## Abstract

Many famous mathematicians have found relatively simple functions that model the behavior of π(is 24739954309690414 while the actual value is 24739954287740860; the relative error is about 1 part in a billion.

*x*), the number of primes below*x*. This graph shows the error, up to one million, of three such approximations: Legendre and Chebyshev used logarithms (blue graph; beyond 10^{12}, Chebyshev is better than Legendre), Gauss (red) used the integral of the reciprocal of the logarithm, and Riemann (green) enhanced Gauss’s integral with an infinite series. As an example of the power of such formulas, note that Gauss’s estimate for π(10^{18}),$$\int_2^{10^{18}} {\frac{1}{{\log t}}dt}$$

## Keywords

Prime Number Riemann Hypothesis Fibonacci Number Congruence Class Prime Number Theorem
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.

## Copyright information

© Springer Science+Business Media, LLC 2010