulae, expressing sums taken over the complex number zeroes of a given L-function, typically in terms of quantities studied by number theory by use of the theory of special functions. ... Fourier analysis, named after Joseph Fouriers introduction of the Fourier series, is the decomposition of a function in terms of a sum of sinusoidal basis functions (vs. ...

The Riemann hypothesis can be generalized by replacing the Riemann zeta-function by the formally similar, but much more general, global L-functions. In this broader setting, one expects the non-trivial zeros of the global L-functions to have real part 1/2, and this is called the generalized Riemann hypothesis (GRH). It is this conjecture, rather than the classical Riemann hypothesis only for the single Riemann zeta-function, which accounts for the true importance of the Riemann hypothesis in mathematics. In other words, the importance of 'the Riemann hypothesis' in mathematics today really stems from the importance of the generalized Riemann hypothesis, but it is simpler to refer to the Riemann hypothesis only in its original special case when describing the problem to people outside of mathematics.^{[citation needed]} The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary number theory. ... The Riemann hypothesis is one of the most important conjectures in mathematics. ...

For many global L-functions of function fields (but not number fields), the Riemann hypothesis has been proven. For instance, the fact that the Gauss sum, of the quadratic character of a finite field of size q (with q odd), has absolute value In algebraic geometry, the function field of an irreducible algebraic variety is the field of fractions of the ring of regular functions. ... 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, quadratic Gauss sums are certain sums over exponential functions with quadratic argument. ...

√q

is actually an instance of the Riemann hypothesis in the function field setting.

Consequences and equivalent formulations of the Riemann hypothesis

The practical uses of the Riemann hypothesis include many propositions which are stated to be true under the Riemann hypothesis, and some which can be shown to be equivalent to the Riemann hypothesis. One is the rate of growth in the error term of the prime number theorem given above.

Growth rate of Möbius function

One formulation involves the Möbius function μ. The statement that the equation The classical Möbius function is an important multiplicative function in number theory and combinatorics. ...

is valid for every s with real part greater than ½, with the sum on the right hand side converging, is equivalent to the Riemann hypothesis. From this we can also conclude that if the Mertens function is defined by In number theory, the Mertens function is where &#956;(k) is the Möbius function. ...

then the claim that

for every,

is equivalent to the Riemann hypothesis. This puts a rather tight bound on the growth of M, since even with no hypothesis we can conclude

(For the meaning of these symbols, see Big O notation.) For other uses, see Big O. In computational complexity theory, big O notation is often used to describe how the size of the input data affects an algorithms usage of computational resources (usually running time or memory). ...

Growth rates of multiplicative functions

The Riemann hypothesis is equivalent to certain conjectures about the rate of growth of other multiplicative functions aside from μ(n). For instance, if σ(n) is the divisor function, given by 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). ... Divisor function σ0(n) up to n=250 Sigma function σ1(n) up to n=250 Sum of the squares of divisors, σ2(n), up to n=250 Sum of cubes of divisors, σ3(n) up to n=250 In mathematics, and specifically in number theory, a divisor function is...

then

for n > 5040. This is known as Robin's theorem and was given by Guy Robin in 1984. A related bound was given by Jeffrey Lagarias in 2002, who proved that the Riemann hypothesis is equivalent to the statement that Divisor function up to n=250 Sigma function up to n=250 In mathematics, and specifically in number theory, a divisor function is an arithmetical function related to the divisors of an integer. ... Jeffrey Lagarias is a professor at University of Michigan. ...

for every natural number n, where H_n is the n-th harmonic number. In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... The harmonic number with (red line) with its asymptotic limit (blue line). ...

Riesz criterion, binomial sums

The Riesz criterion was given by Marcel Riesz in 1916, to the effect that the relation Riesz(x) for x from 0 to 50 In mathematics, the Riesz function is an entire function defined by Marcel Riesz in connection with the Riemann hypothesis, by means of the power series If we set we may define it in terms of the coefficients of the Laurent series development... Marcel Riesz (November 16, 1886 &#8211; September 4, 1969) was a mathematician who was born in Györ, Austria-Hungary (now Hungary) and died in Lund in Sweden. ...

holds for all ε > 0 if and only if RH holds^[3].

