At quite the end of the movie A Beautiful Mind, John Nash tells a student "I am making progress" (towards proving the Riemann hypothesis (RH)). Actually, how much did Nash contribute to the proof of the RH?
Nash was known to have been captivated by RH at an early age after reading E.T. Bell's Men of Mathematics. He had confided in some friends and colleagues that he had an idea that might work involving pseudoprimes, so there was a great deal of anticipation surrounding the announcement of his 1959 lecture at Columbia University sponsored by the American Mathematical Society.
From Karl Sabbagh's Dr Riemann's Zeros :
There have been several occasions in the past hundred years when mathematicians have announced that they have proved the Riemann Hypothesis. But none can be stranger than the events that took place in New York in 1959 .... So when two-hundred and fifty or so mathematicians gathered to listen to Nash, there was an air of expectation in the lecture hall. Perhaps he really had done it? But as Nash got deeper and deeper into his 'proof', the audience of mathematicians became more and more aghast. It wasn't just wrong - it was nonsense.
According to Donald Newman, 'One word didn't fit with the other. Everyone knew something was wrong. It was his chatter. The maths was just lunatic. ... Nash's talk wasn't good or bad. It was horrible.
There is no record of Nash demonstrating any significant progress toward a proof prior to the onset of his mental illness.