In the period of approximately 100 years, which followed the year 1650, the Theory of Numbers fell asleep. This period was marked by an indescribable development in science due to the emergence of Calculus and its consequent development called Mathematical Analysis by Issac Newton (1646-1716), Gottfried Wilhelm Leibniz (1646-1716), the Bernoulli family (Jacob, 1655- 1705; Johann I, 1667-1748; Nicholas II, 1687-1759; Daniel 1700-1792), and Leonhard Euler (1707-1783).

At the end of the 16th century the Italian scientist Galileo Galilei inaugurated Science in the modern sense of the term, as he was the first to perform experiments in an orderly and systematic manner, assuming that Nature obeys mathematical laws, thus discovering some laws of nature formulated. mathematically.

Galileo followed the conception of the philosophers of Ancient Greece and, therefore, for him, Mathematics meant Euclidean Geometry, while Science meant "Natural Philosophy". Under this view mathematical problems were geometrized, that is, solutions were sought in terms of geometric constructions.

By Galileo's time, however, algebra had already been introduced to Europe. Developed by Persian Islamic philosophers, who in turn had learned from Indian mathematicians, the word Algebra was derived from Arabic al-gabr (“to bind together”) and this conception was summarized in the following process:

“*reducing the number of unknown quantities linked to the problem to be solved and then linking them together in a system called the equation. The next step is to find the solution to the equation.*”.

It was up to the brilliant philosopher and mathematician René Descartes to unify Geometry and Algebra. Thus, different conceptions created to solve mathematical problems that came from different cultures gave rise to one of the greatest inventions of mathematics:

“The Analytical Geometry”.

Analytical geometry consisted of a method that made algebraic equations "visible" as geometric shapes. For example, the equation *x* + *y* = 1, now had a geometric representation, that is, the equation was represented graphically by a line.

The graphical representation of an algebraic equation corresponded to an entity of geometry. A line was no longer located on Euclid's plane, but on the Cartesian plane formulated by Descartes. Similarly, equations involving *x* and *y* corresponded to the curves in the Cartesian plane. For example, to the equation *x*^{2} + *y*^{2 } = 1 corresponded in the Cartesian plane to a center circumference at the origin (0,0) and unit radius.

This invention of Descartes allowed Galileo to formulate the laws of mechanics he discovered, algebraically and geometrically. However, one problem remained: how do you find an equation that describes the motion of an animated body of varying speed, accelerated or decelerated? That is, Galileo and his contemporaries were not able to mathematically express the exact velocity of a body with acceleration at any given moment, because the velocity changed at each instant.

It was up to the genius of classical science Isaac Newton, and the German mathematician and philosopher Gottfried Wilhelm Leibniz, to solve this question that has plagued mathematicians and philosophers for twenty centuries since the Greek sophists.

Independently, Newton and Leibniz, a century after Galileo, invented a genius method that would put an end to this question. The Differential and Integral Calculus was born that would prove to be one of the greatest intellectual achievements in the western world.

The methods of mathematical analysis have always played a fundamental role in research in number theory. This partnership between Analysis and Number Theory has its origins in Euler's work and was largely developed by the mathematician L. P. G. Dirichlet (1805-1859).

Leonhard Euler was one of the greatest mathematicians of all time. He published approximately 500 articles in his lifetime and approximately 350 posthumous articles appeared. Although he was blind in one eye as a young man, and completely blind at sixty, he worked in virtually every area of mathematics and physics. In addition, he has written notable books on Algebra, Trigonometry, Calculus, Mechanics, Dynamics, Variation Calculus, Astronomy, Artillery, Optics, and more.

His original research was instrumental in eighteenth- and nineteenth-century mathematics because of his creativity, inspiration, and ingenious ability to unify and systematize all the knowledge so far produced.

Euler was the first mathematician to apply the ideas of analysis to problems of number theory. In fact, as noted later, he was using techniques of Complex Function Theory. In this way, it attacked two fundamental problems of Number Theory.

The first problem of number theory in which Euler applied the analytical methods concerns whole solutions of equations. To determine the integer solutions of a linear equation, Euler created a method that became known as the Generating Functions Method.

The method of generating functions proved so genius that it led to the creation of the Hardy-Littlewood-Ramanujan Circle Method, whose development, in turn, led to one of the basic methods of Contemporary Number Analytic Theory: Trigonometric Sums of Vinogradov (Method of Trigonometric Sums). This idea led to the creation of the branch of Analytical Number Theory known as the "Additive Number Theory."

The other problem related to the behavior of the prime number sequence in the set of positive integers. Euler gave a fresh demonstration of Euclid's Theorem about the existence of infinite prime numbers based on analytical arguments. Euler's idea proved very fruitful and gave impetus to the development of an important line of research in Analytical Number Theory: the “Multiplicative Number Theory”.

The formula

was discovered by Euler approximately in 1735.

Euler was delighted to demonstrate the mysterious fact that this sum is related to the number . In fact, the above identity represents a special value calculated in a class of functions called zeta functions.

Note that if we demonstrate that z defines a function, then

.

Euler demonstrated that for every real number *s* > 1, the series

define the function

called *zeta function*.

Euler demonstrated the existence of an infinite number of prime numbers using the analytical properties of this function. There is a relationship between the Zeta Function and the set of prime numbers called the Euler Product. Euler's product is an analytical expression of the single factorization of integers as a product of prime numbers:

for s> 1, where the product on the right is taken for all prime numbers.

It is interesting to note that the Euler Product implies, in particular, that

for s> 1.

The Zeta Function introduced by Euler has proved to be one of the most important characters in Number Theory, as it has precious arithmetic properties. Some mathematicians often say that Number Theory is the study of Zeta Functions.

In the nineteenth century the mathematician Bernard Riemann defined the Zeta Function in the set of complex numbers and, due to his numerous and fundamental contributions to the study of this function, today it is known as the "Riemann Zeta Function".

A new journey began with Riemann. He made a conjecture called *Riemann Hypothesis* which to this day is one of the biggest challenges for the most brilliant mathematicians. The Riemann Hypothesis will be the subject of our next columns.

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