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During 2002 and 2003 Russian mathematician Grigori Perelman published three papers over the Internet that gave a “sketchy” proof of the Poincaré conjecture. His basic proof was expanded by several mathematicians and universally accepted as valid by 2006. That year Perelman was awarded a Fields Medal, which he refused. Because Perelman published his papers over the Internet rather than in a peer-reviewed journal, as required by the CMI rules, he was not offered CMI’s award, though representatives for the organization indicated that they might relax their requirements in his case. Complicating any such decision was uncertainty over whether Perelman would accept the money; he publicly stated that he would not decide until the award was offered to him. In 2010 CMI offered Perelman the reward for proving the Poincaré conjecture, and Perelman refused the money.
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
real part
A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram called an Argand diagram, representing the complex plane. Re is the real axis, Im is the imaginary axis, and i is the "imaginary unit", that satisfies i2 = −1.
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.
The Riemann hypothesis, considered one of the greatest unsolved problems in mathematics, asserts that all non-trivial zeros are on the critical line. In 1989, Conrey proved that more than 40% of the non-trivial zeros of the Riemann zeta function are on the critical line.
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.
The simplest explanation is that the Riemann zeta function is an analytic function version of the fundamental theorem of arithmetic. That theorem says: Every positive integer is a unique product of finite many prime numbers. That is, n=pk11pk22…
Individual zeros determine correlations between the positions of the primes. The Riemann zeta function encodes information about the prime numbers —the atoms of arithmetic and critical to modern cryptography on which e-commerce is built.
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.
To date, the only Millennium Prize problem to have been solved is the Poincaré conjecture. The Clay Institute awarded the monetary prize to Russian mathematician Grigori Perelman in 2010.
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.
Most mathematicians believe that the Riemann hypothesis is indeed true. Calculations so far have not yielded any misbehaving zeros that do not lie in the critical line.
Prime numbers are the most fundamental objects in mathematics, and wrangling them involves a paradoxical blend of simplicity and complexity that has become a lifelong obsession for James Maynard, one that has consumed him to the point where he sometimes strips everyday life of all distractions.
So if the Riemann hypothesis is proven correct in that all of the solutions to the Riemann zeta function do have the form ½ + bi, we will gain insight into the locations of the prime numbers and how much they deviate from the functions that the Prime Number Theorem presents.
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.
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.
Prime numbers are fundamental to the field of cryptography due to their unique mathematical properties, which provide a foundation for creating secure cryptographic algorithms.
The function converges for all s > 1. Its relation to prime numbers stems from an identity, which Euler discovered, that expresses the zeta function as the repeated product of a term evaluated only for primes.
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