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Riemann hypothesis
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In mathematics, the Riemann hypothesis, due to , is a conjecture about the distribution of the zeros of the Riemann zeta-function stating that all non-trivial zeros of the Riemann zeta function have real part 1/2. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields.
The Riemann hypothesis implies results about the distribution of prime numbers that are in some ways best possible.
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In mathematics, the Riemann hypothesis, due to , is a conjecture about the distribution of the zeros of the Riemann zeta-function stating that all non-trivial zeros of the Riemann zeta function have real part 1/2. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields.
The Riemann hypothesis implies results about the distribution of prime numbers that are in some ways best possible. Along with suitable generalizations, it is considered by many mathematicians to be the most important unresolved problem in pure mathematics . It has since withstood concentrated efforts from many outstanding mathematicians, though Selberg's proof of the Riemann hypothesis for the Selberg zeta function of a Riemann surface, Deligne's proof of the Riemann hypothesis over finite fields, and extensive computer calculations verifying that the first 10 trillion zeros lie on the critical line, all suggest that it is probably true.
The Riemann zeta-function ?(s) 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 hypothesis is part of Problem 8, along with the Goldbach conjecture, in Hilbert's list of 23 unsolved problems, and is also one of the Clay Mathematics Institute Millennium Prize Problems.
There are several popular books on the Riemann hypothesis, such as , , ,
. The books , and give mathematical introductions, while
is an advanced monograph.
The Riemann zeta function
The Riemann zeta function is given for complex s with real part greater than 1 by
Euler showed that it is given by the Euler product
where the infinite product extends over all prime numbers p, and again converges for complex s with real part greater than 1. The convergence of the Euler product shows that ?(s) has no zeros in this region, as none of the factors have zeros.
The Riemann hypothesis discusses zeros outside the region of convergence of this series, so it needs to be analytically continued to all complex s. This can be done by expressing it in terms of the Dirichlet eta function as follows. If s has positive real part, then the zeta function satisfies
where the series on the right converges whenever s has positive real part (though if the real part is less than 1
the convergence is excruciatingly slow). Thus, this alternative series extends the zeta function from Re(s) > 1 to the larger domain Re(s) > 0.
In the strip 0 < Re(s) < 1 the zeta function satisfies the functional equation
We may then define ?(s) for all remaining complex numbers s by assuming that this equation holds outside the strip as well, and letting ?(s) equal the right-hand side of the equation whenever s has non-positive real part. If s is a negative even integer then
?(s) = 0 because the factor sin(ps/2) vanishes; these are the trivial zeros of the zeta function.
(If s is a positive even integer this argument does not apply because the zeros of sin are cancelled by the poles of the gamma function.) The functional equation also implies that the zeta function has no zeros other than the trivial zeros with negative real part, so all non-trivial zeros lie in the critical strip where s has real part between 0 and 1.
The zeros are arranged symmetrically around the critical line with real part 1/2: if 1/2+ß+i? is a zero then so are 1/2+ß−i?, 1/2 −ß+i?, and 1/2−ß−i?. Riemann checked that the first few zeros
are all on the critical line (with ß=0), and suggested that they all are; this is the Riemann hypothesis.
History
In his 1859 paper On the Number of Primes Less Than a Given Magnitude Riemann found an explicit formula for the number of primes p(x) less than a given number x. His formula was given in terms of the related function
which counts primes where a prime power pn counts as 1/n of a prime. The number of primes can be recovered from this function by
Riemann's formula is then
involving a sum over the non-trivial zeros ? of the Riemann zeta function. The sum is not absolutely convergent, but may be evaluated by taking the zeros in order of the absolute value of their imaginary part. The function Li occurring in the first term is the (unoffset) logarithmic integral function given by the principal value of the divergent integral
The terms Li(x?) involving the zeros of the zeta function need some care in their definition as Li has branch points at 0 and 1, and are defined by analytic continuation in the complex variable ρ in the region x>1 and Re(ρ)>0. The other terms also correspond to zeros: the dominant term Li(x) comes from the pole at s = 1, considered as a zero of multiplicity −1, and the remaining small terms come from the trivial zeros. This formula says that the zeros of the Riemann zeta function control the oscillations of primes around their "expected" positions. (For graphs of the sums of the first few terms of this series see .) 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. He checked that a few of the zeros lay on the critical line with real part 1/2 and suggested that they all do; this is the Riemann hypothesis.
and 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 proofs of the prime number theorem.
Consequences of the Riemann hypothesis
The practical uses of the Riemann hypothesis include many propositions which
are known to be true under the Riemann hypothesis, and some which can be
shown to be equivalent to the Riemann hypothesis.
