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Has the Riemann Hypothesis Been Finally Solved? By Brian Simpson

     Most mathematics teachers across the universe have not been sleeping following news that the infamous Riemann hypothesis may have been solved by British mathematician Michael Atiyah: “The Riemann Hypothesis,” a copy which I had sent to me, with no link, the paper being leaked. Anyway, here is a definition of the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. It was proposed by Bernhard Riemann (1859), after whom it is named. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields. The Riemann hypothesis implies results about the distribution of prime numbers. Along with suitable generalizations, some mathematicians consider it the most important unresolved problem in pure mathematics (Bombieri 2000). The Riemann hypothesis and some of its generalizations, along with Goldbach's conjecture and the twin prime conjecture, comprise Hilbert's eighth problem in David Hilbert's list of 23 unsolved problems; it is also one of the Clay Mathematics Institute's Millennium Prize Problems.

     The Riemann zeta function ?(s) is a function whose argument s may be any complex number other than 1, and whose values are also complex. It has zeros at the negative even integers; that is, ?(s) = 0 when s is one of -2, -4, -6, .... These are called its trivial zeros. However, the negative even integers are not the only values for which the zeta function is zero. The other ones are called non-trivial zeros. The Riemann hypothesis is concerned with the locations of these non-trivial zeros, and states that:

     The real part of every non-trivial zero of the Riemann zeta function is 1/2.

     The proof by Atiyah is based on using the so-called Todd functions to derive a proof by contradiction that the Riemann hypothesis is true, taking up only 14 lines of proof, less than most of our high school proofs in number theory. Without going in to technicalities, it is highly likely that the old mathematician has got it wrong, since the proof makes no reference at all to prime numbers and their distribution. There is also no precise theory of the Todd functions, which need to be defined before the proof can get off the ground.

     I would like it to be true that a 14 line proof cracked mathematics toughest problem, but it may not turn out to be so. If any readers do have a proof remember that there is $ 1 million awaiting for you.



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Tuesday, 26 May 2020
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