Riemann's hypothesis

The invention of Differential and Integral Calculus provoked one of the greatest advances in Western thought. The monumental work done by Newton and Leibniz led to the advance of science in all its areas. The Swiss mathematician Leonhard Euler (1707-1783) was one of the pioneers in applying Calculus methods to problems of Number Theory giving rise to Analytic Number Theory. However, the German mathematician G. F. B. Riemann (1826-1866) is recognized as the true founder of the Analytical Number Theory and as possessing one of the most original and profound brilliant minds of the nineteenth century.

Riemann revolutionized Mathematical Analysis, Geometry and Mathematical Physics. In Analytical Number Theory, as well as in other areas of mathematics, its fundamental ideas still have a profound influence. Riemannian Varieties, Riemann Surfaces, Cauchy - Riemann Equations, Riemann Hypothesis, and many other subjects are among his works.

Riemann had a powerful and precise intuition, but despite his genius and creativity, his life was extremely modest. Riemann died prematurely of tuberculosis. His shyness, his lack of ability as a speaker, and his innate talent for mathematics, kept him from pursuing his career as a theologian, contrary to his father's will. The German mathematician Lejeune Dirichlet (1805-1859) was his teacher and had a great influence on his work.

In 1851 Riemann completed his doctorate under the guidance of the great German mathematician K. F. Gauss (1777-1855) who stated: "Riemann possesses a gloriously fertile originality." A peculiar fact is that the key to some of the most essential contemporary problems lies in a conjecture made by Riemann.

Called the Riemann Hypothesis, this conjecture represents one of the most important problems in mathematics.

It all began when Euler defined in 1740 a function denoted by the Greek letter ς (read "zeta"). Euler's zeta function associates every real number greater than 1 with a new real number


It is interesting to note that by replacing s at number 2, Euler found that (2) = π2/ 6. He noted that this function would give information about the pattern of prime numbers, and thus was born the Analytical Number Theory, that is, the study of prime numbers through Calculus applied to the investigation of properties of some complex functions.

Complex functions are functions defined in the set of complex numbers that assume complex values. You can't see a graph of such a function because it has dimension four. However, it is possible with the help of good software to obtain the graphs of the real and imaginary parts of such a function.

Note that there are numerous zeta functions and some mathematicians often say that number theory is the study of zeta functions. However, what is the relationship between prime numbers and Euler's zeta function?

Euler demonstrated the impressive theorem that states that for any real number s greater than 1, the zeta function is expressed as an infinite product of factors of the form

whatever the prime number P, that is,


This function was investigated by Riemann in detail when he replaced the real number s by a complex number, which made the zeta function a complex function. That is, ς (s) is the complex number:

, for re(s) > 1.

Re(s) means the real part of the complex number.

The zeta function is not defined for all complex numbers. However, Riemann realized, using a technique of Complex Function Theory, that it was possible to extend the zeta function to all but the complex number. z = 1. Thus, the zeta function is now called the Riemann zeta function.

In 1859 Riemann published a brilliant eight-page article, his only article in Number Theory, where he used the zeta function to investigate the pattern of cousins. His goal was to demonstrate the Gauss Conjecture, now known as the Prime Number Theorem, which stated that the number of prime numbers between 1 and x, When x it's too big it's about x divided by the natural logarithm of x, this is,

x / ln x.

Although Riemann was unsuccessful, his work was very important for the development of Analytical Number Theory. Several results were obtained by him when investigating the properties of this function. Riemann showed that properties of this function are closely linked to the distribution of prime numbers, that is, to the natural sequence of prime numbers in the set of positive integers.

Riemann outlined the path of future progress in this investigation into a series of well-founded conjectures, including the famous Riemann Hypothesis. In 1896, the French mathematician J. Hadamard and the Belgian mathematician C. J. de la Vallée - Poussin independently demonstrated the Prime Number Theorem using the ideas developed by Riemann.

Consider the equation ς (s) = 0. So any complex number s that solves this equation is called a “zero” equation.

Riemann first noted that the even negative integers -2, -4 -6,… are zeros of the function. He then observed that there should be infinite complex zeros, and then boldly conjectured that any other complex zero of the zeta function has a real part equal to ½, that is, they have the form s = ½ + B i.

Therefore, all zeros of the zeta function that are not real numbers will be on the vertical line. x = ½. This line is often called the critical line.

The first thing to note is that the zeros of the critical line are not real, they are placed symmetrically in relation to the real axis and also in relation to the critical line itself. This is Riemann's famous hypothesis. This is undoubtedly a very important problem, as knowledge of zeros of the zeta function translates into a deeper understanding of the distribution of prime numbers.

In the first half of 2004 the proof of this conjecture was announced by the French mathematician Louis de Branges de Bourcia and is under examination by specialists. This mathematician had previously announced that he had demonstrated this famous conjecture, but errors were found in his demonstrations.

Mathematics exerts great fascination in men, and some millionaires, while not mathematicians, stimulate mathematical research. This is the case of American, investment fund magnate and math lover, Landon Clay, who created in Cambridge, Massachussets, a non-profit organization to promote and fund research in mathematics: the Clay Mathematics Institute (CMI). ).

At a meeting held at the Collège de France in Paris in May 2000, the WCC announced an offer of seven prizes, each worth $ 1 million, for finding solutions to each of the seven most important issues, toughest and most challenging math. A small committee of today's leading mathematicians has chosen these problems, which are now called the "Millennium Problems."

The Riemann Hypothesis was considered one of the problems of the millennium, as it is the most important problem of unresolved mathematics that has consequences in physics and profound repercussions in information theory, such as, for example, the issue of Internet security. These consequences, which represent an essential component of today's life, will be the subject of our next column.

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