Mathematicians attended Roger Appery’s lecture at the French National Center for Scientific Research conference in June 1978 with great skepticism. This presentation was titled “On the irrationality of ξ(3)” and caused a great deal of discussion among experts.
The value of the zeta function ζ(3) has been an open question for more than 200 years. The genius Swiss mathematician Leonhard Euler tried his best to solve this problem, but was unable to solve it. Now, the French mathematician Appéry, then in his 60s and relatively unknown, claimed to have solved this centuries-old mystery. Many in the audience had doubts.
Appery’s lecture did not improve their opinion. He spoke in French, occasionally joked, and omitted important explanations related to the proof. For example, first he wrote down an equation that no one present knew but was the core of his proof. When asked where this equation came from, Appery is said to have replied, “It grows in my garden,” which allegedly caused many in the audience to get up and walk out of the room.
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However, someone in attendance had an electronic calculator, a rarity at the time, and used a short program to check Appelli’s equation and ensure that it was correct. With this, Appery once again captured the attention of the audience. “Appery’s incredible proof seems like a mixture of miracle and mystery,” wrote Alfred van der Poelten, a mathematician who attended the lecture.
It took several weeks for experts to understand and confirm the details of the proof. Appery didn’t really make their job easier. For example, at one meeting he started talking about the current state of French instead of focusing on mathematics. But about two months later, it became clear that Appelli had succeeded in what Euler had failed to do 200 years earlier. He was able to show that ζ(3) is an irrational number.
Connection with prime numbers
The history of the zeta function goes back a long way. In 1644, Italian mathematician Pietro Mengoli wondered what would happen if we added the reciprocals of all square numbers: 1 + 1⁄4 + 1⁄9 + … But he could not calculate the result. Other experts also failed in this task, including the famous Bernoulli family of scientists in Basel, Switzerland. In fact, it took another 90 years for another resident of that city, the then 27-year-old Euler, to find a solution to the so-called Basel Problem. Euler calculated that the infinite sum is π.2⁄6.
But Euler decided to concentrate on the more general problem at hand. He was interested in a whole class of problems, such as finding the sum of the reciprocals of third orders and raising numbers to the fourth power. To do this, Euler uses the so-called zeta function ζ(s), which includes an infinite sum.
The Basel problem is just one of many zeta functions, corresponding to the value ζ(2). Euler tried to find the following solution all Zeta function value. And I actually succeeded in calculating the even value result. s = 2k. in this case,
where p and q is an integer, so the answer is always an irrational number.
But Euler was unable to clarify how the results would change if: s is an odd number. Although he was able to calculate the result to one decimal place, he could not calculate the exact number. He could not determine whether the odd zeta function also assumed irrational values or whether the result could be expressed as a fraction.
In the years and decades that followed, the zeta function received a great deal of attention and became intertwined with one of the greatest mysteries in mathematics today. In the 19th century, German mathematician Bernhard Riemann did more than evaluate the zeta function of the natural numbers. s The same goes for complex numbers. A real number that can contain the square root of a negative number. In 1859, that change allowed him to express what later became known as the famous Riemann hypothesis. This can, in principle, be used to determine the distribution of prime numbers along a number line. The stakes are high in this mystery, because understanding prime numbers is not only essential for number theory, but also has applications in fields such as cryptography, which relies on generating prime numbers for secure encryption. Anyone who can solve the Riemann Hypothesis could win a $1 million prize.
Despite all this attention paid to the zeta function, no one has succeeded in determining the exact value of zeta(3). Moreover, it was not possible to find a formula that is generally valid for all odd values of the zeta function, as Euler succeeded for the even numbers. . Things got especially interesting after ζ(3) appeared in physics in the 20th century.
Riemann zeta function in physics
In the early 20th century, physicists discovered quantum mechanics. This is a radical theory that completely overturns our previous understanding of nature. Here the boundary between particles and waves becomes blurred. Certain values, such as energy, appear only in fragments (quantization), and the formulas of natural laws contain uncertainties that arise from the mathematics itself rather than from measurement errors.
In the 1940s, researchers were able to formulate a quantum theory of electromagnetism. Among other things, it specifies that a vacuum is never truly empty. Instead, it may contain short-lived particle-antiparticle pairs, a veritable fireworks-like show of matter that appears to be created out of thin air but quickly disappears again. there is.
If we want to describe an electrodynamic process such as the scattering of two electrons, we need to account for this constant flare-up of particles. This is because the transient particle-antiparticle pair can divert the electron’s trajectory. If we want to explain this effect, we can see that an infinite sum of cube reciprocals appears, ζ(3).
For physical calculations, it is sufficient to know the value of ζ(3) to the decimal place. But mathematicians wanted to know more about this number.
proof of aperi
Apéry was able to determine that ζ(3) is an irrational number, similar to the even-valued zeta function. His proof was based on a previously unknown series representation of ζ(3). This is a strange equation that he claims to have found in his garden.
This representation allowed him to use the condition of irrationality derived by the German mathematician Gustav Lejeune Dirichlet in the 19th century. It states that the number χ is irrational if there are an infinite number of integers. p and q Use different parts to ensure that the following inequality is satisfied.
here c δ represents a constant value. Although this formula looks complicated, it essentially means that χ can be easily approximated by a fraction, but there is no fraction corresponding to χ. Apéry succeeded in deriving this inequality for ζ(3). Since then, ξ(3) has been shown to be irrational.
In honor of this French mathematician’s work, the value of ζ(3) now bears his name and is known as Appery’s constant. However, this does not answer all questions. Experts still want a definite number for ζ(3) that can be expressed using known constants, similar to the case when ζ(2) = π.2For example /6. However, we are still far from this dream today.
This article was first published Wissenschaft spectrum Reprinted with permission.
