The Riemann hypothesis. Distribution of prime numbers

In 1900, one of the greatest scientists of the last century, David Hilbert compiled a list consisting of 23 unsolved problems in mathematical science. Work on them had a tremendous impact on the development of this area of human knowledge. After 100 years, the Clay Mathematical Institute presented a list of 7 problems known as the Millennium Challenge. For the decision of each of them the prize in 1 million dollars has been offered.

The only problem that turned out to be among both lists of puzzles, has been for more than one century of unsuspecting scientists, has become the Riemann hypothesis. She is still waiting for her decision.

Brief biographical note

Georg Friedrich Bernhard Riemann was born in 1826 in Hanover, in the large family of a poor pastor, and lived only 39 years. He managed to publish 10 works. However, already during his lifetime Riemann was considered the successor of his teacher Johann Gauss. At the age of 25, the young scientist defended the thesis "Foundations of the theory of functions of a complex variable". Later he formulated his own hypothesis, which became famous.

Prime numbers

Mathematics appeared when a person learned to count. At the same time, the first ideas about numbers appeared, which later were attempted to classify. It was noted that some of them have common properties. In particular, among the natural numbers, that is, those that were used to calculate (number) or designate the number of objects, a group of those that were divided only into unity and into themselves was singled out. They were called simple. An elegant proof of the theorem of infinity of the set of such numbers was given by Euclid in his "Beginnings". At the moment, their search continues. In particular, the largest of the already known is the number 2 ^{74 207 281} - 1.

Euler's formula

Along with the notion of the infinity of the set of prime numbers, Euclid also defined the second theorem on the only possible prime factorization. According to it, any positive integer is the product of only one set of primes. In 1737 the great German mathematician Leonard Euler expressed Euclid's first theorem on infinity in the form of the formula presented below.

It is called the zeta function, where s is a constant, and p takes all the simple values. From it directly follows and Euclid's statement about the uniqueness of the decomposition.

The Riemann zeta function

The Euler formula on closer inspection is quite surprising, since it sets the ratio between simple and integer numbers. After all, in its left part infinitely many expressions are multiplied, depending only on simple ones, and the right is the sum associated with all positive integers.

Riemann went further than Euler. In order to find the key to the problem of the distribution of numbers, he proposed to determine the formula for both the real and complex variables. It was later called the Riemann zeta function. In 1859, the scientist published an article under the title "On the number of prime numbers that do not exceed a given value", where he summarized all his ideas.

Riemann proposed using the Euler series, convergent for any real s> 1. If the same formula is applied to complex s, then the series will converge for any values of this variable with the real part greater than 1. Riemann applied the procedure of analytic continuation, extending the definition of zeta (s) to all complex numbers, but "throwing out" the unit. It was excluded, because for s = 1 the zeta function increases to infinity.

Practical meaning

A natural question arises: what is interesting and important is the zeta function, which is key in Riemann's work on the null hypothesis? As is known, at the moment there is no simple pattern that would describe the distribution of prime numbers among natural numbers. Riemann succeeded in discovering that the number pi (x) of prime numbers that did not exceed x is expressed by the distribution of the nontrivial zeros of the zeta function. Moreover, the Riemann hypothesis is a necessary condition for proving time estimates of the operation of some cryptographic algorithms.

The Riemann hypothesis

One of the first formulations of this mathematical problem, which has not been proven to this day, sounds like this: non-trivial 0 zeta functions are complex numbers with real part equal to ½. In other words, they are located on the line Re s = ½.

There is also a generalized Riemann hypothesis, which is the same assertion, but for generalizations of zeta-functions, which are usually called Dirichlet L-functions (see photo below).

In the formula, χ (n) is some numerical character (modulo k).

The Riemannian statement is considered the so-called null hypothesis, since it has been checked for consistency with the already available sample data.