Distribution of prime numbers
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, so the Riemann hypothesis says that these oscillations are as small as possible. proved that the Riemann hypothesis is equivalent to the "best possible" bound for the error of the prime number theorem; a precise version of his result, due to , says that the Riemann hypothesis is equivalent to
where p(x) is the prime-counting function and log(x) is the natural logarithm of x.
Growth of arithmetic functions
The Riemann hypothesis implies strong bounds on the growth of many other arithmetic functions, in addition to the primes counting function above.
One example involves the Möbius function µ. The statement that the equation
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
then the claim that
for every,
is equivalent to the Riemann hypothesis. (For the meaning of these symbols, see Big O notation.) This puts a rather tight bound on the growth of M, since the stronger Mertens conjecture
has been disproven .
The Mertens function is used in Denjoy's probabilistic argument for the Riemann hypothesis. If ?(x) is a random seequence of 1's and −1's then
-
satisfied the bound
for every,
,
with probability 1. In other words, the Riemann hypothesis is in some sense equivalent to saying that µ(x) behaves in certain ways like a random sequence of coin tosses. When µ(x) is non-zero its sign gives the parity of the number of prime factors of x,
so informally the Riemann hypothesis says that the parity of the number of prime factors of an integer behaves randomly. Unfortunately these sorts of informal probability arguments are often very hard to make rigorous.
The Riemann hypothesis is equivalent to many other conjectures about the rate of growth of other arithmetic function aside from µ(n). A typical example is Robin's theorem , which states that if s(n) is the divisor function, given by
then
for all n > 5040 if and only if the Riemann hypothesis is true.
Another example was found by showing that the Riemann hypothesis is equivalent to a statement that the terms of the Farey sequence are fairly regular. More precisely, if Fn is the Farey sequence of order n, beginning with 1/n and up to 1/1, then the claim that for all e > ½
is equivalent to the Riemann hypothesis. Here is the number of terms in the Farey sequence of order n.
For an example from group theory, if g(n) is Landau's function given by the maximal order of elements of the symmetric group Sn of degree n, then showed that the Riemann hypothesis is equivalent to the bound
for all sufficiently large n.
Riesz criterion
The Riesz criterion was given by , to the effect that the bound
holds for all if and only if the Riemann hypothesis holds.
Later (~1918) Hardy provided an integral equation for the left hand side using a variant of Borel resummation with Mellin transform.
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.
The derivative of the Riemann zeta function
proved that the Riemann hypothesis is equivalent to the statement that , the derivative of , has no zeros in the strip
That ? has only simple zeros on the critical line is equivalent to its derivative having no zeros on the critical line.
Lindelöf hypothesis and growth of the zeta function
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.
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
so the growth rate of ?(1+it) and its inverse would be known up to a factor of 2 .
Large prime gap conjecture
Another conjecture is the large prime gap conjecture. The prime number theorem implies that on average, the gap between the prime p and its successor is . Cramér proved that, assuming the Riemann hypothesis, the gap is . Cramér's conjecture implies that it is which is far smaller (and consistent with numerical evidence). Often the best bound for something that can currently be proved assuming the Riemann hypothesis is close to the best possible bound; this gives a case where this is not so.
Generalizations and analogues of the Riemann hypothesis
Dirichlet L-series and other number fields
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. It is these conjectures, 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.
The generalized Riemann hypothesis extends the Riemann hypothesis to all Dirichlet L-functions.
In particular it implies the conjecture that Siegel zeros (zeros of L functions between 1/2 and 1) do not exist.
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.
Function fields and zeta functions of varieties over finite fields
introduced global zeta-functions of (quadratic) function fields and conjectured an analogue of the Riemann hypothesis for them, which has been proven by Hasse in the genus 1 case and by in general. 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
is actually an instance of the Riemann hypothesis in the function field setting. This led to conjecture a similar statement for all algebraic varieties; the resulting Weil conjectures were proven by
Selberg zeta functions
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.
Montgomery's pair correlation conjecture
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. showed that this is supported by large scale numerical calculations of these correlation functions.
Attempts to prove the Riemann hypothesis
Several mathematicians have addressed the Riemann hypothesis, but none of their attempts have yet been accepted as correct solutions.
Operator theory
Hilbert and Polya suggested that one way to derive the Riemann hypothesis would be to find a self-adjoint operator, from the existence of which the statement on the real parts of the zeroes of ?(s) would follow when one applies the criterion on real eigenvalues.
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 of the classical Hamiltonian so that
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 and the natural numbers as spectrum of the quantum harmonic oscillator. The crucial point is that the Hamiltonian should be a self-adjoint operator so that the quantization would be a realization of the Hilbert–Pólya program.
