Number Theoretic Functions

TL;DR: In this article, the problem of approximating the number theoretic function π(x) with respect to all prime numbers p ≤ x by means of the function x/log x was studied.

Abstract: Until now the approximation of individual real numbers has been studied with the help of integers, and for this the methods of Analysis and Geometry have been seen to be helpful. From now on the reverse problem will be attacked: number theoretic functions, the calculation of which for large values is severely limited because of the irregularity of their behaviour, will be represented approximately by known functions from differential and integral calculus with easily described behaviour. Moreover the methods of analysis are also available for these approximation problems. For example in Chapter 5 we prove the prime number theorem, which has for its subject the approximation of the number theoretic function π(x), which counts all prime numbers p ≤ x by means of the function x/log x. In this chapter we develop the notation and necessary techniques, discuss how good an approximation is, and present some simple examples.