The Riemann Hypothesis

Riemann's Hypothesis was one of the 23 problems - milestones that David Hilbert suggested in 1900, at the 2nd International Conference on Mathematics in Paris, that they should define research in mathematics for the new century (and indeed, it is not an exaggeration to say that modern mathematics largely come from the attempts to solve these 23 problems). It is the most famous open question today, especially after the proof of Fermat's Last Theorem.

The Riemann zeta function is of central importance in the study of prime numbers. In its first form introduced by Euler, it is a function of a real variable x:

This series converges for every x > 1 (for x=1 it is the non-corvergent harmonic series).

Euler showed that this function can also be expressed as an infinite product which involves all prime numbers pn, n=1,…:

Riemann studied this function extensively and extended its definition to take complex arguments z. So the function  bears his name. Of particular interest are the roots of :

• Trivial zeros are at z= -2, -4, -6, …
• Nontrivial zeros are at z such that 0 < Re(z) < 1 and there are infinite ones. Infinitely many lie in particular on Re(z)=1/2 (proved in 1914 by the English number theorist Godfrey Hardy)
• There are no zeros for Re(z)  1. The case Re(z)=1 was proved in 1896 by Hadamard and de la Vallée-Poussin and used in their proof of the Prime Number Theorem.

Riemann conjectured that all nontrivial zeros are at Re(z)=1/2. Although this has been shown to be true for more than the first billion nontrivial zeros, the conjecture remains open. A proof would establish new results in number theory, for example on the distribution of primes. The fact that Riemann Hypothesis holds for billions of nontrivial zeros does not guarantee anything. As noted by I. Good and R. Churchhouse in 1968, in the theories of zeta function and of primes distribution, one frequently meets terms like log log x, a function which increases extremely slow. The first nontrivial root not on Re(1/2) might have an imaginary part y such that log log y is of the order say 10. Then y would be 1010,000, a definitely unreachable number, computationally.