Riemann hypothesis
The Riemann Hypothesis Riemann hypothesis or ( after Bernhard Riemann ) is an assumption about the zeros of the Riemann zeta function. It says that all nontrivial zeros of the complex-valued function have real part ½. Whether the assumption is correct or not, is one of the most important unsolved problems of mathematics.
In 2000, the problem of the Clay Mathematics Institute ( CMI) in Cambridge (Massachusetts ) was placed on the list of the Millennium problems. The Institute in Massachusetts has offered a prize of one million U.S. dollars for a conclusive solution to the problem in the form of a mathematical proof.
- 4.1 Recent proof attempts
The Riemann zeta function
The Riemann zeta function is a complex-valued function defined for real parts by the following infinite sum:
The variable is a complex number.
One of the most important properties of the Riemann zeta function is their connection with the primes. It represents a relationship between complex analysis and number theory ago (see: analytic number theory) and is the starting point of the Riemann Hypothesis. The following expression, which dates back to Leonhard Euler ( 1748), illustrates the relationship is formulaic as
With an infinite product over all primes represents. The expression follows immediately from the theorem on the uniqueness of prime factorization and the summation formula for geometric series.
The function can be beyond the original domain of convergence of Euler's sum and product formula addition to the entire complex plane - with the exception of - clearly continue analytically. This gives a meromorphic function: At the point it has a simple pole.
Wherein the gamma function and the Bernoulli numbers.
Note: For the definition of the Bernoulli numbers used here applies:
Riemann Hypothesis
In the following, the Riemann zeta function is considered in analytic continuation. In this form, the zeta function has revealed so-called "trivial zeros ", which decreases from the set of poles of the gamma function to the set of poles of the brackets by lifting. This is the amount of the negative even numbers.
A central finding of Riemann in his famous work of 1859 was the finding that all possible non-trivial zeros in the so-called critical strip
Must be located.
The famous - and to this day neither refuted nor proven - said of Bernhard Riemann conjecture that all non-trivial zeros on the middle line
Lie.
Riemann came on his assumption in the study of the product of the zeta function with the gamma function
Which is at the interchange of with invariant, that is, it satisfies the functional equation:
Riemann himself used and thus received to all:
Is thus the straight line in the complex plane with the real part 1/2 in this reflection also invariant. Riemann himself writes about the zeros:
" [ ... ] And it is very likely that all roots are real. However thereof would be desirable for a rigorous proof; I indeß the exploration of the same provisionally released after some fleeting futile attempts at hand, since he seemed unnecessary for the next objective of my investigation. "
With " real roots " said Riemann that for a critical strip in the equation
Only for real, so be to resolve.
Importance
The non- trivial zeros and primes
This insight is crucial Riemann was the relationship between primes and the zeros of its zeta function. In his work he dealt with finding an analytical expression for the prime function. As a starting point for this he used the well-known formula
The relationship between primes and the zeta function underpins fundamental. This can be transformed by simply taking logarithms in the following expression:
About the integral
Riemann could now bring the expression into a closed mold. For this purpose he had with the number-theoretic function
, which thus, for each prime power, which is less than the fraction accumulated. A simple example would be
Moreover, a step function. A pure integral expression for is:
Riemann was a master of Fourier analysis and thus he succeeded to the next forming a milestone of analytic number theory. About an inverse Mellin transformation he deduced an analytical expression for:
With a. In the next steps of his work Riemann pointed to the representation of the product named after him, the Riemann function, which by itself
Defined way. This product presentation runs over all non-trivial zeros of the zeta function and has the form of a polynomial factored to infinity (similar to the factorization of the sine or cosine ).
This one wins without much effort in the truest sense a non-trivial expression for the second.
The last part of Riemann's work only deals throughout with the substitution of this second expression for the equation
Despite the difficult evaluation Riemann came to the result:
With the on the Möbius inversion inferred (using the Möbius function) connection between and, namely
A deep connection between primes and the zeros of the zeta function was created.
Note: with a numerical calculation of Riemann 's formula should the expression be in the sum replaced by, where the ( complex ) Integralexponentialfunktion called because not always true on the main branch of the complex logarithm in the evaluation of and thus the result would be distorted.
Conclusions
From the Riemann Hypothesis a remainder estimate in the prime number theorem, for example, follows ( Helge von Koch 1901):
Is
The logarithmic integral. The result of cooking is even equivalent to the Riemann Hypothesis. It can be written also
Is a constant, and a somewhat weaker form:
For arbitrary.
