Many famous mathematicians have found relatively simple functions that model the behavior of π(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 1012, 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(1018),
$$\int_2^{10^{18}} {\frac{1}{{\log t}}dt}$$
is 24739954309690414 while the actual value is 24739954287740860; the relative error is about 1 part in a billion.


Prime Number Riemann Hypothesis Fibonacci Number Congruence Class Prime Number Theorem 
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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Stan Wagon
    • 1
  1. 1.Department of Mathematics and Computer ScienceMacalester CollegeSt. PaulUSA

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