Millennium Problem | Riemann Hypothesis, Prime Numbers & Zeta Function (2024)

FAQs

What is the relationship between the zeta function and the prime numbers? ›

Knowing that the zeta function converges when Re(s) > 1, we can rewrite it as an Euler Product. where the product runs over all prime numbers. This is the product expansion of ζ(s), which is known as the Euler Product.

To date, the only Millennium Prize problem to have been solved is the Poincaré conjecture.

In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 12. Many consider it to be the most important unsolved problem in pure mathematics.

The Riemann zeta function plays a pivotal role in analytic number theory, and has applications in physics, probability theory, and applied statistics. Leonhard Euler first introduced and studied the function over the reals in the first half of the eighteenth century.

The Riemann hypothesis, formulated by Bernhard Riemann in an 1859 paper, is in some sense a strengthening of the prime number theorem. Whereas the prime number theorem gives an estimate of the number of primes below n for any n, the Riemann hypothesis bounds the error in that estimate: At worst, it grows like √n log n.

Clay “to increase and disseminate mathematical knowledge.” The seven problems, which were announced in 2000, are the Riemann hypothesis, P versus NP problem, Birch and Swinnerton-Dyer conjecture, Hodge conjecture, Navier-Stokes equation, Yang-Mills theory, and Poincaré conjecture.

The Riemann Hypothesis is a mathematical conjecture proposed by the German mathematician Bernhard Riemann in 1859 that has puzzled mathematicians for over 150 years. It states that every nontrivial zero of the Riemann zeta function has a real part of ½.

It's called a Diophantine Equation, and it's sometimes known as the “summing of three cubes”: Find x, y, and z such that x³+y³+z³=k, for each k from one to 100.

For positive k, ZetaZero[k] represents the zero of on the critical line that has the k. smallest positive imaginary part. For negative k, ZetaZero[k] represents zeros with progressively larger negative imaginary parts.

What is Grigori Perelman's IQ? ›

Top 10 highest IQ. Grigori Perelman. As mentioned, the most recent title of the highest IQ—238—goes to this incredible mathematician. His greatest academic contribution to date is Thurston's Geometrization Conjecture: the solution to the famously challenging mathematical hypothesis, the Poincaré Conjecture.

The Riemann Hypothesis holds one of the seven unsolved problems known as the Millennium Prize Problems, each carrying a million-dollar prize for a correct solution. Its inclusion in this prestigious list further emphasizes its status as an unparalleled mathematical challenge.

Riemann zeta function, function useful in number theory for investigating properties of prime numbers. Written as ζ(x), it was originally defined as the infinite series ζ(x) = 1 + 2^−x + 3^−x + 4^−x + ⋯. When x = 1, this series is called the harmonic series, which increases without bound—i.e., its sum is infinite.

In either case, for every positive integer n, there is at least one prime bigger than n. The conclusion is that the number of primes is infinite.

The expression states that the sum of the zeta function is equal to the product of the reciprocal of one minus the reciprocal of primes to the power s. This astonishing connection laid the foundation for modern prime number theory, which from this point on used the zeta function ζ(s) as a way of studying primes.

Properties of Prime Numbers

Every even positive integer greater than 2 can be expressed as the sum of two primes. Except 2, all other prime numbers are odd. In other words, we can say that 2 is the only even prime number. Two prime numbers are always coprime to each other.

A factorial prime is a prime number that is one less or one more than a factorial (all factorials greater than 1 are even). The first 10 factorial primes (for n = 1, 2, 3, 4, 6, 7, 11, 12, 14) are (sequence A088054 in the OEIS): 2 (0! + 1 or 1!

Euler's totient function

This function has the following properties: If p is prime, then φ § = p – 1 and φ (pa) = p a * (1 – 1/p) for any a. If m and n are coprime, then φ (m * n) = φ (m) * φ (n). For example, to find φ(616) we need to factorize the argument: 616 = 23 * 7 * 11.

Prime numbers are fundamental to the field of cryptography due to their unique mathematical properties, which provide a foundation for creating secure cryptographic algorithms.