Many other results of analytic number theory, but also about the important in cryptography fast primality tests, can be proved only under the assumption of the Riemann hypothesis so far. Are the complex zeros of the zeta function, as Michael Berry wrote that the fluctuations around the rough logarithmic asymptotic distribution of primes, the prime number theorem describes the encoding. If we know the exact distribution, you can find them more precise statements about the probability of how many prime numbers are to be found in one area.
But the real reason that many mathematicians have been looking hard for a solution, is - apart from the fact that this is the last still-unproved statement of Riemann 's famous essay - that in this exceptionally perfect symmetry of an otherwise very chaotic function (eg B. universality theorem of Voronin: the zeta function can be any analytical nonzero function within a circle of radius 1/4 arbitrarily approximate ) probably hides the tip of the iceberg of a fundamental theory, such as behind the Fermatvermutung the parametrization of elliptic curves by modular functions hid, part of the Langlands program.
History
The Riemann Hypothesis was in 1859, mentioned only in passing by Bernhard Riemann in a famous work that laid the foundations of analytic number theory, because - as he wrote - for the immediate continuation of the investigation of his essay was not essential. He secured his conjecture from extensive numerical calculations of the zeros as Carl Ludwig Siegel found out in the 1930s in the study of Riemann's estate. 1903 Jørgen Pedersen Gram published numerical approximations for the first 15 in the critical region lying zeros. Support (but do not prove ) the Riemann Hypothesis, as well as all other zeros that were later found and the number of the early eighties of the 20th century crossed the 100 million mark. In 2001 it was shown by means of mainframes, that the first ten billion zeros of the complex zeta function satisfy all the Riemann Hypothesis, that is, they are all on the line with real part.
A further milestone in the numerical search presented the launched in August 2001, Zeta - Grid project dar. Using the method of distributed computing, which was attended by many thousands of Internet users were after three years found about 1 trillion zeros. The project has now been set.
The two French mathematician Gourdon and Demichel started with the process of Odlyzko and Schönhage in 2004, a new trial and had checked the first 10 trillion zeros in October 2004 without finding a counterexample. Although it is numerical methods on all invoices show this exactly and not approximately, that the investigated zeros on the critical line are.
Many famous mathematicians have attempted to translate the Riemann Hypothesis. Jacques Hadamard claimed in 1896 without further explanations in his work Sur la distribution of zeros de la fonction ζ ( s ) et ses Conséquences Arithmétiques, in which he proved the prime number theorem that the then recently deceased Stieltjes had proved the Riemann Hypothesis without the evidence publish. Stieltjes claimed in 1885 in an essay in the Compte Rendu of the Academie des Sciences to have proved a theorem on the asymptotic behavior of the Mertens function, from which the Riemann conjecture follows ( see below). The famous British mathematician Godfrey Harold Hardy wont send off a telegram before crossing the English Channel in bad weather, in which he claimed to have a proof found, following the example of Fermat, the traditional on the edge of a book for posterity, he would have for a proof which is unfortunately too long to find his guess on the edge of space. His colleague John Edensor Littlewood was in Cambridge in 1906 as a student even the Riemann hypothesis as a function theoretical problem of his professor Ernest William Barnes asked, without connection to the distribution of prime numbers - this relationship had Littlewood discover for themselves and proved in his Fellowship dissertation that the prime number theorem from the following hypothesis, which was already known for some time in continental Europe. As he admitted in his book A mathematician 's miscellany, this threw a bad light on the contemporary state of mathematics in England. But Littlewood made soon made important contributions to analytic number theory in connection with the Riemann hypothesis. The problem was declared in 1900 by David Hilbert in his list of 23 mathematical problems than age-old problem, with Hilbert itself it to be less difficult than, for example sorted the Fermat problem: in a lecture in 1919, he expressed the hope that a proof too would his lifetime found in the case of Fermat 's conjecture perhaps during his lifetime young audiences, for the most difficult he held the transcendence evidence in his list of problems - a problem that was solved in the 1930s by Gelfond and Schneider. Meanwhile, many of the problems are solved Hilbert's list, but the Riemann conjecture resisted all attempts. Since there is no proof of the Riemann hypothesis was found in the 20th century, the Clay Mathematics Institute has this project again declared one of the most important mathematical problems in 2000 and offered a prize of one million U.S. dollars to a conclusive proof, but not for a counterexample.
There is also the Riemann conjecture analogous conjecture for other zeta functions, some of which are also well supported numerically. In the case of the zeta function of algebraic varieties (the case of the function body ) over the complex numbers, the presumption in the 1930s by Helmut Hasse for elliptic curves and in the 1940s by André Weil for abelian varieties and algebraic curves ( also over finite fields ) was proved. As formulated, the Weil conjectures, which include an analogue of the Riemann hypothesis for algebraic varieties (also of higher dimension as curves ) over finite fields. The proof has been furnished by the development of modern methods of algebraic geometry in the Grothendieck school in the 1970s by Pierre Deligne.
