Riemann hypothesis

The Riemann conjecture or Riemann hypothesis is one of the most important unsolved problems in mathematics. It was first formulated in 1859 by Bernhard Riemann in his paper Über die Anzahl der Primzahlen unter einer gegebenen Größe. After being included by David Hilbert on his list of 23 important century problems as early as 1900, it was added to the list of seven millennium problems in mathematics by the Clay Mathematics Institute in 2000. The institute in Cambridge (Massachusetts) thus offered a prize of one million US dollars for a conclusive solution of the problem in the form of a mathematical proof.

Simply put, the Riemann conjecture states that the sequence of prime numbers 2, 3, 5, 7, 11 ... behaves "as randomly as possible". This should manifest itself, for example, by the sequence of events that a number has an even number of prime factors, such as , or has an odd number of prime factors, such as , in the long run exhibits behavior that a frequently repeated coin toss with heads and tails might also have. A theory that would solve the Riemann conjecture and thus provide a deeper explanation for this randomness among primes could therefore entail a fundamentally new understanding of numbers in general from the mathematicians' point of view.

Translating this into the technical language of analytic number theory, the Riemann conjecture is equivalent to the statement that all complex zeros of the Riemann zeta function in the so-called critical strip have the real part ^1⁄2. It is already known and proved that the zeta function has real zeros (the so-called "trivial" zeros), as well as infinitely many nonreal zeros with real part ^1⁄2. Thus, Riemann's conjecture states that there are no other zeros beyond that, i.e., that all nontrivial zeros of the zeta function lie on a straight line in the number plane parallel to the imaginary axis.

The Riemann conjecture is very significant for modern mathematics. Many important proofs of a number of hitherto unsolved problems, especially from number theory, could be inferred from it. This concerns problems from basic research, such as those of the distribution of prime numbers in the setting of the prime number theorem or the open Goldbach conjecture, as well as applied mathematics, such as fast prime number tests. At the same time, it is also considered extremely difficult to prove. One reason for this is that, from an expert point of view, mankind does not yet have the necessary mathematical tools to be able to attack it at all. Previous attempts at proof by prominent mathematicians have all failed.

Through extensive use of computers, it has been possible to verify the Riemann conjecture for the first 10 trillion zeros of the zeta function. However, since it can be shown that there are infinitely many non-real zeros with real part ^1⁄2, it could only be disproved in this way by giving an explicit counterexample, but not proved. A counterexample would be a non-real zero in the critical strip with real part not equal to ^1⁄2.

Introduction

Prime numbers

At the centre of number theory, that branch of mathematics which deals with the properties of the natural numbers 1, 2, 3, 4 ... , are the prime numbers 2, 3, 5, 7, 11 ... . These are distinguished by the property of having exactly two divisors, namely 1 and themselves. The 1 is not a prime number. Already Euclid could show that there are infinitely many prime numbers, which is why the list 2, 3, 5, 7, 11 ... will never end. His result is called Euclid's theorem.

The prime numbers are, in a sense, the atoms of the integers, since every positive integer can be uniquely decomposed into such multiplicatively. For example, 21 = 3 - 7 and 110 = 2 - 5 - 11. Despite this elementary property, after several millennia of mathematical history, there is still no known simple pattern to which the prime numbers conform in their sequence. Their nature is one of the most important open questions in mathematics.

The prime number theorem

Even if the detailed understanding of the sequence 2, 3, 5, 7, 11 ... of prime numbers is considered to be unattainably distant, one can look for patterns if one broadens one's view. This is helped, for example, by the idea that statistical methods can often be used to describe the behaviour of very large numbers of people (for example, in terms of consumption and voting behaviour) with surprising precision, even though a single person is extremely complex. Roughly speaking, this has to do with the fact that larger and larger relevant data sets provide increasingly reliable information. In the case of prime numbers, such an expansion leads, among other things, to the question of how many primes there are below a fixed chosen number.

