June 20, 2024

Prime numbers: new developments in the Riemann conjecture

It is the most important open question in number theory – if not in all of mathematics. The Riemann conjecture has astounded experts for more than 160 years. It is the only problem that appeared in David Hilbert’s groundbreaking speech in 1900 and in the “Millennium Problems” formulated a hundred years later, the solution of which carries a $1 million prize.

But the Riemann conjecture is difficult to solve: despite huge interest and attractive prize money, little progress has been made in this field. But now, decades later, mathematicians Larry Guth of MIT and James Maynard of the University of Oxford have come up with a solution. published an exciting new result. “The authors have improved a result that seemed intractable for more than 50 years,” says numerologist Valentin Blumer from the University of Bonn. This is a “remarkable breakthrough” Mathematician and Fields Medalist Terence Tao writes about ‘mastodons’“, “Even if they are still far from fully proving the conjecture.”

The reason for interest in this conjecture is that it relates to the basic units of natural numbers. It deals with prime numbers, those values ​​that are divisible only by one and themselves. Examples include 2, 3, 5, 7, 11, 13, and so on. Any other number, such as 15, can be clearly factored into the product of prime numbers: 15 = 3 · 5. The problem: Prime numbers do not seem to follow a simple pattern; they appear randomly among the natural numbers. The Riemann conjecture deals with this peculiarity. It explains how prime numbers are distributed on a number line – at least from a statistical perspective.

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Periodic table of numbers

Thus the solution of the famous conjecture would provide mathematicians with nothing less than a kind of “periodic table of numbers.” Just as the basic building blocks of matter (such as quarks, electrons, or photons) help understand the universe and our world, prime numbers also play an important role – not just in number theory, but in almost all areas of physics and mathematics. . There are now many theories based on the Riemann conjecture. Proving this conjecture would prove a number of other theories. This has motivated many experts in the past (and still does today) to address the persistent problem.

Interest in prime numbers goes back thousands of years. Euclid proved as early as 300 BC. BC that there are an infinite number of prime numbers. Although interest in prime numbers continued, it took until the 18th century for more important insights into these basic building blocks. When he was 15 years old, Gauss realized that the number of prime numbers decreases along a number line: the so-called prime number theory (which was proven only 100 years later) states that in the interval from 0 to n almost nNationality law (n) Prime numbers appear.

However, the exact number of primes may differ from the estimate given by prime number theory. For example: According to prime number theory, in the interval between 1 and 100 there is approx 100Nationality Law(100) &almost; 22 is a prime number, but in reality you can find 25; So there is a deviation of 3. The Riemann conjecture states that this deviation cannot become arbitrarily large. More precisely: the difference is measured at most with a root n (√n); That is, the root of the length of the period under consideration.