If proved, it would immediately solve many other open problems in number theory and refine our understanding of the behavior of prime numbers. Roughly speaking, the prime number theorem states that the number of primes less than n is proportional to n divided by the number of digits in n.
A greater proportion of two-digit numbers are prime than three-digit numbers, which are themselves more likely to be prime than four-digit numbers, and so on; the prime number theorem quantifies that decreasing relationship. The Riemann hypothesis, formulated by Bernhard Riemann in an paper, is in some sense a strengthening of the prime number theorem.
Though mathematicians usually use the letter z to represent a complex variable, they defer to Riemann and use the variable s in the zeta function. When the value of a is less than or equal to 1, the infinite series above does not yield a well-defined, finite value. But mathematicians have a trick up their sleeves: analytic continuation. If mathematicians add the constraint that the extended function must connect smoothly to the original function, there is a unique way to do it, so mathematicians refer to the analytic continuation of a function.
Defining the Riemann zeta function via analytic continuation is important because it enables mathematicians to use techniques from a field called complex analysis, which deals with continuous functions on the complex plane, to draw conclusions about the infinite sums that motivated the definition of the function in the first place. The other zeros are the mysteries. It was an exciting time to be working in number theory. Moreover, if you could improve their result by any amount at all, you would prove that primes infinitely often differ by some bounded constant.
And this would be a huge leap toward solving the notoriously difficult twin primes conjecture , which predicts that there are infinitely many pairs of primes that differ by 2.
By the end of the week, the experts agreed that it was basically impossible to improve the GPY method to get bounded prime gaps. Fortunately, Yitang Zhang did not attend this meeting. Almost a decade later, after years of incredibly hard work in relative isolation, he found a way around the impasse and proved the experts wrong. Get highlights of the most important news delivered to your email inbox.
Whatever the reasons may be, it is clear such a short list is incomplete and does not claim to be a comprehensive list of the most important problems to solve. However, each of the problems solved is extremely central, important, interesting, and hard. Some of these problems have direct consequences, for instance the Riemann hypothesis.
There are many many many theorems in number theory that go like "if the Riemann hypothesis is true, then blah blah", so knowing it is true will immediately validate the consequences in these theorems as true. In contrast, a solution to some of the other Millennium problems is highly likely not going to lead to anything dramatic.
The reason it's an important question is not because we don't philosophically already know the answer, but rather that we don't have a bloody clue how to prove it.
It means that there are fundamental issues in computability which is a hell of an important subject these days that we just don't understand. But that is about as likely as it is that the Hitchhiker's Guide to the Galaxy is based on true events.
I think three-dimensional space is very important, so if we can't answer a very fundamental question about it, then we don't understand it well. I'm not an expert on Perelman's solution, nor the field to which it belongs, so I can't tell what consequences his techniques have for better understanding three-dimensional space, but I'm sure there are. Explaining the true mathematics behind the Riemann Hypothesis requires more text that I'm allotted took most of my undergraduate degree in mathematics to even touch the surface; required all of graduate school to fully appreciate the beauty.
In very simple terms, the Riemann Hypothesis is mostly about the distribution of prime numbers. The idea is that mathematicians have some very good approximations emphasis on approximate for the density of the primes so you give me an integer, and I can use these approximate functions to tell you roughly how many primes are between 0 [really 2] and that integer.
The reason we use these approximations is that no [known] function exists that efficiently and perfectly computes the number of primes less than a given integer we're talking numbers with literally millions of zeros. Since we can't determine the exact values again, I'm simplifying a lot of this the problem mathematicians want to know is exactly HOW good are these approximations.
This is where the Riemann Hypothesis comes in to play. For well over a century, mathematicians have known that a special form of the polylogarithm function again, more fun math if you're bored is a really great approximation for the prime counting function and it's way easier to compute.
The Riemann Hypothesis, if true, would guarantee a far greater bound on the difference between this approximation and the real value. But for now, it is only a conjecture. A zeta function is a function of a complex variable a combination of real and imaginary variable.
This function becomes zero at all the negative even integers called the trivial zeros but, more importantly, it also becomes zero at other points called the non-trivial zeros. The conjecture states that all non-trivial zeros of the zeta function occur on a single straight line — the so-called critical line — of the complex plane. However, there is serious danger to the modern banking system if the Riemann Hypothesis is proved.
Prime numbers large , considered to be the atoms of arithmetic, play a key role in modern cryptography and e-commerce. Codes in e-banking and credit card transactions are based on the mechanical process of multiplying large prime numbers.
If the Riemann Hypothesis is proved, one would find faster ways to locate prime numbers. This would be a disaster for cryptographic systems and their codes, making them vulnerable.
India needs free, fair, non-hyphenated and questioning journalism even more as it faces multiple crises. But the news media is in a crisis of its own.
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