For example, only 4 prime numbers, namely 2, 3, 5 and 7, are smaller than the number 10. In the case of 50, there are already 15 smaller prime numbers, namely

One question in number theory is whether there is a universal and simple principle for at least estimating how many prime numbers there are under a given bound. Such a principle was first recognized in 1792/93 by the then 15-year-old Carl Friedrich Gauss, after he had studied logarithm tables. Gauss conjectured that the number of all primes from 2 to a large number x is approximately equal to the area between the x-axis and the function in the interval from 2 to x. Here the natural logarithm. Thus the approximation

Number of primes to x

The integral to the right cannot be calculated elementary closed, because the reciprocal logarithm has no elementary root function. It thus defines an "independent" function, also known as the integral logarithm. Gauss did not present a mathematical proof of his conjecture, and it was over 100 years before such a proof was provided - independently by Jacques Hadamard and Charles-Jean de La Vallée Poussin - in 1895. Here, proof does not mean that all conceivable values have been tried through, which is impossible with infinitely many numbers, but that a logical argument based on the mathematical axioms proves the facts in full generality. The theorem thus demonstrated is still called the prime number theorem.

The approximation given in the prime number theorem provides quite good values. For example, below the number 73 893 there are exactly 7293 prime numbers, and the following applies

The prime number theorem captures the average behavior of prime number distances. One interpretation of its statement is that a random number between 2 and a very large n with approximate probability is a prime number. However, it does not give arbitrarily detailed information about the prime sequence.

Riemann's ideas

Original work from 1859

In 1859, in gratitude for his admission to the Berlin Academy of Sciences, Bernhard Riemann wrote a 9-page paper that was later to lay the foundations for modern analytic number theory. His work aimed at proving and further deepening Gauss's conjecture on the prime number theorem. However, since the essay was extremely sketchy and many of the statements made in it were not rigorously proved, it was to be some time before mathematicians accepted the assertions made there. To this day, all of Riemann's statements in his paper, with the exception of the Riemann conjecture formulated there in a subordinate clause, are considered proven.

The Riemann zeta function

One possible tool for proving this formula is the Riemannzeta function. Here it is exploited that it expresses the law of unique prime factorization in the language of analysis. So the properties of the prime numbers are stored hidden in this function. The decisive characteristics, which allow conclusions about the prime numbers, are the zeros of the zeta function, i.e. all points, at which it takes the value 0. These generate a correction term of the above formula, which converts it into an exact expression. So this resulting exact formula knows the distribution of the prime numbers to the last detail. However, this does not mean that the questions about prime numbers have been solved: the computational effort increases very strongly with increasing values and thus practical calculations with this formula are not effective. In contrast, modern prime number tests are more suitable for numerical research. However, the exact formula is of theoretical interest: namely, it conceals the error gap between the simple prediction and the actual prime number distribution. It is assumed that this error is as small as possible (within the spectrum of all possibilities). Within the exact formula, which is to output the number of primes under the number , terms with summed up, where ρ the zeros. Now, if the real part of ρ becomes larger, this also increases the size of which would have the consequence that the distance between the estimate of the prime set and the actual distribution would also be larger. It can be shown that the real part of infinitely many values of ρ equal to , so the error will in any case have a minimum magnitude of . However, the Riemann conjecture now states that there are no other zeros that behave differently from those previously known in the critical strip.

Deciphering the error is not relevant to numerics. Rather, pure mathematics strives to learn the hitherto hidden reason why the error (if true) turns out to be as small as possible. Mathematicians hope for a fundamental insight into the nature of numbers behind the formal justification of this regularity.

The Riemann zeta function

The Riemann zeta function is a complex-valued function which, for a complex number with a real part by the infinite sum

is defined.

One of the most important properties of the Riemann zeta function is its connection with prime numbers. It establishes a relationship between complex analysis and number theory (see analytic number theory) and forms the starting point of the Riemann conjecture. The following expression, which goes back to Leonhard Euler (1748), represents the connection formulaically as

where is an infinite product over all primes . The expression follows immediately from the theorem on the uniqueness of prime decomposition and the summation formula for the geometric series.

The function can be uniquely analytically continued beyond the original convergence region of Euler's sum or product formula to the entire complex plane - except for . One obtains a meromorphic function

where Γ is the gamma function and are the Bernoulli numbers. At the point it has a simple pole. The other singularities of this representation are all liftable because the whole function has a simple zero at each of these points .