math.NT — Number Theory
REAL ZEROS OF L-FUNCTION DERIVATIVES: DENSITY-ONE ASYMPTOTICS IN ARITHMETIC FAMILIES
We study real zeros of the derivative of an uncompleted L-function on [1/2, 1]. The limiting stationary process has covariance sech2((t− s)/2) after the change of variables σ = 1/2+e^−t. Its nonzero-level Rice intensity gives the constant e^−1/2/(π√2) for quadratic Dirichlet characters, a specified family of quadratic Hecke characters over Q(i), and the unweighted level-one holomorphic family; the resulting density-one asymptotics are unconditional. For a fixed holomorphic modular form and its quadratic twists, and for non-Galois cubic Artin L- functions, we obtain the same constant under the respective family GRH. The real cyclic cubic augmentation instead has constant 1/(π√2). In these three GRH families a uniform quantile estimate controls all three counts: distinct real zeros, zeros with multiplicity, and strict sign changes. For an individual complex cubic character, the derivative has o(log log q) real zeros in density under cubic-character GRH. We also prove a uniform mean asymptotic and exponential concentration for the random Euler model.