Riemann hypothesis

Unsolved problem in mathematics:

Is the real part of a non-trivial zero of the Riemann zeta function always ½?

(more unsolved problems in mathematics)

In mathematics, the Riemann hypothesis (also called the Riemann zeta-hypothesis), first formulated by Bernhard Riemann in 1859, is one of the most famous unsolved problems. 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.

The Riemann hypothesis 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 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.

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 polar 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 is one of the most important open problems of contemporary mathematics; a $1,000,000 prize has been offered by the Clay Mathematics Institute for a proof. 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.)

History

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, so 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. He said of the problem: "If I were to awaken after having slept for five hundred years, my first question would be: Has the Riemann hypothesis been proven?" 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 and Helge von Koch proved in 1901 that the Riemann hypothesis is equivalent to the following considerable strengthening of the prime number theorem: there exists a constant C such that

$\left|\pi (x)-\int _{2}^{x}{\frac {\mathrm {d} t}{\ln(t)}}\right|\leq C\,{\sqrt {x}}\,\ln(x)$

for all sufficiently large x, where π(x) is the prime-counting function, and ln(x) is the natural logarithm of x. By a strengthening due to Lowell Schoenfeld, we may take C = 1/(8π) for all x ≥ 2657.

The zeroes 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^1/2 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

$M(x)=\sum _{n\leq x}\mu (n)$

then the claim that

$M(x)=O(x^{e})\,$

for every exponent

e > ½

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

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

(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

$\sigma (n)\leq H_{n}+\ln(H_{n})e^{H_{n}}$

for every natural number n, where $H_{n}$ is the harmonic number.

Other functions, such as the Riesz function, have conjectured rates of growth equivalent to the Riemann hypothesis as well.

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

$\sum _{i=1}^{m}|F_{n}(i)-i/m|=O(n^{e})$

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, and e>½. Similarly, equivalent to the Riemann hypothesis is

$\sum _{i=1}^{m}(F_{n}(i)-i/m)^{2}=O(n^{e}),$

where now

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, 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 $\zeta '(s)$ has no zeros in the strip

$0<\Re (s)<{\frac {1}{2}}$.

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 $0<\Re (s)\leq {\frac {1}{2}}$. This approach has been fruitful; refining it allowed Norman Levinson to prove his strengthening of the critical line theorem.

Li's criterion

Li's criterion is a statement that the positivity of a certain sequence of numbers is equivalent to the Riemann hypothesis.

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

$\sum _{n=1}^{M}n^{-s}$

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.

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,

$\zeta \left({\frac {1}{2}}+it\right)=O(t^{e}),$

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,

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.

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

Attempted proofs of the Riemann hypothesis

The French-American mathematician Louis de Branges de Bourcia of Purdue University is known for his insistance in trying to prove the Riemann Hypothesis, as well as for his past failed attempts that were announced prematurely [1], and therefore received with scepticism from the mathematical community. In June 2004 he published another such purported proof in his website[2], which has been nevertheless ignored by the mathematical community. As of 2006, the papers (his proof and two ancillary manuscripts [3][4]) are the only standing attempt at a proof of the Riemann Hypothesis made by a serious professional. It must be noted that while premature claims have been a constant in de Branges' career, his most important result yet, the solution of the long-standing Bieberbach conjecture, was also preceded by several false claims on his part.

Matthew Watkins has a collection of proposed proofs [5].

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.

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 however. As of 2006, 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

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).
• Karl Sabbagh, The Riemann Hypothesis: the greatest unsolved problem in mathematics, (2003) Farrar, Straus and Giroux, ISBN 0374250073. Also (2004) First American paperback edition.
• Schoenfeld, Lowell. "Sharper bounds for the Chebyshev functions θ(x) and ψ(x). II." Mathematics of Computation 30 (1976), no. 134, 337--360.

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 037542136X.
• John Derbyshire, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, Joseph Henry Press (April 23, 2003), ISBN 0309085497. (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 January 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 an episode, 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) (In one episode, Charlie Epps (David Krumholtz) determines that a mathematician's child was kidnapped because he apparently solved the Riemann Hypothesis and wanted the million dollar prize 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