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Searching for sequences: Irrationality beyond Apery

Date: September 19, 2024 Speaker: Frank Calegari, University of Chicago Title: Searching for sequences: Irrationality beyond Apery Abstract: In 1978, Apery found a "miraculous" proof that zeta(3) is irrational, by finding an explicit pair of sequences of rational numbers a_n and b_n satisfying a recurrence relation so that their ratio a_n/b_n converged to zeta(3) "too quickly" for zeta(3) to be rational. Given another such pair of sequences, it is easy to verify experimentally whether or not the same "miracle" occurs. The problem is, it seems very hard in practice to find such miraculous sequences whose ratio converges to other interesting Dirichlet L-values. In recent work with Vesselin Dimitrov and Yunqing Wang, we have found (more or less) a weaker condition on the sequences a_n and b_n which implies rationality, and applied this to show that such numbers like L(2,chi_{-3}) = 1/1^2 - 1/2^2 + 1/4^2 - 1/5^2 + 1/7^2 - 1/8^2 + ,,, are irrational. The goal of this talk will be to sketch the basic idea, but the main point of the talk will be to explain how to detect sequences which could at least plausibly establish irrationality of interesting constants, and also where one might try to look for them.