As Riemann reasoned

The remark of the German mathematician was originally formulated rather casually. The fact is that at that time the scientist was going to prove the theorem on the distribution of prime numbers, and in this context this hypothesis had no special significance. However, its role in solving many other issues is enormous. This is why the assumption of Riemann at the moment by many scientists is recognized as the most important of unproven mathematical problems.

As already mentioned, to prove the distribution theorem, the complete Riemann hypothesis is not needed, and it is sufficient to logically justify that the real part of any nontrivial zero of the zeta function lies in the interval from 0 to 1. It follows from this property that the sum over all 0-m Zeta functions, which appear in the exact formula given above, is a finite constant. For large values of x, it can be lost altogether. The only member of the formula that remains unchanged even for very large x is x itself. The remaining composite terms asymptotically vanish in comparison with it. Thus, the weighted sum tends to x. This circumstance can be considered as a confirmation of the truth of the theorem on the distribution of prime numbers. Thus, the zeros of the Riemann zeta-function have a special role. It consists in proving that such values can not make a significant contribution to the expansion formula.

Followers of Riemann

The tragic death from tuberculosis did not allow this scientist to bring his program to its logical conclusion. However, he was taken over from the battalion of Sh. De la Vallee Poussin and Jacques Hadamard. Regardless of each other, they derived a theorem on the distribution of prime numbers. Hadamard and Poussin succeeded in proving that all non-trivial 0 zeta functions are within the critical band.

Thanks to the work of these scientists, a new direction in mathematics - analytic number theory - emerged. Later, other researchers obtained somewhat more primitive proofs of the theorem over which Riemann worked. In particular, Pal Erdes and Atle Selberg discovered even a very complex logical chain that confirmed it, which did not require the use of complex analysis. However, by this time, several important theorems have already been proved by means of the Riemann idea, including the approximation of many functions of number theory. In this regard, the new work of Erdos and Atle Selberg practically did not affect anything.

One of the simplest and most beautiful proofs of the problem was found in 1980 by Donald Newman. It was based on the well-known theorem of Cauchy.

Does the Riemannian hypothesis threaten the basics of modern cryptography?

Data encryption arose with the advent of hieroglyphs, more precisely, they themselves can be considered the first codes. At the moment there is a whole line of digital cryptography, which is developing encryption algorithms.

Simple and "semisimple" numbers, that is, those that divide only by 2 other numbers from the same class, underpin a public-key system known as RSA. It has the widest application. In particular, it is used when generating an electronic signature. If to speak in terms accessible to "teapots", the Riemann hypothesis asserts the existence of a system in the distribution of prime numbers. Thus, the robustness of cryptographic keys, on which the security of online transactions in the field of e-commerce depends, is significantly reduced.

Other unresolved mathematical problems

Finish the article worthwhile by dedicating a few words to other tasks of the millennium. These include:

• Equality of classes P and NP. The problem is formulated as follows: if a positive answer to a particular question is checked for polynomial time, is it true that the answer itself to this question can be found quickly?
• Hodge hypothesis. In simple words, it can be formulated as follows: for some types of projective algebraic varieties (spaces), Hodge cycles are combinations of objects that have a geometric interpretation, that is, algebraic cycles.
• The Poincaré conjecture. This is the only one of the millennium's problems that have been proven to date. According to it, any 3-dimensional object that has specific properties of a 3-dimensional sphere must be a sphere up to a deformation.
• The assertion of the quantum Yang-Mills theory. It is required to prove that the quantum theory advanced by these scientists for the space R ⁴ exists and has the 0-th mass defect for any simple gauge compact group G.
• The Birch-Swinnerton-Dyer conjecture. This is another problem related to cryptography. It concerns elliptic curves.
• The problem of the existence and smoothness of solutions of the Navier-Stokes equations.

Now you know the Riemann hypothesis. In simple words, we also formulated some of the other tasks of the millennium. The fact that they will be resolved or it will be proved that they do not have a solution is a matter of time. And it is unlikely that this will have to wait too long, as mathematics increasingly uses the computing capabilities of computers. However, not everything is subject to technology, and for solving scientific problems, intuition and creativity are first of all required.