For even exponents, the function ζ always returns an irrational (and even transcendental) number. But until Apéry’s discovery in 1979, no one knew what odd natural numbers looked like. His argument was the first to show the irrationality of a strange exponent. To this day, Apéry’s method of proof has not been applied to other numbers. However, we know that the function returns an irrational value for at least one of the numbers 5, 7, 9, or 11.
Prime numbers, integrals, and photon density
Not surprisingly, the Apéry constant is also associated with prime numbers. After all, the function is at the center of the famous Riemann conjecture, one of the greatest unsolved problems in mathematics, which deals with the distribution of prime numbers. The relationship between Apéry’s constant and prime numbers can be seen very directly in an alternative representation, where the constant is given as the infinite product of the expression 1/(1−)s-3) on the set of prime numbers s Is written. You can also represent Apéry’s constant as an integral, which in turn means that this constant also appears in the analysis. You can also meet them in other surprising areas, such as when you want to determine the average density of photons in the cosmic background radiation. But it can also be found in the mechanics of fluid flow, in the investigation of sources of thermal radiation (such as stars) and in various other areas of physics and astronomy.
Carl Ludwig Siegel (1896–1981), one of the most important mathematicians of the twentieth century, commented on Abery’s proof of irrationality with the following words: “You can only hold the proof in front of you like a crystal.” In this is understandable. It is to be hoped, however, that mathematicians will not only carry the proof in front of them in the future, but will continue to analyze it. Because who knows what ideas the constant has to offer.
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