Later (1918) Hardy provided an integral equation for

using a variant of Borel resummation with Mellin transform

Other functions related to the multiplicative functions have growth rates equivalent to the Riemann hypothesis as well.

There are several relations on binomial sums that are equivalent to RH. For example, let

Báez-Duarte^[4][5] and Flajolet and Vallée ^[6] have shown that RH holds if and only if Brigitte Vallée is a French mathematician and computer scientist, professor at Université de Caen[1], specializing in computational number theory[2] and analysis of algorithms. ...

for all ε > 0. Similarly, let

then Flajolet and Vepstas show^[7] that RH holds if and only if

| d_n | < C_εn^{1 / 2 + ε}

for all ε > 0 and some constant C_ε depending on ε. Entering into the proof is the Mobius function μ(n), and so similar results hold for binomial sums over ζ(s − 1) / ζ(s), and so on, which correspond to Dirichlet series for Euler's totient function, the divisor function, and so on. The classical Möbius function is an important multiplicative function in number theory and combinatorics. ... In mathematics, a Dirichlet series, one of a number of concepts named in honor of Johann Peter Gustav Lejeune Dirichlet, is a series of the form The most famous of Dirichlet series is which is the Riemann zeta function. ... In number theory, the totient &#966;(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. ... Divisor function σ0(n) up to n=250 Sigma function σ1(n) up to n=250 Sum of the squares of divisors, σ2(n), up to n=250 Sum of cubes of divisors, σ3(n) up to n=250 In mathematics, and specifically in number theory, a divisor function is...

Weil's criterion, Li's criterion

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. In mathematics, Weils criterion is a criterion of André Weil for the Generalized Riemann Hypothesis to be true. ... In mathematics, in the area of number theory, Lis criterion is a particular statement about the positivity of a certain series that is completely equivalent to the Riemann hypothesis. ...

Relation to Farey sequence

Two other equivalent statements to the Riemann hypothesis involve the Farey sequence. If F_n is the Farey sequence of order n, beginning with 1/n and up to 1/1, then the claim that for all e > ½ In mathematics, the Farey sequence of order n is the sequence of completely reduced fractions between 0 and 1 which, when in lowest terms, have denominators less than or equal to n, arranged in order of increasing size. ...

is equivalent to the Riemann hypothesis. Here is the number of terms in the Farey sequence of order n. Similarly equivalent to the Riemann hypothesis is

for all e > −1.

Relation to group theory

The Riemann hypothesis is equivalent to certain conjectures of group theory. For instance, if g(n) is the maximal order of elements of the symmetric group S_n of degree n, known as Landau's function, then the Riemann hypothesis is equivalent to the bound, for all n greater than some M, of Group theory is that branch of mathematics concerned with the study of groups. ... In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ... Landaus function g(n) is defined for every natural number n to be the largest order of an element of the symmetric group Sn. ...

Critical line theorem

The Riemann hypothesis is equivalent to the statement that ζ'(s), the derivative of ζ(s), has no zeros in the strip

.

That ζ has only simple zeros on the critical line is equivalent to its derivative having no zeros on the critical line, so under the usual hypotheses on the Riemann zeta-function we can extend the zero-free region to . This approach has been fruitful; refining it allowed Norman Levinson to prove his strengthening of the critical line theorem. Norman Levinson (August 11, 1912 - October 10, 1975) was an American mathematician. ... In mathematics, the critical line theorem tells us that a positive percentage of the nontrivial zeros of the Riemann zeta function lie on the critical line. ...

