Dedicated to the memory of John Forbes Nash Jr.

You do remember what “simple numbers” are? These numbers are not divided into any other than themselves and 1. And now I ask a question that is already 3000 years old:

- 2, 3, 5, 7, 11, 13, 17, 19, 23, 29,
**p**. What is**p**? 31. What will be the following**p**? 37. And the following**p**? 41. And the following? 43. Yes, but ... how do we know what the next value will be?

Come up with a judgment or a formula that (at least with a sin in half) predicts what the next prime number will be (in any given series of numbers) and your name will forever be associated with one of the greatest achievements of the human brain. You will be on a par with Newton, Einstein and Gödel. Understand the behavior of prime numbers, and then you can rest on your laurels all your life.

The properties of primes have been studied by many great people in the history of mathematics. From the first proof of the infinity of Euclidean primes to the formula for the product of Euler, which relates primes to the zeta function. From the formulation of the Gauss and Legendre prime numbers theorem to its proof, invented by Hadamard and Valle-Poussin. Nevertheless, Bernhard Riemann is still considered the mathematician who made the single largest discovery in the theory of prime numbers. In his article, published in 1859, which consisted of only eight pages, new, previously unknown discoveries about the distribution of prime numbers were made. This article is still considered one of the most important in number theory.

After publication, the Riemann article remained the main work in the theory of primes and in fact became the main reason for the proof in 1896 of the

Let's start with the simple. We all know that a number is either

Euclidean theorem

Suppose that the set of primes is not infinite. Create a list of all prime numbers. Then let P be the product of all the primes of the list (multiply all the primes from the list). We add to the result 1: Q = P +1. Like all numbers, this Q number must be either simple or compound:

- If Q is simple, then we find a prime number that is not in our "list of all prime numbers."
- If Q is not simple, then it is composite, i.e. composed of prime numbers, one of which, p, will be a divisor of Q (because all composite numbers are products of prime). Each prime p from which P is composed is obviously a divisor of P. If p is a divider for both P and Q, then it must be a divisor for their difference, that is, units. No single number is a divisor of 1, so the number p cannot be in the list — another contradiction to the fact that the list contains all prime numbers. There will always be another simple p that is not in the list and is a divisor of Q. Consequently, there are infinitely many primes.

The fact that any newcomer understands the above problem, eloquently speaks of its complexity. Even the arithmetic properties of primes, despite the active study, are poorly understood by us. The scientific community is so confident in our inability to understand the behavior of primes that factoring large numbers (defining two primes whose product is a number) remains one of the fundamental foundations of encryption theory. You can look at it as follows:

We understand compound numbers well. These are all non-prime numbers. They consist of prime numbers, but we can easily write a formula that predicts and/or generates composite numbers. Such a “composite number filter” is called a

One of the functions that seriously simplifies our work will be 6n ± 1. This simple function returns all prime numbers, with the exception of 2 and 3, and removes all numbers that are multiples of 3, as well as all even numbers. We substitute n = 1,2,3,4,5,6,7 and we get the following results: 5,7,11,13,17,19,23,25,29,31,35,37,41,43. The only non-prime numbers generated by the function are 25 and 35, which can be factorized by 5 x 5 and 5 x 7. The next non-simple numbers, as you might guess, will be 49 = 7 x 7, 55 = 5 x 11, and so on. Everything is easy, right?

To visualize this, I used what I call the “ladder of composite numbers” —a convenient way to show how the composite numbers generated by the function are located and combined. In the first three columns of the image below, we see how primitive 5, 7, and 11 climb each ladder of composite numbers, up to 91. Chaos occurs in the fourth column, which shows how the sieve removed everything except prime numbers — excellent an illustration of why primes are so hard to understand.

So how does this all relate to a concept that you could hear about - the “Riemann hypothesis”? Well, to put it simply, in order to understand more about primes, mathematicians in the 19th century stopped trying to predict the location of primes with absolute precision, and instead began to consider the phenomenon of primes in general. Riemann became the master of this

Harmonic rows are infinite rows of numbers that were first explored by Nikolai Orem in the 14th century. His name is associated with the concept of musical harmonics - overtones that are higher than the frequency of the fundamental tone. The rows are as follows:

Orem proved that this sum is non-convergent (that is, it does not have a finite limit; it does not approach and does not tend to any particular number, but aims at infinity).

