Riemann Hypothesis in detail

#What is Riemann Hypothesis?

The Riemann Conjecture is a mathematical conjecture, first proposed in 1859 and still unproven as of 2024. It is arguably the most famous of all unsolved mathematical problems, known as "𝐓𝐡𝐞 𝐡𝐨𝐥𝐲 𝐠𝐫𝐚𝐢𝐥 𝐨𝐟 𝐦𝐚𝐭𝐡𝐞𝐦𝐭𝐢𝐜𝐬𝐬𝐬𝐢𝐭". Although it is related to many areas of mathematics, it is usually concerned with the distribution of prime numbers.

 #Who is Riemann?

 Bernhard Riemann (full name Georg Friedrich Bernhard Riemann, 1826–1866) was a shy, modest German mathematician who made significant contributions to several areas of mathematics, including analysis and differential geometry. He only wrote one paper on number theory, but it contained the statement of his conjecture, so it is easily one of the best number theory papers ever published. As well as this, his work on differential geometry paved the way for Einstein's mathematical foundations of general relativity.

 #How is it related to prime numbers?

Of course answering that question would require a lot of advanced math. So I can only provide an outline here.

 Prime numbers appear throughout the counting sequence but fail to show any clear pattern. However, they show a tendency to thin (split), and the "average rate" at which they thin is described by the prime number theorem. It was first proposed in the late 1700s, but not proven for another hundred years. To prove the PNT, mathematicians needed to study a mathematical function called the Riemann zeta function. The zeta function was introduced in Riemann's 1859 paper and was shown to control (in a sense) the fluctuations of prime numbers around their "normal" behavior. The zeta function operates on a two-dimensional "number plane" called the complex plane, which involves an infinite set of "non-zero" points (usually called "zeta zeros" or "Riemann zeros"). . The positions of these zeros on the complex plane can be related to an infinite set of wave-like entities that collectively govern the fluctuation of the primitives. Riemann was able to calculate all zeros on a vertical line, and he assumed that all (non)zeros of the zeta function lie on this "critical line". That is the Riemann hypothesis. From his writings, it seems he didn't realize how important this casual remark would be—he went on to say that he believed it to be true, but that it was not directly relevant to his investigations.

 Riemann was able to prove certain things about zeta zeros, including that they must all lie in a vertical column one unit wide (the "critical column") centered on the aforementioned "critical line." In the 1800s, it was shown that the Prime Number Theorem is true if it can be shown that all zeta zeros lie properly within the critical column, that is, not at its edges. In 1896, the mathematicians Hadamard and de la Vallee Poussin proved this almost simultaneously, thereby proving the PNT.

Further narrowing the bar where all zeta zeros are known to be false leads to more accurate information about the distribution of prime numbers. The ultimate achievement is to reduce this bar to its midline (the "critical line"), as narrow as it can get. If this is possible, then RH is proven and we know that the prime numbers are as "well-behaved" as possible. If RH is false, there are zeta zeros that do not lie on the critical line, and the wave-like entities associated with these cause large fluctuations in the a priori distribution, thereby disrupting some "balance" in the number. A system that the mathematical community almost universally expects and believes works.

 #What does this say about a $1,000,000 prize?

 The non-profit Clay Mathematics Institute, founded in 1998, published its seven "Millennium Prize Problems" in 2000, each offering a million-dollar prize. Naturally, the Riemann hypothesis is one of these problems. This caused great popular interest in the problem, but since its proof was already the ultimate prize for mathematicians, the million dollars was unlikely to make much of a difference to them. Needless to say, the prize has not yet been claimed.

 is it? Why is something important? Most people's lives are completely unaffected by RH being proven (or disproved). However, it is very important in mathematics. Because of the fundamental role prime numbers play in the number system, RH can relate to many different areas of mathematics. There are hundreds of theorems whose statements begin with the assumption that RH is true. Consequently, if RH is disproved, all these theorems will break down, and if it is proved, they will hold. A falsification of RH would be a disaster for mathematics, as we currently understand it.

 Also, the failure of more than 150 years of dedicated effort to produce a proof means what mathematicians call a "gap in our understanding," or a large gap between where we are now, mathematically, and where we want to be. RH must be proven.This suggests that to prove the hypothesis, some major new ideas are needed, ideas that could fundamentally change our understanding of the number system. So tracking the achievement of RH is important in that sense.

 It should be added that it is various "generalizations" of RH. Their proof or disproof has a really major impact on mathematics.

 Explaining the importance of the Riemann hypothesis to mathematics is almost as difficult as explaining what it is.🥲