Disproven conjectures

Stronger conjectures than the Riemann hypothesis have also been formulated, but they have a tendency to be disproven. Paul Turan showed that if the sums Paul (Pál) Turán (August 28, 1910&#8211;September 26, 1976) was a Hungarian mathematician who made contributions in number theory and group theory. ...

have no zeros when the real part of s is greater than one then the Riemann hypothesis is true, but Hugh Montgomery showed the premise is false. Another stronger conjecture, the Mertens conjecture, has also been disproven. Hugh Montgomery is an American mathematician, working in the fields of analytic number theory and mathematical analysis. ... The Mertens conjecture is a statement about the behaviour of a certain function as its argument increases. ...

Weaker conjectures

Lindelöf hypothesis

The Riemann hypothesis has various weaker consequences as well; one is the Lindelöf hypothesis on the rate of growth of the zeta function on the critical line, which says that, for any e > 0,

as t tends to infinity.

Denoting by p_n the n-th prime number, a result by Albert Ingham, shows that the Lindelöf hypothesis implies that, for any e > 0, Albert Edward Ingham (3 April 1900–6 September 1967) was an English mathematician. ...

p_n+1 - p_n < p^1/2+e,

if n is sufficiently large. However, this result is worse than that of the large prime gap conjecture, stated below. In mathematics, the phrase sufficiently large is used in contexts such as: is true for sufficiently large which is actually shorthand for: there exists an such that is true for all . ...

Large prime gap conjecture

Another conjecture is the large prime gap conjecture. Cramér proved that, assuming the Riemann hypothesis, the gap between the prime p and its successor is . On average, the gap is merely O(lnp) and numerical evidence does not suggest it can grow nearly as fast as the Riemann hypothesis seems to allow, much less as fast as the best that can at present be shown without it. Harald Cramér (September 25, 1893 - October 5, 1985) was a Swedish mathematician and statistician, specialised in mathematical statistics. ... The n-th prime gap (short for prime number gap), denoted gn, is the difference between the n+1-th and n-th prime number, pn. ...

Attempted proofs of the Riemann hypothesis

Several teams of mathematicians have addressed RH over decades, and a few purported proofs go unverified as of 2007. However, these have been received with skepticism by the mathematical community, and professionals at large do not believe them to be true. Matthew R. Watkins from the University of Exeter has a compilation of such claims (serious and ludicrous alike)^[8], and a few others may be found in the arXiv database. 2007 is a common year starting on Monday of the Gregorian calendar. ... The University of Exeter (usually abbreviated as Exon. ... arXiv (pronounced archive, as if the X were the Greek letter χ) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ...

Possible connection with operator theory

Main article: Hilbert–Pólya conjecture

It has long been speculated that the correct way to derive the Riemann hypothesis has been to find a self-adjoint operator, from the existence of which the statement on the real parts of the zeroes of ζ(s) would follow when one applies the criterion on real eigenvalues. This has led to many investigations; but has not yet proven fruitful. In mathematics, the Hilbert-Pólya conjecture is a possible approach to the Riemann hypothesis, by means of spectral theory. ... On a finite-dimensional inner product space, a self-adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...

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. This gives some support to the Hilbert–Pólya conjecture. In probability theory and statistics, a random matrix is a matrix-valued random variable. ...

In 1999, Michael Berry and Jon Keating conjectured that there is some unknown quantization of the classical Hamiltonian H = xp so that Sir Michael (Victor) Berry, born 14 March 1941, is a mathematical physicist at the University of Bristol. ...

and even more strongly, that the Riemann zeros coincide with the spectrum of the operator . This is to be contrasted to canonical quantization which leads to the Heisenberg uncertainty principle [x,p] = 1 / 2 and the natural numbers as spectrum of the quantum harmonic oscillator. The crucial point is that the Hamiltonian should be a Hermitian operator (or more precisely closed self adjoint operator) so that the quantisation would be a realisation of the Hilbert–Pólya program. In physics, canonical quantization is one of many procedures for quantizing a classical theory. ... In quantum physics, the Heisenberg uncertainty principle, sometimes called the Heisenberg indeterminacy principle, expresses a limitation on accuracy of (nearly) simultaneous measurement of observables such as the position and the momentum of a particle. ... 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... The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. ... On a finite-dimensional inner product space, a self-adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose. ... On a finite-dimensional inner product space, a self-adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose. ...

