For pleasurable background reading, permit me to suggest the oft-told tale of the encounter between Freeman Dyson and Hugh Montgomery. One source is The Riemann hypothesis: the greatest unsolved problem in mathematics by Karl Sabbagh, Chapter 9, available via Google Books here.
...when Hugh Montgomery was visiting Princeton in 1972 and was introduced to ... Freeman Dyson over tea, he answered perfectly truthfully when Dyson asked him conversationally what he was working on. His answer struck a chord with Dyson, who then supplied a piece of information that indirectly led to what today is seen as the most promising approach to proving the Riemann Hypothesis.
The same story is told in the book Stalking the Riemann Hypothesis: The Quest to Find the Hidden Law of Prime, by Dan Rockmore; Google Books here. A bit more technical detail is in the Wikipedia article on Montgomery's conjecture. And here is a discussion by John Baez in his This Week's Finds column.
