The Struggle for Riemann Hypothesis

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riemannDavid Hilbert in in his 1900 address to the Paris International Congress of Mathematicians, listed the Riemann Hypothesis as one of his 23 problems for mathematicians of the twentieth century to work on. Bernhard Riemann was a German mathematician who made important contributions to geometry, number theory, topology, mathematical physics and the theory of complex variables. Now we find it is up to twenty-first century mathematicians! The Riemann Hypothesis (RH) has been around for more than 140 years. The Riemann hypothesis, proposed by Bernhard Riemann (1859), is a conjecture that the nontrivial zeros of the Riemann zeta function all have real part 1/2.

Hilbert’s Contradictory Views

hilbert3David Hilbert seems to have had somewhat contradictory views about the difficulty of Riemann Hypothesis. On one occasion he compared three unsolved problems: the transcendence of 2^{\sqrt{2}}, Fermat’s Last Theorem and the Riemann Hypothesis. In his view, Riemann Hypothesis would likely to be solved in a few years, Fermat’s last Theorem possibly in his lifetime and the transcendence question possibly never. Things are turning out to be in reverse order. The transcendence question was resolved a few years later by Gelfond and Schneider. Andrew Wiles proved Fermat’s Last Theorem in 1993. On the other hand, at another time Hilbert remarked:

If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann Hypothesis been proven?

Littlewood and Hardy

hardy_littlewoodIn England in the early 1900s the difficulty of the question was not yet appreciated. Barnes assigned Riemann Hypothesis to Littlewood as a thesis problem. Littlewood independently discovered some of the developments that had already occurred on the continent. Hardy, Littlewood, Ingham, and other British mathematicians were responsible for many of the results on the zeta-function in the first quarter of the century. In the 1920s, the British mathematician G. H. Hardy wrote a postcard to his friend, listing six New Year’s wishes:

  1. Prove the Riemann Hypothesis.
  2. Score well at the end of an important game of cricket.
  3. Find an argument for the nonexistence of God that convinces the general public.
  4. Be the first man at the top of Mount Everest.
  5. Be the first president of the USSR, Great Britain, and Germany.
  6. Murder Mussolini.

Hardy grew to love the problem. He and Littlewood wrote at least ten papers on the zeta-function.


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