Newer proof attempts
In June 2004, Louis de Branges de Bourcia has published repeatedly alleged evidence that has been critically examined. Some years earlier, however, Eberhard Freitag a counterexample for an established claim in the proof shown, so that the proof is now seen as wrong.
Generalized Riemann conjecture
As a generalized or general Riemann conjecture the following statement is commonly referred to:
From the generalized Riemann conjecture follows the Riemann conjecture as a special case. Andrew Granville was able to show that the (strong ) Goldbach's conjecture is essentially equivalent to the generalized Riemann conjecture.
Related assumptions and equivalent formulations
In analytic number theory, there are other assumptions that are related to the Riemann Hypothesis. The Mertenssche conjecture states that for all. Here is the Möbius function and the so-called Mertens function. The Mertenssche assumption is stronger than the Riemann hypothesis was disproved in 1985.
In this context, there is also the probabilistic interpretation of the Riemann conjecture by Arnaud Denjoy. Be a random sequence of values ( 1, -1) (i.e., they have the same probability), then for each of the sum (using the Landau - symbols), that is, the amount of deviation from the mean to zero asymptotically grows most as strong as. If, for the Möbius function, then the Riemann hypothesis is equivalent to the statement that this asymptotic growth behavior and for their sum ( the Mertens function) applies ( Littlewood 1912). The Riemann hypothesis can then be interpreted as a statement that the distribution of the Möbius function ( that is, whether numbers without double prime factors of an even or odd number of prime factors have ) is completely random.
As mentioned above, follow from the Riemann hypothesis by Helge von Koch barriers for the growth of the error term of the prime number theorem. But the result of cooking is also equivalent to the Riemann hypothesis. from
Follows the Riemann hypothesis.
The Lindelöfsche guess about the order of the zeta function along the critical line is weaker than the Riemann Hypothesis, but still unproven.
Marcel Riesz was 1916, the equivalent to a conjecture about the asymptotic behavior of the Riesz function. Jerome Franel 1924 proved the equivalence to a statement of Farey series.
Jeffrey Lagarias presented in 1992 an equivalent to the Riemann conjecture conjecture of elementary number theory.
Proof ideas from physics
New ideas for the proof of the conjecture came from physics. Even David Hilbert and George Polya was noticed that the Riemann hypothesis would follow, if the zeros of eigenvalues of an operator (1/ 2 i T ) would be, where T is a Hermitian (ie, self-adjoint ) operator, so the only real eigenvalues , similar to the Hamiltonians in quantum mechanics. In the 1970s, then was Hugh Montgomery in a conversation with Freeman Dyson, that the distribution of the ( normalized ) distances of consecutive zeros of a similar distribution as the eigenvalues of unitary random matrices showed what Andrew Odlyzko confirmed by numerical calculations. In the 1990s, then began physicists like Michael Berry to look for such an underlying system, such as in the theory of quantum chaos. For additional assistance, these considerations into an analogy of the " explicit formulas " in the theory of Riemann zeta function with the Selberg trace formula relating the eigenvalues of the Laplace -Beltrami operator on a Riemann surface with the lengths of closed geodesics in relationship, and the Gutzwiller trace formula in quantum chaos theory. This connects the eigenvalues ( energies) of the quantum mechanical version of a classical chaotic system with the lengths of the periodic orbits in the classical case. In all of these trace formulas (trace formulas ) are identities between the sums of the respective zeros Trajectory period lengths, eigenvalues , etc.
One of the Fields Medals winners Alain Connes 1996 specified operator fits "almost". Previously Connes could not rule out the existence of additional zeros off the line but.
Another idea from physics, which was discussed in the context of the Riemann conjecture, the " Yang -Lee zeros " of the complex analytically continued partition function in models of statistical mechanics. Chen Ning Yang and Tsung- Dao Lee proved using a result of George Polya from the theory of the zeta function to which they drew attention Mark Kac, that in certain models of the zeros were on a circle with other models they lie on a straight line. The location of the zeros determines the behavior in phase transitions similar to the zeros on the critical line control the fine distribution of the primes.
All these ideas is an analogy based on the simplified can be described as follows: the primes are " elementary ", which enter via the multiplication in interaction and thus build up the composite numbers. Simultaneously, the "particles" are arranged by the addition. In the zeta function both aspects (additive / multiplicative natural numbers and / primes ) are now joined together in the form of a sum or product formula.
A compound of the Riemann conjecture for one-dimensional quasicrystals suggested Freeman Dyson 2009.