The analogy with the Riemann hypothesis over finite fields suggests that the Hilbert space
containing eigenvectors corresponding to the zeros should be some sort of first cohomology group of the spectrum Spec(Z) of the integers. described some of the attempts to find such a cohomology theory.
constructed a natural space of invariant functions on the upper half plane which has eigenvalues under the Laplacian operator corresponding to zeros of the Riemann zeta function, and remarked that if one could show the existence of a suitable positive definite inner product on this space the Riemann hypothesis would follow.
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 .
Turán's result
Stronger statements than the Riemann hypothesis have also been formulated, but they have a tendency to be disproven. showed that if the functions
have no zeros when the real part of s is greater than one then
for all x>0
where ?(n) is the Liouville function given by (−1)r if n has r prime factors,
and showed that this in turn implies that the Riemann hypothesis is true. However Haselgrove proved that that this inequality is false for some x, and showed by numerical calculation that the finite Dirichlet series above for M=19 has a zero with real part greater than 1. Turán also showed that a somewhat weaker statement about the zeros of the finite Dirichlet series would also imply the Riemann hypothesis, but this statement was disproved by .
Non-commutative geometry
has described a relationship between the Riemann hypothesis and non-commutative geometry,
and shows that a suitable analogue of the Selberg trace formula for the action of the idèle class group on the adèle class space would imply the Riemann hypothesis.
Hilbert spaces of entire functions
De Branges showed that the Riemann hypothesis would follow from a positivity condition on a certain Hilbert space of entire functions.
However showed that the necessary positivity conditions are not satisfied.
Location of the zeros
Number of zeros
The functional equation combined with the argument principle implies that the number of zeros of the zeta function with imaginary part between 0 and T is given by
for s=1/2+iT, where the argument is defined by varying it continuously along the line with Im(s)=T,
starting with argument 0 at ∞+iT.
This is the sum of a large but well understood term
and a small but rather mysterious term
So the density of zeros with imaginary part near T is about log(T)/2π, and the function S describes the small deviations from this.
Selberg showed that S(T)2 behaves in some ways like log log T times a Gaussian random variable, so it is usually about this size, but occasionally much larger. The exact order of growth of S(T) is not known, but is suspected to be around log(T)1/2 possibly multipled by some power of log log T. It is very small as far as it has been calculated.
Zeros on the critical line
and showed there was an infinity of zeros on the critical line, by considering moments of certain functions related to the zeta function. proved that a (small) positive proportion of zeros lie on the line. improved this to one-third of the zeros by relating the zeros of the zeta function to those of its derivative, and improved this further to two-fifths.
Most zeros lie close to the critical line. More precisely, showed that for any positive e, all but an infinitely small proportion of zeros lie within a distance e of the critical line.
Zero-free regions
de la proved that if s+it is a zero of the Riemann zeta function, then 1-s = C/log(t) for some positive constant C. In other words zeros cannot be too close to the line s=1: there is a zero-free region close to this line. This zero-free region has been enlarged by several authors.
gave a version with explicit numerical constants: ?(s + it) ? 0 whenever |t| = 3 and
Numerical calculations
The function
- p−s/2G(s/2)?(s)
has the same zeros as the zeta function in the critical strip, and is real on the critical line, so one can prove the existence of zeros exactly on the real line between two points by checking numerically that the function has opposite signs at these points. By finding many intervals where the function changes sign one can show that there are many zeros on the critical line.
To verify the Riemann hypothesis up to a given imaginary part T of the zeros, one also has to check that there are no further zeros off the line in this region. This can be done by calculating the total number of zeros in the region and checking that it is the same as the number of zeros found on the line. The total number of zeros can be found either by calculating the function S(T) above, or by using a method due to Turing depending on the fact that S(T) has average value 0. This allows one to verify the Riemann hypothesis computationally up to any desired value of T (provided all the zeros of the zeta function in this region are simple and on the critical line).
stated that in unpublished notes dating from the 1850s Riemann used the Riemann-Siegel formula to compute rough approximations for the first 3 zeros.
The earliest published calculations are due to who used the Euler-Maclaurin summation formula to evaluate the zeta function found the first 15 zeros (with imaginary part less than 50). lie on the critical line with real part 1/2. He found that the imaginary part of the first few zeros are about
- 14.13, 21.02, 25.01, 30.42, 32.93, 37.58, ...
He showed that all zeros in this range lie on the critical line with real part 1/2 by computing the sum of the inverse 10th powers of the roots he found.