Searching for ζ-function zeroes

Absolute value of the ζ-function

There is a long history of computational attempts to explore as many zeroes of the ζ-function as possible. One notable such attempt was ZetaGrid, a distributed computing project, which checked over a billion zeros a day when it was running. The project was shut down in November 2005. As of 2006, no computational project has succeeded in finding a counterexample to the Riemann hypothesis. Image File history File links Download high-resolution version (1000x1000, 23 KB) Summary Map of the Riemann zeta function absolute value. ... Image File history File links Download high-resolution version (1000x1000, 23 KB) Summary Map of the Riemann zeta function absolute value. ... There is a long history of computational attempts to explore as many zeroes of the Riemann ζ-function as possible. ... Distributed computing is a method of computer processing in which different parts of a program run simultaneously on two or more computers that are communicating with each other over a network. ... 2006 is a common year starting on Sunday of the Gregorian calendar. ...

In 2004, Xavier Gourdon and Patrick Demichel verified the Riemann hypothesis through the first ten trillion non-trivial zeros using the Odlyzko-Schönhage algorithm.

Michael Rubinstein has made public an algorithm for generating the zeros. Michael Oded Rubinstein is a Canadian mathematician at the University of Waterloo who works in number theory, computational mathematics and random matrix theory. ...

References

1. ^ Devlin, Keith J. (2002), The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time, Basic Books, ISBN 0-465-01729-0.
2. ^ Helge von Koch, "Sur la distribution des nombres premiers", Acta Mathematica 24 (1901), pp. 159–182.
3. ^ M.Riesz, "Sur l'hypothèse de Riemann", Acta Mathematica, 40 (1916) pp185-190.
4. ^ Luis Báez-Duarte, "A New Necessary and Sufficient Condition for the Riemann Hypothesis" (2003) ArXiv math.NT/0307215
5. ^ Luis Báez-Duarte, "A sequential Riesz-like criterion for the Riemann hypothesis", Internation Journal of Mathematics and Mathematical Sciences, 21, pp. 3527-3537 (2005)
6. ^ Philippe Flajolet and Brigitte Vallée, "Continued fractions, comparison algorithms and fine structure constants", In Micheal Théra, Constructive, Experimental and Nonlinear Analysis volume 27 of Canadian Mathematical Society Conference Proceedings (2000) pp.53-82 AMS, Providence RI
7. ^ Philippe Flajolet and Linas Vepstas, "On differences of zeta values", ArXiv math.CA/0611332
8. ^ Proposed proofs of the Riemann Hypothesis (2007-07-18).

Brigitte Vallée is a French mathematician and computer scientist, professor at Université de Caen[1], specializing in computational number theory[2] and analysis of algorithms. ...

Historical references

• Bernhard Riemann, Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse, (1859) Monatsberichte der Berliner Akademie. (This site provides both a facsimile of the original manuscripts, as well as English translations.)
• Jacques Hadamard, Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques, Bulletin Société Mathématique de France 14 (1896) pp 199-220.

Modern technical references

• H. M. Edwards, Riemann's Zeta Function, Academic Press, 1974. (Reprinted by Dover Publications, 2001 ISBN 0-486-41740-9)
• E. C. Titchmarsh, The Theory of the Riemann Zeta Function, second revised (Heath-Brown) edition, Oxford University Press, 1986
• Jeffrey Lagarias (2002). "An Elementary Problem Equivalent to the Riemann Hypothesis". American Mathematical Monthly 109: 534-543.  (A relationship in terms of Harmonic numbers.)
• (no author credited), Computation of zeros of the Zeta function (2004). (Reviews the GUE hypothesis, provides an extensive bibliography as well).
• Schoenfeld, Lowell. "Sharper bounds for the Chebyshev functions θ(x) and ψ(x). II." Mathematics of Computation 30 (1976), no. 134, 337--360.
• Conrey, J. Brian. "the Riemann Hypothesis" Notices of the American Mathematical Society, March 2003, 341-353. Available free http://www.ams.org/notices/200303/fea-conrey-web.pdf

