👨🏽‍🎤 👩🏻‍🤝‍👨🏿 🐯 Riemann hypothesis. Prime distribution 🤟🏼 ⛸️ 🧚

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

The only task that turned out to be among both lists of puzzles that have been haunting scientists for centuries has been the Riemann hypothesis. She is still waiting for her decision.

Brief curriculum vitae

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

Prime numbers

Mathematics appeared when a person learned to count. At the same time, the first ideas about numbers arose, which they later tried to classify. It has been observed that some of them have common properties. In particular, among natural numbers, that is, those that were used in counting (numbering) or designating the number of objects, a group of those that were divided only by unity and by themselves was distinguished. They were called simple. An elegant proof of the theorem of infinity of the set of such numbers was given by Euclid in his Principles. 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 infinity of the set of primes, Euclid also defined the second theorem on the only possible 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 the first Euclidean infinity theorem in the form of the formula below.

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

Riemann Zeta Function

Euler's formula, upon closer examination, is absolutely amazing, since it defines the relation between primes and integers. Indeed, in its left part infinitely many expressions are multiplied, depending only on simple ones, and on 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 defining a formula for both real and complex variables. It was she who later received the name of the Riemann zeta function. In 1859, the scientist published an article entitled "On the number of primes that do not exceed a given value", where he summarized all his ideas.

Riemann proposed the use of the Euler series, converging for any real s> 1. If the same formula is used for complex s, then the series will converge for any value of this variable with the real part greater than 1. Riemann applied the procedure of analytic continuation, expanding 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 logical question arises: what is interesting and important is the zeta function, which is key in the work of Riemann on the null hypothesis? As you know, at the moment there is no simple pattern that would describe the distribution of primes among natural numbers. Riemann was able to find that the number pi (x) of primes that did not exceed x is expressed through the distribution of non-trivial zeros of the zeta function. Moreover, the Riemann hypothesis is a necessary condition for proving temporary estimates of the operation of some cryptographic algorithms.

Riemann hypothesis

One of the first formulations of this mathematical problem, which has not been proved to this day, is as follows: non-trivial 0 zeta functions are complex numbers with the 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 statement, but for generalizations of zeta functions, which are commonly called Dirichlet L-functions (see photo below).

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

The Riemann assertion is considered the so-called null hypothesis, since it was checked for consistency with existing sample data.

As Riemann reasoned

The remark of the German mathematician was initially formulated quite casually. The fact is that at that time the scientist was going to prove a theorem on the distribution of primes, and in this context this hypothesis did not have much significance. However, its role in solving many other issues is enormous. That is why Riemann’s assumption is currently recognized by many scientists 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 enough to justify logically that the real part of any non-trivial zero of the zeta function is in the range from 0 to 1. From this property it follows that the sum over all 0th The zeta functions that appear in the exact formula given above are a finite constant. For large values of x, it can generally be lost. The only term in the formula that remains unchanged even at very large x is x itself. Other complex terms in comparison with it disappear asymptotically. Thus, the weighted sum tends to x. This circumstance can be considered a confirmation of the truth of the theorem on the distribution of primes. Thus, the zeros of the Riemann zeta function have a special role. It consists in proving that such values cannot make a significant contribution to the decomposition formula.

Followers of Riemann

The tragic death from tuberculosis did not allow this scientist to bring his program to its logical conclusion. However, he took the baton from Sh. de la Valle Poussin and Jacques Hadamard. Independently of each other, they derived a theorem on the distribution of primes. Hadamard and Poussin managed to prove that all non-trivial 0 zeta functions are within the critical band.

Thanks to the work of these scientists, a new direction in mathematics has appeared - analytical number theory. Later, other researchers obtained several more primitive proofs of the theorem on which Riemann worked. In particular, Pal Erdös and Atle Selberg even discovered a very complex logical chain confirming it, which did not require the use of complex analysis. However, at this point, through the idea of Riemann, several important theorems had already been proved, including the approximation of many functions of number theory. In this regard, the new work of Erds and Atle Selberg practically did not affect anything.

One of the simplest and most beautiful evidence of the problem was found in 1980 by Donald Newman. It was based on the famous Cauchy theorem.

Does the Riemann 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 area of digital cryptography, which is engaged in the development of encryption algorithms.

Prime and “semi-simple” numbers, that is, those that are divisible only by 2 other numbers from the same class, underlie the public key system known as RSA. It has the widest application. In particular, it is used when generating an electronic signature. Speaking in terms available to “dummies,” the Riemann hypothesis states the existence of a system in the distribution of primes. Thus, the strength of the cryptographic keys, on which the security of online transactions in the field of electronic commerce depends, is significantly reduced.

Riemann hypothesis. Prime distribution