Reimann hypothesis commemorated 150 years later

The Riemann Hypothesis, named after 19^th-Century mathematician Bernhard Riemann, is one of the most influential and compelling problems in mathematics. The conjecture, with its simple statement and numerous ramifications, has intrigued mathematicians since its publication 150 years ago, but remains unproven. To celebrate what is often considered the most important open-problem in mathematics, two lectures were given on the Mines campus on November 18. Dr. Paul Martin spoke on the topic of the historical context of Riemann's work and Dr. Mark Coffey discussed the hypothesis itself. Often stated in English as “non-trivial zeroes of the Riemann zeta function have real part one-half,” the proof of Riemann's hypothesis would have major consequences in the field of Number Theory. Despite this, according to Martin, “This was the only thing he wrote that has anything to do with number theory.” However, Riemann's work primarily focused on complex analysis. According to Martin, the eponymous hypothesis “is really steeped in the context of functions of a complex variable.”
In Riemann's time, concepts relating to complex numbers were not well defined and mostly due to Augustin Cauchy. According to Martin, Riemann was able to improve Cauchy's work. “Riemann's work emphasized the geometric aspect [of complex variables]. Cauchy was suspicious of using geometrical ideas.” Riemann first made these improvements in his PhD thesis. “This first part [of his thesis] is a synthesis of what's known... he knew about Cauchy's work. Then he introduced the idea of Riemann sheets and surfaces... and then he had the idea of a Riemann mapping. That's incredibly powerful in the context of conformal mapping.”
All of the above are very important concepts in complex number theory. However, the last segment of this thesis, that of analytic continuation, inspired his most important hypothesis. “The question is this: supposing you are given a function and you know it's analytic [differentiable] in some region. Is it possible to extend this region so that it's still analytic,” Martin explained about this concept. “The answer is that usually you can... but not always.”
When analytic continuation works, there are often points in the region you're trying to extend to which are still not analytic; this is where Riemann hypothesis comes in. Riemann's zeta function converges to the right of x=1. Riemann wanted to use analytic continuation theory to extend it to the left of 1. When he tried to do this, he found that where the zeroes of the zeta function fall correspond to the distribution of prime numbers. Thus, by proving the Riemann Hypothesis, one can also know more about the distribution of prime numbers.
In his talk on the subject, Mark Coffey addressed how the hypothesis is related to the distribution of prime numbers. He showed charts of the zeros mathematicians have found; at least the first 10 trillion nontrivial zeros have real part 1/2. According to Coffey, “If you're a believer in the hypothesis, this makes you quite optimistic.”
Whether or not the proof or disproof of the hypothesis is ever found, the conjecture is one of the most important in mathematics. “Certainly the hypothesis itself has proved important to this day... it started this whole field,” according to Coffey, who continued, “In some sense, this is the paper of the millennium.”