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Explore advanced research in analytic number theory through this 34-minute conference talk that presents improved bounds for the Fourier uniformity conjecture. Delve into the properties of the Liouville function λ(n) = (-1)^Ω(n), where Ω(n) represents the number of prime factors of n with multiplicity, and examine its conjectured pseudo-random statistical behavior. Learn about the connection between partial sums of the Liouville function and the Riemann Hypothesis, specifically the square-root cancellation estimate. Discover the Fourier uniformity conjecture's relationship to the Chowla and Sarnak conjectures, focusing on pseudo-random behavior in short intervals. Examine Walsh's 2023 breakthrough result for intervals of length exp((log X)^(1/2+ε)) ≤ H ≤ X and the non-correlation estimate for exponential sums involving the Liouville function. Follow the presentation of new improvements that extend the valid range to intervals of length H ≥ exp((log X)^(2/5+ε)), representing significant progress toward proving the full conjecture for arbitrarily slowly growing interval lengths.
