Taming the Infinite by Ian Stewart

Author:Ian Stewart
Language: eng
Format: epub
Tags: MATHEMATICS / History & Philosophy
Publisher: Quercus
Published: 2015-04-07T04:00:00+00:00

is equal to the product, over all primes p, of the series

Here we must take s > 1 for the series to converge.

In 1848 Pafnuty Chebyshev made some progress towards a proof of Gauss’s conjecture, using a complex function related to Euler’s series, later called the zeta function ζ(z). The role of this function was made clear by Riemann in his 1859 paper On the Number of Primes Less Than a Given Magnitude. He showed that the statistical properties of primes are closely related to the zeros of the zeta function, that is, the solutions, z, of the equation ζ(z) = 0.

In 1896 Jacques Hadamard and Charles de la Vallée Poussin used the zeta function to prove the Prime Number Theorem. The main step is to show that ζ(z) is non-zero for all z of the form 1+ it. The more control we can gain over the location of the zeros of the zeta function, the more we learn about primes. Riemann conjectured that all zeros, other than some obvious ones at negative even integers, lie on the critical line. z = ½ + it.

In 1914 Hardy proved that an infinite number of zeros lie on this line. Extensive computer evidence also supports the conjecture. Between 2001 and 2005 Sebastian Wedeniwski’s program ZetaGrid verified that the first 100 billion zeros lie on the critical line.

The Riemann Hypothesis was part of Problem 8 in Hilbert’s famous list of 23 great unsolved mathematical problems, and is one of the Millennium prize problems of the Clay Mathematics Institute.

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