Riemann hypothesis

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

This shows the values of ζ(1/2+it) in the complex plane for 0 ≤ t ≤ 34. (For t=0, ζ(1/2) ≈ -1.460 corresponds to the leftmost point of the red curve.)

Millennium Prize Problems
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The Poincaré conjecture
The Riemann hypothesis
Yang–Mills existence and mass gap
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The Riemann hypothesis (also called the Riemann zeta-hypothesis), along with suitable generalizations, is considered by many mathematicians to be the most important unresolved problem in pure mathematics^[1]. First formulated by Bernhard Riemann in 1859, it has withstood concentrated efforts from many outstanding mathematicians for almost 150 years (as of 2008).

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:

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, where t is a real number and i is 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.

The Riemann hypothesis implies a large body of other important results.^[2] Most mathematicians believe the Riemann hypothesis to be true.^[3] A $1,000,000 prize has been offered by the Clay Mathematics Institute for the first correct proof.^[4]

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.

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.

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.

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.

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

$\left|\pi(x) - \int_0^x \frac{\mathrm{d}t}{\ln(t)}\right| = \mathcal{O}(x^{1/2+\varepsilon}),$

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.^[5] A non-asymptotic version, due to Lowell Schoenfeld, says that the Riemann hypothesis is equivalent to

$\left|\pi(x) - \int_0^x \frac{\mathrm{d}t}{\ln(t)}\right| < \frac{1}{8\pi} \sqrt{x} \, \ln(x), \qquad \text{for all } x \ge 2657.$

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.

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.

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

√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

$\frac{1}{\zeta(s)} = \sum_{n=1}^\infty \frac{\mu(n)}{n^s}$

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

$M(x) = \sum_{n \le x} \mu(n)$

then the claim that

$M(x) = \mathcal{O}(x^{1/2+\varepsilon}) \,$

for every,

$\varepsilon > 0, \,$

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

$M(x) \ne o(x^\frac{1}{2}).$

(For the meaning of these symbols, see Big O notation.)

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

$\sigma(n) = \sum_{d\mid n} d \,$

then

$\sigma(n) < e^\gamma n \ln \ln n \,$

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

$\sigma(n) \le H_n + \ln(H_n)e^{H_n}$

for every natural number n, where $H_n$ is the n-th harmonic number.

Riesz criterion, binomial sums

The Riesz criterion was given by Marcel Riesz in 1916, to the effect that the relation

$-\sum_{k=1}^\infty \frac{(-x)^k}{(k-1)!\,\zeta(2k)}= \mathcal{O}\left(x^{1/4+\epsilon}\right)$

holds for all $\epsilon>0$ if and only if the RH holds.^[6]

Later (~1918) Hardy provided an integral equation for

$-\sum_{k=1}^\infty \frac{(-x)^k}{(k-1)!\,\zeta(2k)}=f(x)$ 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

$c_k = \sum_{j=0}^k (-1)^j {k \choose j} \frac {1}{\zeta(2j+2)}.$

Báez-Duarte^[7]^[8] and Flajolet and Vallée^[9] have shown that RH holds if and only if

$c_k \ll k^{-3/4+\epsilon}$

for all $\epsilon>0$ . Similarly, let

$d_k = \sum_{j=2}^k (-1)^j {k \choose j} \frac {1}{\zeta(j)}$

then Flajolet and Vepstas show^[10] that RH holds if and only if

$|d_n| < C_\epsilon n^{1/2+\epsilon}$

for all $\epsilon>0$ and some constant $C_\epsilon$ depending on $\epsilon$ . Entering into the proof is the Möbius function $\mu(n)$ , and so similar results hold for binomial sums over $\zeta(s-1)/\zeta(s)$ , $\zeta^\prime(s)/\zeta(s)$ and so on, which correspond to Dirichlet series for Euler's totient function, the divisor function, and so on.

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.

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

$\sum_{i=1}^m|F_n(i) - i/m| = \mathcal{O}(n^e)$

is equivalent to the Riemann hypothesis. Here $m = \sum_{i=1}^n\phi(i)$ is the number of terms in the Farey sequence of order n. Similarly equivalent to the Riemann hypothesis is

$\sum_{i=1}^m(F_n(i) - i/m)^2 = \mathcal{O}(n^e),$

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

$\ln g(n) < \sqrt{\operatorname{Li}^{-1}(n)}.$

Critical line theorem

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

$0 < \Re(s) < \frac12.$

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 $0 < \Re(s) \le \frac12$ . This approach has been fruitful; refining it allowed Norman Levinson to prove his strengthening of the critical line theorem.

