In 1900, one of the greatest scientists of the last century, David Hilbert, compiled a list of 23 unsolved problems in mathematics. Work on them had a tremendous impact on the development of this area of human knowledge. 100 years later, the Clay Mathematical Institute presented a list of 7 problems known as the Millennium Problems. Each of them was offered a prize of $1 million.
The only problem that appeared among both lists of puzzles that have been haunting scientists for more than one century was the Riemann hypothesis. She is still waiting for her decision.
Short biographical note
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, already during his lifetime, Riemann was considered the successor of his teacher Johann Gauss. At the age of 25, the young scientist defended his dissertation "Fundamentals of the theory of functions of a complex variable." Later he formulatedhis famous hypothesis.
Prime numbers
Mathematics appeared when man learned to count. At the same time, the first ideas about numbers arose, which they later tried to classify. Some of them have been observed to have common properties. In particular, among the natural numbers, i.e., those that were used in counting (numbering) or designating the number of objects, a group was singled out that were divisible only by one and by themselves. They are called simple. An elegant proof of the theorem of infinity of the set of such numbers was given by Euclid in his Elements. At the moment, their search continues. In particular, the largest number already known is 274 207 281 – 1.
Euler formula
Along with the concept of the infinity of the set of primes, Euclid also determined the second theorem on the only possible decomposition into prime factors. According to it, any positive integer is the product of only one set of prime numbers. In 1737, the great German mathematician Leonhard Euler expressed Euclid's first infinity theorem as the formula below.
It's called the zeta function, where s is a constant and p takes all prime values. Euclid's statement about the uniqueness of the expansion directly followed from it.
Riemann Zeta Function
Euler's formula, on closer inspection, is completelysurprising because it defines the relationship between primes and integers. After all, infinitely many expressions that depend only on prime numbers are multiplied on its left side, and the sum associated with all positive integers is located on the right.
Riemann went further than Euler. In order to find the key to the problem of the distribution of numbers, he proposed to define a formula for both real and complex variables. It was she who subsequently received the name of the Riemann zeta function. In 1859, the scientist published an article en titled "On the number of prime numbers that do not exceed a given value", where he summarized all his ideas.
Riemann suggested using the Euler series, which converges 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 a real part greater than 1. Riemann applied the analytic continuation procedure, extending the definition of zeta(s) to all complex numbers, but "thrown out" the unit. It was excluded because at s=1 the zeta function increases to infinity.
Practical sense
A logical question arises: why is the zeta function, which is key in Riemann's work on the null hypothesis, interesting and important? As you know, at the moment no simple pattern has been identified that would describe the distribution of prime numbers among natural numbers. Riemann was able to discover that the number pi(x) of primes that did not exceed x is expressed in terms of the distribution of non-trivial zeros of the zeta function. Moreover, the Riemann hypothesis isa necessary condition for proving time estimates for the operation of some cryptographic algorithms.
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 statement, but for generalizations of zeta functions, which are commonly called Dirichlet L-functions (see photo below).
In the formula χ(n) - some numerical character (modulo k).
The Riemannian statement is considered the so-called null hypothesis, as it has been tested for consistency with existing sample data.
As Riemann argued
The remark of the German mathematician was originally formulated rather carelessly. 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 was of no particular importance. However, its role in solving many other issues is enormous. That is why Riemann's assumption is now recognized by many scientists as the most important of the unproven mathematical problems.
As already mentioned, the full Riemann hypothesis is not needed to prove the distribution theorem, and it is enough to justify logically that the real part of any non-trivial zero of the zeta function is inbetween 0 and 1. It follows from this property that the sum over all 0's of the zeta function that appears in the exact formula above is a finite constant. For large values of x, it may be lost altogether. The only member of the formula that remains the same even for very large x is x itself. The remaining complex terms vanish asymptotically in comparison with it. So the weighted sum tends to x. This circumstance can be considered 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 cannot make a significant contribution to the decomposition formula.
Followers of Riemann
Tragic death from tuberculosis did not allow this scientist to bring his program to its logical end. However, Sh-Zh took over from him. de la Vallée Poussin and Jacques Hadamard. Independently of each other, they deduced a theorem on the distribution of prime numbers. Hadamard and Poussin managed to prove that all nontrivial 0 zeta functions are within the critical band.
Thanks to the work of these scientists, a new direction in mathematics has appeared - the analytic theory of numbers. Later, several more primitive proofs of the theorem Riemann was working on were obtained by other researchers. In particular, Pal Erdős and Atle Selberg even discovered a very complex logical chain confirming it, which did not require the use of complex analysis. However, by this point, several importanttheorems, including approximations of many number theory functions. In this regard, the new work of Erdős 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 famous Cauchy theorem.
Does the Riemannian hypothesis threaten the foundations of modern cryptography
Data encryption arose along with the appearance 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 developing encryption algorithms.
Prime and "semi-prime" numbers, i.e. those that are only divisible 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 accessible to dummies, the Riemann hypothesis asserts the existence of a system in the distribution of prime numbers. Thus, the strength of cryptographic keys, on which the security of online transactions in the field of e-commerce depends, is significantly reduced.
Other unresolved math problems
It is worth finishing the article by devoting a few words to other millennium targets. These include:
- Equality of classes P and NP. The problem is formulated as follows: if a positive answer to a particular question is checked in polynomial time, then is it true that the answer to this question itselfcan be found quickly?
- Hodge's conjecture. 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, i.e., algebraic cycles.
- Poincaré's conjecture. This is the only Millennium Challenge that has been proven so far. According to it, any 3-dimensional object that has the specific properties of a 3-dimensional sphere must be a sphere up to deformation.
- Affirmation of the quantum theory of Yang - Mills. It is required to prove that the quantum theory put forward by these scientists for the space R 4 exists and has a 0th mass defect for any simple compact gauge group G.
- Birch-Swinnerton-Dyer hypothesis. This is another issue related to cryptography. It touches elliptic curves.
- The problem of the existence and smoothness of solutions to the Navier-Stokes equations.
Now you know the Riemann hypothesis. In simple terms, we have formulated some of the other Millennium Challenges. That they will be solved or it will be proved that they have no solution is a matter of time. And it is unlikely that this will have to wait too long, since mathematics is increasingly using the computing capabilities of computers. However, not everything is subject to technology, and first of all, intuition and creativity are required to solve scientific problems.
