# How difficult is number theory

## Number theory

In fact, Euler had only shown that the sum of the reciprocal values ​​becomes infinitely large; Nevertheless, the right-hand side is interesting: Euler obtained it from his equation

whose both sides for all complex numbers s with real part > 1 converge by having the border crossing point s → 1 examined. The equation (3) was later Euler identity and forms the basis for the relationships between the distribution of prime numbers and the Riemannian ζ -Function. Euler's mathematical intuition was also evident in his use of the quadratic reciprocity law, which was only fully proven by Gauss, and in his contributions to the theory of elliptic curves.

Lagrange's work on binary square forms was "Recherches d’arithmétiques", inspired by Euler, albeit going beyond that, in that some of it was proven in what Euler had discovered empirically. Based on this book, Gauss called his first book on number theory "Disquisitiones arithmeticae". According to Gauss, for whom number theory the Queen of the Mathematical Sciences As the arsenal of methods increases, the distinction between different branches of number theory according to the methods used is gradually becoming meaningful, although the distinction is not always unambiguous.

According to today's terminology, the close connection between algebraic expressions on the one hand and properties of natural numbers on the other falls within the scope of algebraic number theory. It turned out that number theoretic problems are a strong motivation for developing algebraic concepts and methods. Many basic concepts of today's algebra developed with "the discovery of the arithmetic laws of higher number fields under the hands of Gauss, Dirichlet, Kummer, Kronecker, Dedekind and Hilbert ”. Behind this stands the striving "for a more comprehensive, conceptual clarity that always seeks the father of the thought behind the variety of number-theoretical phenomena" . From the fact that certain properties of algebraic expressions can best be expressed in geometric language, there is also a connection to geometry via the Diophantine equations, for which the expression is used today arithmetic geometry used. A current high point of such a combination of different mathematical disciplines is the proof of the Fermat Hypothesis (Wiles 1995):

Is n any natural number ≥ 3so there is no triple (x, Y Z) from natural numbers, which is the equation xn + yn & equals; zn Fulfills.

The application of functional theory methods in number theory led to analytical number theory. This was first initiated by Euler through his virtuoso handling of power series, and received significant new impulses from Riemann's 1859 eight-page work "About the number of prime numbers under a given quantity". This work is the only publication on number theory by Riemann; it is written very briefly, consists of numerous statements and mostly very vague references to evidence, contains the Riemann Hypothesis, and is generally very difficult to understand . This explains why Riemann's ideas were only taken up again more than 30 years later. Nevertheless, this paper exerted a great influence on the development of number theory; Riemann's ideas were z. B. Landau, Hardy, Siegel, Polya, Selberg, Artin, Hecke and many others taken up, carefully examined, and brought to a deeper understanding - however, no one has yet succeeded in proving or refuting the Riemann Hypothesis. An important result in this context is the prime number theorem, proved by Hadamard and de la Vallée Poussin in 1896 (independently of one another and in different ways) using Riemann's ideas:

Denotes π(x) the number of prime numbers below x, then the asymptotic equality applies

With the help of this theorem it can be proved that Euler guessed the correct order of divergence of the sum in his formula (2) with the expression "log log ∞".

From today's point of view, the prime rate should be seen more as the starting point than the end point of the mathematical results on questions of prime number distribution. If one wants to investigate further questions about prime twins or about the Goldbach problems, much more subtle methods are required . Not only functional theory methods, but also methods from probability theory and asymptotic analysis are used . A proof of the Riemann Hypothesis (with whatever method) would give a deeper insight into the regularity of the distribution of prime numbers (Mertens Hypothesis). Incidentally, the distribution of prime numbers is not just a problem that is detached from everyday life: some (frequently used) cryptographic methods are related to the fact that one would know more about their reliability if the Riemann Hypothesis (or a generalization of it) were proven.

What is remarkable in this context is the fruitful interplay between the interest in algorithms and computer technology that emerged in the 20th century on the one hand and number theory on the other. The interest in algorithms gave rise to new number theoretic questions, e. Take the Collatz problem, for example, which raised a host of new questions, few of which have been answered today. Computer technology made it possible to make extensive calculations, which gave rise to the need to put them on a solid mathematical basis. So came z. For example, in Monte Carlo simulations (Monte Carlo method), number theoretic methods very soon came into play (e.g. Lehmer's congruence method for generating pseudo random numbers). This development is by no means over: The so-called quasi-Monte-Carlo methods, whose applications include technical simulations as well as risk analyzes in financial mathematics, require deep-seated number theoretic considerations such as the algebraic-geometric investigation of function fields. Such methods are also used in the construction of error-correcting codes for the secure transmission of information. In cryptographic processes for hiding information from unauthorized access, not only questions about the prime number distribution play a role, but number theoretic processes are also used. These concern z. B. primality tests, factoring natural numbers, elliptic curves (encryption by means of elliptic curves), or also class groups of algebraic number fields . These examples clearly show that the common distinction between “pure” and “applied” mathematics does not make sense with regard to number theory.