Is the Riemann Hypothesis Math’s Greatest Mystery?

Mathematics is full of puzzles — problems that still lack solutions. The most challenging one is considered to be the hypothesis formulated by Bernhard Riemann, aimed at deciphering the distribution code of prime numbers.

Many brilliant scientists have grappled with it, but it still remains on the list of unsolved mathematical problems. What is the Riemann Hypothesis and what does it have to do with prime numbers? Below, I will answer this question.

Ancient Greeks already knew that prime numbers stretch to infinity. Despite knowledge dating back to antiquity, many mysteries surrounding them remain unsolved. The most significant challenge for modern mathematics is determining the meaning of their distribution. However, to crack the code hidden by prime numbers, one must first prove the Riemann Hypothesis.

The Riemann Hypothesis and Prime Numbers

Mathematics is a science governed by order. Patterns emerge among numbers, which can be expressed by specific formulas. Just look at the Fibonacci sequence (0, 1, 1, 2, 3, 5, 8, 13, 21, 34…) to see that each subsequent number is the sum of the two preceding ones. However, when prime numbers are included in the set, no regularity or dependence can be observed. Their occurrence seems chaotic, entirely random. Yet, prime numbers are the foundation upon which all of mathematics rests. Should we then conclude that order emerges from chaos?

Many brilliant mathematicians have argued that this chaos is only apparent, and beneath the mask of randomness lies a specific order. One such mathematician who delved into the nature of prime numbers was Bernhard Riemann. In November 1859, he presented a hypothesis regarding the zeta function (in an article titled “On the Number of Primes Less Than a Given Magnitude”), which is intended to solve the puzzle troubling mathematicians.

Leonhard Euler was the first to define the zeta function, but the Swiss mathematician considered its value only for real variables. It was Riemann who expanded its definition to include all complex numbers, conducted the proof of the function’s meromorphicity, demonstrated the functional equation that described the function across the entire complex plane, and showed the relationship between the distribution of its zeros and the number of prime numbers.

What is the Riemann Hypothesis, and what are its basic assumptions?

In his hypothesis, Bernhard Riemann stated that all nontrivial (meaning non-real) zeros of the zeta function have a real part equal to 1/2. The German mathematician identified four zeros of the function and noticed that they all lie on a single half-line, intersecting the horizontal axis in the region of the value 1/2. This is the so-called critical line.

Riemann believed that all nontrivial zeros of the zeta function should lie on the critical line. Therefore, we speak of the first regularity contradicting the chaotic nature of prime numbers. However, an important question arises here: why is Riemann’s conclusions referred to as a hypothesis rather than a theory? Because, in mathematical terms, computing the four zeros does not allow us to conclude that there are no exceptions to the presented rule.

Therefore, Riemann’s work does not constitute final proof, but it suggests that prime numbers’ distribution is not governed by chance. It is no wonder that mathematicians persist in their efforts to prove the validity of its assumptions.

Attempts to prove the hypothesis

Many geniuses have grappled with prime numbers and the Riemann Hypothesis. The list of mathematicians who undertook to prove the hypothesis formulated in 1859 includes the names of the most eminent scientists. Godfrey Hardy, John Littlewood, Atle Selberg, John Nash, Louis de Branges de Bourcia — each of them took on the challenge of the “cursed numbers”.

None of them managed to present a convincing proof that beneath the apparent chaos lies a deeper meaning. For some, the problem of prime numbers literally shattered their careers, yet there is no shortage of brave souls willing to tackle this problem.

As early as 1900, the Riemann Hypothesis was considered one of the most important challenges in mathematics. At the International Congress of Mathematicians, David Hilbert presented 23 key problems requiring solutions. The zeta function hypothesis was ranked eighth on this list. Many attempted to find a proof, but all failed.

A century later, the Clay Mathematics Institute announced a list of mathematical problems for which a reward was offered. Not just any reward, but a hefty sum of one million dollars. Among the seven so-called Millennium Prize Problems, only one item from Hilbert’s list was included — the Riemann Hypothesis.

By 2024, only one problem from the Millennium list had been solved: the Poincaré Conjecture. This was accomplished in 2006 by the Russian mathematician Grigori Perelman. The remaining items, including the Riemann Hypothesis, still remain unsolved.

Has the Riemann Hypothesis been proven?

The monumental task of proving the 1859 hypothesis was undertaken by the aforementioned Louis de Branges de Bourcia. The French-American mathematician devoted his entire career to understanding the hidden meaning in prime numbers. He presented proofs of the Riemann Hypothesis several times (most recently in 2014), but each was rejected.

The last proof was presented by Michael Atiyah. In 2018, he announced that while working on calculating the constant of subtle structure, he managed to confirm the zeta function hypothesis. The British mathematician based his proof on the work of Dirac, Hirzebruch, and von Neumann. When he announced that he had confirmed the Riemann Hypothesis using a radically new approach, the entire scientific community held its breath. Unfortunately, it seems that the problem formulated by Bernhard Riemann will continue to haunt mathematicians. Atiyah’s proof was rejected, just like every previous one.

Or perhaps this means that the zeta function hypotheses simply cannot be defended? Well, no, because while no one has yet managed to prove it, no one has been able to refute Riemann’s theorem either.

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