# Prime Numbers

• Olivier Bordellès
Part of the Universitext book series (UTX)

## Abstract

This chapter is devoted to the study of the distribution of prime numbers, from Euclid to Erdős. The text follows the lines of these mathematicians who paved the way for all branches of modern number theory. After recalling the basic tools essentially due to Euclid, we investigate Chebyshev’s reasoning in his attempt to give a proof of the Prime Number Theorem. The latter will finally be shown with the theory of functions building on Riemann’s idea to extend an old-fashioned function to the whole complex plane, nowadays called the Riemann zeta-function. In the section Further Developments, we provide a proof of the PNT as a consequence of deep estimates of ζ near the line σ=1 and summation formulae. It is also the opportunity to provide explicit estimates of the classic results, yet quite rare in the literature, and to explore some of the consequences of the famous Riemann hypothesis.

## Keywords

Prime Number Dirichlet Series Arithmetic Progression Critical Line Primitive Root
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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