At least 1/9 of zeros of L(s, Π₀ × χ) lie on the critical line for large Q, with new power-saving mean square asymptotics for PGL(3) twists (unconditional if self-dual) and stronger results for PGL(2).
arXiv:2507.20653 [math.NT] 18, 20
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For random hyperbolic surfaces of large genus, the variance of the weighted count of closed geodesics in short intervals [X, X+H] with H=o(X) is asymptotically 2H log X as genus tends to infinity.
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Critical Zeros and Unconditional Mean Value Theorems for twisted $\hbox{PGL}(2)$ and $\hbox{PGL}(3)$ $\mathrm{L}$-functions
At least 1/9 of zeros of L(s, Π₀ × χ) lie on the critical line for large Q, with new power-saving mean square asymptotics for PGL(3) twists (unconditional if self-dual) and stronger results for PGL(2).
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Closed geodesics in short intervals for random hyperbolic surfaces
For random hyperbolic surfaces of large genus, the variance of the weighted count of closed geodesics in short intervals [X, X+H] with H=o(X) is asymptotically 2H log X as genus tends to infinity.