Disproven conjectures

Stronger conjectures than the Riemann hypothesis have also been formulated, but they have a tendency to be disproven. Pál Turán claimed that if the sums

$\sum_{n=1}^M n^{-s}$

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.

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 ε > 0,

$\zeta\left(\frac12 + it\right) = \mathcal{O}(t^\varepsilon),$

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 ε > 0,

$p_{n+1}-p_n<p^{1/2+\epsilon}\,$

if n is sufficiently large. However, this result is worse than that of the large prime gap conjecture, stated below.

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 $\mathcal{O}(\sqrt{p} \ln p)$ . On average, the gap is merely $\mathcal{O}(\ln p)$ 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.

Attempts to prove the Riemann hypothesis

Several mathematicians have addressed the Riemann hypothesis over decades, and a few purported proofs go unverified as of 2007^[update]. 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).^[11] A few others may be found on the arXiv preprint server.

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.

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 1999, Michael Berry and Jon Keating conjectured that there is some unknown quantization $\hat H$ of the classical Hamiltonian $H=xp$ so that

$\zeta (1/2+i\hat H) = 0$

and even more strongly, that the Riemann zeros coincide with the spectrum of the operator $1/2 + i \hat H$ . 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.

Lee-Yang theorem

Main article: Lee-Yang theorem

The Lee-Yang theorem states that the zeros of certain partition functions in statistical mechanics all lie on a "critical line" with real part 0, and this has led to some speculation about a relationship with the Riemann hypothesis (Knauf 1999).

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 2008^[update], no computational project has succeeded in finding a counterexample to the Riemann hypothesis.

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.

References

↑ Bombieri, Enrico. "The Riemann Hypothesis - official problem description". Clay Mathematics Institute. Retrieved on 2008-10-25.
↑ The Extended Riemann Hypothesis and its Application to Computation, Jason Wojciechowski, 22nd January 2003: "... were the ERH to fall, a lot of mathematics would go with it."
↑ (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.)
↑ Devlin, Keith J. (2002), The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time, Basic Books, ISBN 0-465-01729-0 .
↑ Helge von Koch, "Sur la distribution des nombres premiers", Acta Mathematica 24 (1901), pp. 159–182.
↑ M.Riesz, «Sur l'hypothèse de Riemann», Acta Mathematica, 40 (1916) pp185-190.
↑ Luis Báez-Duarte, "A New Necessary and Sufficient Condition for the Riemann Hypothesis" (2003) ArXiv math.NT/0307215
↑ Luis Báez-Duarte, "A sequential Riesz-like criterion for the Riemann hypothesis", Internation Journal of Mathematics and Mathematical Sciences, 21, pp. 3527-3537 (2005)
↑ 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
↑ Philippe Flajolet and Linas Vepstas, "On differences of zeta values", ArXiv math.CA/0611332
↑ "Proposed proofs of the Riemann Hypothesis" (2007-07-18).

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)
Knauf, Andreas (1999), "Number theory, dynamical systems and statistical mechanics", Reviews in Mathematical Physics. A Journal for Both Review and Original Research Papers in the Field of Mathematical Physics 11 (8): 1027–1060, doi:10.1142/S0129055X99000325, MR1714352, ISSN 0129-055X
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. doi:10.2307/2695443. (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
A.A. Karatsuba, S.M. Voronin, The Riemann Zeta-Function, W. de Gruyter, Berlin, 1992.

Popular References

David Morgan-Mar, Irregular Webcomic #1960, A simple introduction to complex numbers leading to the Riemann hypothesis in the Annotation
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
A Beautiful Mind (2001, produced by Ron Howard). 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, (CBS), season 1 episode 5, "Prime Suspect", 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
PopCo, a novel wherein the main character's grandmother used to spend all her free time trying to prove the Riemann Hypothesis
Thomas Pynchon, the novel Against The Day, wherein 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

Cited References

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