Riemann hypothesis is "is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 12". The Riemann zeta function is conventionally
represented as the sum:
Recently, I read in Peter Woit's blog that some researchers have published a paper that describes a Hamiltonian operator H that can be used to possibly solve this problem. This operator has the property that if the eigenfunctions obey a suitable boundary condition, then the associated eigenvalues correspond to the nontrivial zeros of the Riemann zeta function! The paper is also available as a preprint. In a sense, this paper says that one can set up a quantum system whose evolution "solves" Riemann hypothesis. To me this is a reasonable approach to the solution of the problem. And it reminds me of the work of Leonard Adlemam.
Recently I read an article that presented a novel idea by Nicolas Gisin. In a nutshell, Gisin says that
only a certain number of digits of real numbers have physical meaning. After some number of digits, for example, the thousandth digit, or maybe even the billionth digit, their values are essentially random.
This is very interesting because it means that there are no noncomputable numbers. Provided this idea is correct, we can easily decide if for example there are three 4s in the decimal expansion of π! The real problem of course is to agree on the number of significant digits. Once this problem is settled, then we can answer any question about physical real numbers. Another consequence of this idea would be that real numbers might be directly representable in even present computer hardware. What is left is to examine deeply this idea and see if it is actually valid.