Harmonic series are a special case of a more general type of function called

If we substitute n = 1, then we get a harmonic series that diverges. However, for all values of n & gt; The first row

The first connection between zeta functions and prime numbers was established by Euler, when he showed that for two natural (integer and more than zero) numbers

This expression first appeared in a 1737 article entitled

Proving a formula is one of my favorite proofs, so I’ll present it, even though it’s not necessary for our purposes (but it’s just as beautiful!):

Euler starts with a common zeta function.

First he multiplies both parts by the second term:

Then he subtracts the resulting expression from the zeta function:

He repeats this process, further multiplying both sides by the third member

And then subtracts the resulting expression from the zeta function

If you repeat this process ad infinitum, in the end we will have the expression:

If this process is familiar to you, it is because Euler essentially created a sieve, very similar to that of Eratosthenes. It filters out non-simple numbers from the zeta function.

Then we divide the expression into all its members, which are the inverse of prime numbers and we get:

Simplifying the expression, we showed the following:

Was it not beautiful? Substitute s = 1 and find the infinite harmonic series, re-proving the infinity of primes.

August Ferdinand Möbius rewrote the product of Euler, creating a new amount. In addition to the quantities inverse to prime numbers, the Möbius function also contains every positive integer that is the product of an even and an odd number of prime factors. The numbers excluded from its series are those numbers that are divisible by some prime number in the square. Its sum, denoted as

The sum contains the inverse:

- Every prime number;
- Each natural number that is a product of an odd number of different prime numbers, taken with a minus sign; and
- Each natural number that is a product of an even number of different prime numbers, taken with a plus sign;

The first members are shown below:

The sum does not contain those reciprocals, which are divided by the square of one of the primes, for example, 4.8.9, and so on.

The Möbius function

Although for the first time this tricky amount was first formally defined by Möbius, it is noteworthy that 30 years before him he wrote about this amount in notes on the Gauss fields:

“The sum of all primitive roots (a prime number p) or≡0 (when p-1 is divided by a square), or± 1 (mod p ) (when p-1 is the product of unequal primes); if their number is even, then the sign is positive, but if the number is odd, then the sign is negative. "

Let's go back to prime numbers. To understand how prime numbers are distributed when moving up a numerical line, without knowing exactly where they are, it will be helpful to calculate how many they are up to a certain number.

It is this task that the distribution function of primes π (x) proposed by Gauss performs: it gives us the number of primes less than or equal to a given real number. Since we do not know the formulas for finding prime numbers, we only know the distribution formula for prime numbers as a graph, or

The theorem on the distribution of prime numbers, formulated by Gauss (and Legendre independently of him), says:

In ordinary language, this can be stated as follows: “When x moves to infinity, the distribution function of primes π (x) will approach the function x/ln (x)”. In other words, if you get far enough, and the graph of the distribution of primes rises to a very high number

From the point of view of probabilities, the theorem on the distribution of prime numbers says that if you randomly choose a positive integer x, then the probability P (x) that this number is prime is approximately equal to 1/ln (x). This means that the average gap between consecutive primes among the first

The function Li (x) is defined for all positive real numbers, except x = 1. It is given by the integral from

By plotting this function next to the distribution function of primes and the formula from the theorem on the distribution of primes, we see that Li (x) is actually a better approximation than x/ln (x):

To find out how much better this approximation is, we can build a table with large values of x, the number of primes to x and the magnitude of the error between the old (theorem on the distribution of prime numbers) and the new (integral logarithm) functions:

As can be easily seen, the integral logarithm is much better in approximation than the function from the theorem on the distribution of prime numbers; it “made a mistake” upward by only 314,890 primes for x = 10 to the power of 14. However, both functions converge to prime distribution functions π (x). Li (x) converges much faster, but as

The gamma function Γ (z) has become an important object of study since when, in the 1720s, Daniel Bernoulli and Christian Goldbach investigated the problem of generalizing the factorial function to non-integer arguments. This is a generalization of the factorial function

Her schedule is very curious:

The gamma function Γ (z) is defined for all

So that the gamma function Γ (z) can be used, it is usually rewritten as

With this equation, we can get values for z below zero. However, it does not give values for negative integers, because they are not defined (formally, they are degeneracies or simple poles).

The connection between the zeta function and the gamma function is given by the following integral:

After reviewing all the necessary fundamental resources, we can finally begin to establish a connection between prime numbers and the Riemann hypothesis.