Edward Charles (Ted) Titchmarsh (born 1 June 1899 in Newbury died 18 January 1963 at Oxford) was a leading British mathematician. ...

Popular References

• Clay Mathematics Institute, Millennium Problems, (2000) (Announcement of the million dollar rewards for solutions to famous problems in mathematics)
• Marcus du Sautoy, The Music of the Primes, HarperCollins, 2003
• Marcus du Sautoy, "Prime Numbers Get Hitched", Seed Magazine" (03/27/2006)
• Daniel Rockmore, Stalking the Riemann Hypothesis : The Quest to Find the Hidden Law of Prime Numbers, Pantheon Books, New York, 2005. ISBN 0-375-42136-X.
• John Derbyshire, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, Joseph Henry Press (April 23, 2003), ISBN 0-309-08549-7. 448 page book at a non-specialist level, can be read online for free.
• Zetagrid (2002) A distributed computing project that attempted to disprove Riemann's hypothesis; closed in November 2005.
• Ron Howard, A Beautiful Mind (2001). In the film, the mathematician John Nash attempts to prove the Riemann Hypothesis.
• Law & Order: Criminal Intent: In season 2, episode 2, "Bright Boy", the detective played by Vincent D'Onofrio tricks a father into confessing to a murder by telling him his son has solved the Riemann Hypothesis.
• Numb3rs: In season 1 episode 5, "Prime Suspect", (CBS) Criminals kidnap a mathematician's daughter and demand his allegedly complete proof of Riemann's Hypothesis as ransom.
• Ed Pegg, Jr., Ten Trillion Zeta Zeros, (2004) Math Games website. A discussion of Xavier Gourdon's calculation of the first ten trillion non-trivial zeros.
• de Vries, The Graph of the Riemann Zeta function ζ(s) (2004). A simple animated java applet.
• Erica Klarreich, "Prime Time", New Scientist - November 11, 2000, p. 32. A simple introduction to the Riemann Hypothesis, and its connection to prime numbers and quantum systems.
• QEDen A wiki dedicated to solving the millennium problems
• Perplex City, an Alternate Reality Game soliciting answers to various puzzles from its players, is offering 60 points to any player to submit a proof of the Riemann Hypothesis on card 238 of Season 1, entitled "Riemann".
• Proof, a film starring Gwynneth Paltrow and Anthony Hopkins, in which a character is purported to have solved a problem that sounds like the Riemann Hypothesis.
• In the novel PopCo the main character's grandmother used to spend all her free time trying to prove the Riemann Hypothesis.
• In Thomas Pynchon's novel Against The Day the Riemann Hypothesis and the question of non-trivial zeros are recurring tropes and preoccupations of the students and spies gathered at Göttingen University in the years preceding World War I.
• Karl Sabbagh, The Riemann Hypothesis: the greatest unsolved problem in mathematics, (2003) Farrar, Straus and Giroux, ISBN 0-374-25007-3. Also (2004) First American paperback edition. Conversations with mathematicians working on the problem. The reader should not expect to learn anything about the problem itself.
• Dirk L. van Krimpen, Proving the Riemann Hypothesis and other simple things, (2007) The main story of this bundle of math oriented science fiction stories shows the internal geometry of the zeta function in various pictures. From this geometry rules can be deducted proving that indeed for every possible non-trivial zero the real part cannot be anything else than 1/2.

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Cited References

• Bollobas, Bela, foreword to Littlewood's Miscellany, Cambridge University Press, 1986