A Gram point is a value of t such that ?(1/2+it} is a non-zero real; these are easy to find because they are the points where the Euler factor at infinity p−s/2G(s/2) is real at s=1/2+it.
Gram observed that there was often exactly one zero of the zeta function between any two Gram points; Hutchinson called this observation Gram's law.
Gram's hand calculations were extended by who found the 79 zeros with T=200, and introduced a better method of checking all the zeros up to that point are on the line, by studying the argument of the zeta function.
found the 138 zeros with T=300, and found two small exception to Grams law. E. C. Titchmarsh (1935, 1041 zeros) used the more powerful Riemann-Siegel formula instead of the Euler-Maclaurin formula. After this, all calculations were done by computer. The first computer calculations were done by A. M. Turing (1953) who found a more efficient way to check that all zeros up to some point are accounted for by the zeros on the line and checked the first 1104 zeros.
extended the calculations further and discovered a few cases where the the zeta function has zeros
that are "only just" on the line: two zeros of the zeta function are so close together that it is unusually difficult to find a sign change between them. This is called "Lehmer's phenomenon", and first occurs at the Gram point g4763.
A Gram point t is called good if the zeta function is positive at 1/2 + it. A Gram block
consists of two good Gram points such that all the points between them are bad. A refinement of Grams law due to (who checked there were no exceptions in the first 3 million zeros, though there are infinitely many exceptions) says that most Gram blocks have the expected number of zeros in them, even though some of the
individual Gram intervals in the block may not have a unique point.
There are many other computer calculations of zeros of the zeta function. As of 2009, the largest calculation is by who verified the Riemann hypothesis through the first ten trillion non-trivial zeros using the Odlyzko-Schönhage algorithm.
and later Gourdon also computed smaller numbers of zeros of much larger heights, up to about 1024.
For tables of the zeros, see or .
Arguments for and against the Riemann hypothesis
Mathematical papers about the Riemann hypothesis tend to be cautiously noncommittal about its truth. Of authors who express an opinion, most of them, such as or , imply that they expect (or at least hope) that it is true. The few authors who express doubt about it include who lists some reasons for being skeptical, and who flatly states that he believes it to be false, and that there is no evidence whatever for it and no imaginable reason for it to be true.
Some of arguments for (or against) the Riemann hypothesis are listed by and , and include the following reasons (most discussed in more detail in the rest of this article).
- The proof of the Riemann hypothesis for varieties over finite fields by is possibly the single strongest theoretical reason in favor of the Riemann hypothesis. This provides some evidence for the more general conjecture that all zeta functions associated with automorphic forms satisfy a Riemann hypothesis, which includes the classical Riemann hypothesis as a special case.
- The numerical verification that many zeros lie on the line seems at first sight to be strong evidence for it. However analytic number theory has had many conjectures supported by large amounts of numerical evidence that turn out to be false; see Skewes number for a notorious example. The problem is that the behavior is often influenced by very slowly increasing functions such as log log T, that tend to infinity, but do so so slowly that this cannot be detected by computation. Such functions occur in the theory of the zeta function controlling the behavior of its zeros; for example the function S(T) above has average size around (log log T)1/2 . As it jumps by at least 2 at any counterexample to the Riemann hypothesis, one might expect any counterexamples to the Riemann hypothesis to start appearing only when it becomes large. It is never much more than 3 as far as it has been calculated, suggesting that calculations may not have yet reached the region of typical behavior of the zeta function.
- Debye's probabilistic argument shows that the Riemann hypothesis is equivalent to the Moebius function behaving like a random function. Such probabilistic arguments in number theory often give the right answer, but tend to be very hard to make rigorous, and occasionally give the wrong answer.
- Odlyzko's calculations show that the zeros of the zeta function behave very much like the eigenvalues of a random Hermitean matrix, suggesting that they are the eigenvalues of some self adjoint operator which would imply the Riemann hypothesis. However all attempts to find such an operator have failed.
- Lehner's phenomenon where two zeros are sometimes very close is sometimes given as a reason to disbelieve in the Riemann hypothesis. However one would expect this to happen occasionally just by chance even if the Riemann hypothesis were true, and Odlyzko's calculations suggest that nearby pairs of zeros occur just as often as predicted by Montgomery's conjecture.
External links
- American institute of mathematics,
- Poem about the Riemann hypothesis, by John Derbyshire.
- (Reviews the GUE hypothesis, provides an extensive bibliography as well).
- including and
- A discussion of Xavier Gourdon's calculation of the first ten trillion non-trivial zeros
- .*
- de Vries, (2004), a simple animated Java applet* (2002) A distributed computing project that attempted to disprove Riemann's hypothesis; closed in November 2005
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