German mathematician Bernhard Riemann was born in 1826 in Brezelents. As a student of Gauss, Riemann published work in the field of mathematical analysis and geometry. It is believed that he made the greatest contribution in the field of differential geometry, where he laid the foundation for the geometry language, later used by Einstein in the general theory of relativity.

His only work in number theory, the 1859 paper

- The definition of the Riemann zeta function ζ (s) is a zeta function with complex values;
- Analytical continuation of the zeta function to all complex numbers s ≠ 1;
- The definition of the x-function of the Riemann ξ (s) is an entire function associated with the zeta function of the Riemann through the gamma function;
- Two proofs of the functional equation of the Riemann zeta function;
- Determining the distribution function of Riemann prime numbers J (x) using the prime distribution function and the Möbius function;
- Explicit formula for the number of primes less than a given number using the distribution function of Riemann prime numbers, defined using nontrivial zeros of the Riemann zeta function.

This is an incredible example of ingenuity and creative thinking, the likes of which have probably not been seen since. Totally awesome work.

We have seen the close connection between primes and the zeta function shown by Euler in his work. However, with the exception of this connection, little was known about their relationship, and in order to show them, it took the invention of complex numbers.

Riemann was the first to consider the zeta function ζ (s) for the complex variable

This infinite series, called the Riemann zeta function ζ (s), is analytic (that is, it has definable values) for all complex numbers with a real part greater than 1 (Re (s) & gt; 1). In this domain, it converges

To analyze a function in domains outside the usual domain of convergence (when the real part of the complex variable

This new definition of the zeta function is analytic in any part of the half-plane Re (s) & gt; 0, except for s = 1, where it is a degeneracy/simple pole. In this domain, it is called an

In his article, Riemann did not stop at that. He switched to analytic continuation of his zeta function ζ (s) to

His method uses the integral representation of the gamma Γ (z) for complex variables and the Jacobi theta function ϑ (x), which can be rewritten in such a way that the zeta function appears. Deciding on zeta, we get:

In this form, we notice that the term ψ (s) decreases faster than any power of x, which means that the integral converges to all values of s.

Going further still, Riemann noticed that the first term in the brackets (-1/s (1 - s)) is an invariant (does not change) if s is replaced by 1 - s. Due to this, Riemann expanded the utility of the equation even more, eliminating two poles at s = 0 and s = 1, and setting the Riemann xi function ξ (s) without degeneracies:

The roots/zeros of the zeta function, when ζ (s) = 0, can be divided into two types, which are called “trivial” and “non-trivial” zeros of the Riemann zeta function.

Trivial zeros are zeros that are easy to find and explain. They are most noticeable in the following functional form of the zeta function:

This product becomes zero when the sine becomes zero. This occurs at kπ values. That is, with a negative even integer

So, the Riemann zeta function has zeros in every negative even integer s = -2n. These are trivial zeros, and you can see them on the graph of the function:

From the formulation of the Euler zeta, we can instantly see that the zeta ζ (s) cannot be zero in a domain with a real part of

We have found the trivial zetas of the zeta in the negative half-plane, when Re (s) & lt; 0, and showed that in the region of Re (s) & gt; 1 cannot be zeros.

However, the area between these two areas, called the critical band, has been the main focus of analytic number theory for the last hundreds of years.

On the graph shown above, I displayed the real parts of zeta ζ (s) in red, and the imaginary parts in blue. We see the first two trivial zeros in the lower left corner, where the real part of

We defined the Riemann x-function ξ (s) (the kind of functional equation in which all degeneracies are eliminated, that is, it is defined for all values of s) like this:

This function satisfies the ratio

This means that the function is symmetric about the vertical line: Re (

Looking at the two graphs shown above, you can immediately notice that for all non-trivial zeros of the Riemann zeta function ζ (

For non-trivial zeros of the Riemann zeta function ζ (s), the real part has the form Re (s) = 1/2.

This is the modern formulation of an unproved assumption made by Riemann in his famous article. It says that all points where zeta is zero ((s) = 0) on the critical band 0 ≤ Re (s) ≤ 1 have a real part Re (s) = 1/2. If this is true, then all non-trivial zeros of zetas will have the form ζ (1/2 +

The equivalent formulation (stated by Riemann himself) is that all the roots of the Riemann x-function ξ (s) are real.

In the graph below, the line Re (s) = 1/2 is the horizontal axis. The real part of Re (

If the Riemann hypothesis turns out to be true, then all non-trivial zeros of the function will occur on this line as intersections of two graphs.

