In 1965, Sarvadaman Chowla conjectured that ∑_{n≤x} μ(F(n)) = o(x) for every squarefree polynomial F with integer coefficients. For F(x) = x this is equivalent to the prime number theorem, but it remains open for every F of higher degree; taking F to be a product of distinct linear factors recovers the familiar autocorrelation form of the conjecture. In this talk I consider the analogue over F_q[t] for the truncated Möbius function μ_M, which records only the prime factors of degree at most M. I will establish the two-point correlation estimate in this setting, explain the connection to the Erdős–Kac theorem that makes the truncated problem tractable, and describe what obstructs removing the truncation.
https://www.imsc.res.in/~anupdixit/IMSc-CMI-NT-seminar.html
We shall discuss counting reversible elements in the Picard group. This group exhibits a higher count than equidistribution would allow. This exceptional feature can be explained partly by its action on hyperbolic 3-space and partly by its arithmetic. We'll try to explore this and also compare it with the modular group which acts on the hyperbolic 2-space. The counting includes joint work with Debattam Das and Krishnendu Gangopadhyay.