Riemann hypothesis

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In mathematics, the Riemann hypothesis (also called the Riemann zeta hypothesis), first formulated by Bernhard Riemann in 1859, is one of the most famous of all 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 certain so-called "trivial" zeros for s = −2, s = −4, s = −6, ... 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.

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

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

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. He said of the problem: "If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann hypothesis been proven?".

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:

\pi (x)=\int _{2}^{x}{\frac  {{\mathrm  {d}}t}{\ln(t)}}+O\left({\sqrt  x}\,\ln(x)\right)\quad {{\rm {as}}}\quad x\rightarrow \infty

where, π(x) is the prime-counting function, ln(x) is the natural logarithm of x, and the O-notation is the Landau symbol.

The zeroes of the Riemann zeta function and the prime numbers satisfy a certain duality property, known as the explicit formulae which show that in the language of Fourier analysis the zeros of the zeta function can be regarded as the harmonic frequencies in the distribution of primes.

The Riemann hypothesis can be generalized in various ways by replacing the Riemann zeta function by the formally similar global L-functions. None of these generalizations has been proven or disproven. See generalized Riemann hypothesis.

Other consequences 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. Other formulations equivalent to the Riemann hypothesis involve 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)=\Omega (x^{{\frac  {1}{2}}})

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

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 (Guy Robin, 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.

Two other equivalent statements to the Riemann hypothesis involve the Farey sequence. If Fn 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.

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 Sn 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)}}.

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

0<\Re (s)<{\frac  12}.

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 ζ we can extend the zero-free region to 0<\Re (s)\leq {\frac  12}. This approach has been fruitful; refining it allowed Norman Levinson to prove his strengthening of the critical line theorem.

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.

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

\zeta \left({\frac  12}+it\right)=o(t^{e})

where

e > 0;

in other words

\left|\zeta \left({\frac  12}+it\right)\right|

grows more slowly than any positive exponent.

Another conjecture is the large prime gap conjecture; Cramér proved that on the Riemann hypothesis we have that the largest gaps between successive prime numbers 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

In June 2004, Louis De Branges de Bourcia of Purdue University, the same mathematician who solved the Bieberbach conjecture, claimed to have proved the Riemann hypothesis in an "Apology for the proof of the Riemann Hypothesis" [1]. He has in the past announced a proof a number of times, but all of his previous attempts at this proof have failed. [2] The proof's method has been tried before unsuccessfully. Linked is Conrey and Li's counterexample on the problems in the earlier version of his proof. [3]

The example involves a numerical calculation. The authors also give a non-numerical counterexample, due to Peter Sarnak. On the other hand, De Branges's successful proof of the Bieberbach conjecture was also preceded by his failed proofs of it.

Matthew Watkins has a collection of proposed proofs [4].

Possible connection with operator theory

See 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.

Legends

Number theorists as a rule have lived a long and productive life, thus it is said that the person who proves the Riemann hypothesis will actually become immortal. Others say that Riemann's hypothesis has been proved false several times already, and every time the unhappy mathematician was struck dead as soon as he had finished the proof.

Searching for ζ-function zeroes

There is a long history of computational attempts to explore as many zeroes of the ζ-function as possible. As of 2005, the largest of these is ZetaGrid, a distributed computing project, which checks over a billion zeros a day. So far, every single one of them has failed to be a counterexample to the Riemann hypothesis.

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 conséqunces arithmétiques, Bulliten Societé Mathematique 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
  • Template:Journal reference (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.

Popular literature

  • Clay Mathematics Institute, Millennium Problems, (2000) (Announcement of the million dollar rewards for solutions to famous problems in mathematics)
  • 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.)

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