Mathematicians attended Roger Apéry’s lecture at a conference of the French National Center for Scientific Research in June 1978 with great scepticism. The presentation was titled “On the Irrationality of ζ(3)” and it caused quite a stir among experts.
The value of the zeta function ζ(3) has been an open question for more than 200 years. The brilliant Swiss mathematician Leonhard Euler cut his teeth and didn’t fix it. Now the relatively unknown French mathematician Apéry, who was 60 years old at the time, claimed to have solved this centuries-old riddle. Many in the audience had doubts.
Apéry’s speech did not improve their opinion. He spoke French, made occasional jokes, and omitted crucial explanations relevant to the evidence. At first, for example, he wrote an equation that no one in the room knew but that formed the heart of his proof. When asked where this equation comes from, Apéry seems to have answered“They grow in my garden,” which presumably caused many in the audience to stand up and leave the room.
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But someone present had an electronic calculator—an unusual device at the time—and, with a short program, he checked Apéry’s equation and found it correct. With that, Apéry had the room’s attention again. “Apéry’s incredible proof appears to be a mixture of wonder and mystery,” wrote the mathematician Alfred van der Poortenwho went to the conference.
It took several weeks for the experts to understand and verify the details of the evidence. Apéry did not make it easy for them: in one meeting, for example, instead of dedicating it to mathematics, he started talking about the state of French. But after about two months, it became clear that Apéry had succeeded in doing what had eluded Euler 200 years earlier. He was able to show that ζ(3) is an irrational number.
Connection to prime numbers
The history of Zeta functions is long. In 1644, the Italian mathematician Pietro Mengoli asked what would happen if the reciprocal of all square numbers were added together: 1 + 1⁄4 + 1⁄9 + … He could not calculate the result, however. Other experts also failed at this task, including the famous Bernoulli family of scientists from Basel, Switzerland. In fact, another 90 years passed by another resident of that city, then 27 year old Eulerhe found a solution to the so-called The Basel problem: Euler calculated that the infinite sum is π2⁄6.
But Euler decided to devote himself to the more general problem at hand. He was interested in a whole class of problems, including finding the sum of the reciprocals of cubic numbers, numbers to the fourth power, and so on. For this, Euler introduced the so-called zeta functions), which has an infinite sum:
The Basel problem is just one of many zeta functions and corresponds to the value of ζ(2). Euler wanted to find a solution all values of the zeta function. And it actually managed to calculate the result for even values, s = 2k. In this case,
where p and Q are whole numbers, so the answer is always an irrational number.
However, Euler could not explain how the result changes when s is an odd number. He was able to calculate the first decimal places of the results but not the exact numerical value. He could not determine whether the zeta function for odd numbers also takes irrational values or whether the result can be represented as a fraction.
In the years and decades that followed, the function of silk received a lot of attention—and became associated with art. the greatest mysteries of mathematics today In the 19th century, the German mathematician Bernhard Riemann not only evaluated the zeta function of natural numbers. s but also for complex numbers: real values that can have square roots of negative numbers. In 1859 this change allowed him to state what would later become known as the famous Riemann hypothesis. With it, in principle, the distribution of prime numbers along the number line can be determined. Since understanding prime numbers is essential not only to number theory, but also has applications in fields such as cryptography, which relies on generating prime numbers for secure encryption, the stakes surrounding this mystery are high. Anyone who can solve the Riemann hypothesis will win a million dollar prize.
Despite all the attention to the zeta function, no one managed to determine the exact value of ζ(3), let alone find a formula that is generally valid for all odd values of the zeta function, as Euler did for even numbers. . Things became particularly interesting in ζ(3) physics in the 20th century.
Riemann Zeta Function in Physics
At the beginning of the 20th century, physicists discovered quantum mechanics: a radical theory that overthrew our previous understanding of nature. Here the boundary between particles and waves is blurred; some values, such as energy, appear only in bits and pieces (quantized), and the formulas of the laws of nature contain uncertainties that are not based on measurement errors but arise from mathematics itself.
In the 1940s researchers succeeded in formulating the quantum theory of electromagnetism. Among other things, it establishes that the void is never truly empty. Instead, it may contain a veritable firework display of short-lived particle-antiparticle pairs, matter seemingly created out of nothing but immediately destroyed again.
If you want to describe electrodynamic processes, such as the scattering of two electrons, you have to consider this continuous explosion of particles. This is because transient particle-antiparticle pairs can deflect electrons from their path. It turns out that if you want to describe this effect, the infinite sum between the cubes appears, ζ(3).
To perform physical calculations, it is sufficient to know the numerical value of ζ(3) to a few decimal places. But mathematicians wanted to know more about this number.
Apéry’s proof
Apéry was able to determine that ζ(3) is irrational, similar to the even-valued zeta function. His proof was based on a hitherto unknown representation of the ζ(3) series, a strange equation he claimed to have found in his garden:
With this statement, the German mathematician Gustav Lejeune Dirichlet in the 19th century. χ denotes that a number is irrational if there are an infinite number of integers p and Q with different parts, so that the following difference is satisfied:
here c and δ constant values are indicated. Although the formula sounds complicated, it basically means that χ can be easily approximated by fractions, but there is no fractional number corresponding to χ. Apéry managed to derive this inequality for ζ(3). Since then it has become clear: ζ(3) is irrational.
To honor the work of the French mathematician, the value ζ(3) bears his name today and is known as Apéry’s constant. This does not answer all the questions related to it, however. Experts still want a clear numerical value for ζ(3) that can be expressed using known constants, as with ζ(2) = π.2/6, for example. But even today we are far from this dream.
This article originally appeared the spectrum of science and reproduced with permission.