There are many reasons to believe the truth of the Riemann hypothesis regarding the zeros of the zeta function. Probably the most persuasive for mathematicians are the consequences that it will have for the distribution of prime numbers. Numerical testing of the hypothesis at very high values suggests that it is true. In fact, the numerical confirmation of the hypothesis is so strong that in other fields, for example, in physics or chemistry, it could be considered experimentally proven. However, in the history of mathematics there were several hypotheses, tested to very high values, and yet proved to be incorrect.Derbyshire (2004) tells the story of a Scuse number — an extremely large number that indicated an upper limit and thus proved one of the Gauss hypotheses to prove that the integral logarithm of Li (

Taking the truth of the Riemann hypothesis as a basis, Riemann himself began to explore its consequences. In his article, he wrote:

Recall the prime distribution function π (

The first thing that can be noticed in this function is that it is not infinite. With some member, the distribution function will be zero, because there are no primes for

Like the prime distribution function, the Riemann function J (

To associate the value of J (

Recall that the possible values of the Möbius function are

This means that now we can write any distribution function of primes as a distribution function of Riemann prime numbers, which will give us

This new expression is still a finite amount, because J (

If we now look at the example of J (100) again, we get the sum

Which, as we know, is the number of prime numbers below 100.

Riemann then used Euler’s product as a starting point and obtained a method for analytically estimating primes in an incalculable language of mathematical analysis. Starting with Euler:

First, taking the logarithm from both sides and then rewriting the denominators in brackets, he derived the relationship

Then, taking advantage of the well-known Taylor-Maclaurin series, he expanded each logarithmic term on the right-hand side, creating an infinite sum of infinite sums, one for each term in a series of prime numbers.

Consider one such member, for example:

This member, like every other member of the calculation, represents a part of the area under the function J (

In other words, using Euler’s work, Riemann showed that one can represent a discrete stepwise distribution function of prime numbers as a continuous sum of integrals. On the graph below, an example of a member taken by us is shown as part of the area under the graph of the distribution function of Riemann prime numbers.

Thus, each expression in a finite sum that makes up a series of quantities inverse to prime numbers from Euler’s work can be expressed as integrals that make up an infinite sum of integrals corresponding to the area under the distribution function of Riemann prime numbers. For prime 3, this infinite product of integrals has the form:

If we collect all these infinite sums together into one integral, then the integral under the distribution function of Riemann prime numbers J (

Or in a more well-known form

Thanks to this, Riemann managed to link his zeta function ζ (

Having obtained this analytical form of Euler's work, Riemann set about formulating his own theorem on the distribution of prime numbers. He presented it in the following explicit form:

This is the explicit Riemann formula. It became an improvement of the theorem on the distribution of prime numbers, a more accurate estimate of the number of prime numbers up to the number

- The first, or "main" term, is the integral logarithm of Li (
*x*), which is an improved approximation of the distribution function of primes π (*x*) from the theorem on the distribution of simple numbers This is the largest term, and as we have seen, it overestimates the number of primes to a given value of*x*. - The second, or "periodic" term is the sum of the integral logarithm of x to the power of
*ρ*, summarized by*ρ*, which is nontrivial zeros of the Riemann zeta function. This member regulates the overestimation of the values of the primary member. - The third member is the constant -log (2) = -0.6993147 ...
- The fourth and last term is an integral that is zero for
*x*& lt; 2, because there are no primes less than 2. Its maximum value is 2 when its integral is approximately 0.1400101 ....

The influence of the last two terms on the value of the function with increasing

In the graph shown above, I approximated the distribution function of primes π (

Below is the same graph using more non-trivial zeros.

Using the explicit Riemann function, it is possible to approximate the number of prime numbers up to a given number

"... These zeros act like telegraph poles, and the special nature of the Riemann's zeta function exactly orders how the wire (its graph) must hang between them ...", - Dan Rockmore

After the death of Riemann in 1866, just at the age of 39, his pioneering article continues to be a reference point in the field of analytic number theory and the theory of prime numbers. To this day, Riemann's hypothesis about the non-trivial zeros of the Riemann zeta function remains unsolved, despite active research by many great mathematicians. Each year, various new results and conjectures connected with this hypothesis are published in the hope that someday the proof will become real.

Source text: [Translation] Affordable explanation of the Riemann hypothesis