Abstract. We show that the sum of the traces of Frobenius elements of Artin L-functions in a family of G-fields satisfies the Gaussian distribution under certain counting conjectures. We prove the counting conjectures for S4 and S5-fields. We also show central limit theorem for modular form L-functions with the trivial central character with respect to congruence subgroups as the level goes to infinity.

We compute the one-level density for the family of cubic Dirichlet L-functions when the support of the Fourier transform of the associated test function is in (−1, 1). We also establish the Ratios conjecture prediction for the one-level density for this family, and confirm that it agrees with the one-level density we obtain.In this work, we study the low-lying zeros of cubic Dirichlet L-functions. Let φ be an even Schwartz function whose Fourier transform is compactly supported. For a cubic Dirichlet character χ, let ρ denote the nontrivial zeros of L(χ, s) in the critical strip. Define

We prove that the complete L-function associated to any cuspidal automorphic representation of GL 2 (A Q ) has infinitely many simple zeros.

Let C be a conjugacy class of Sn and K an Sn-field. Let nK,C be the smallest prime which is ramified or whose Frobenius automorphism Frobp does not belong to C. Under some technical conjectures, we compute the average of nK,C . For S3 and S4-fields, our result is unconditional. For Sn-fields, n = 3, 4, 5, we give a different proof which depends on the strong Artin conjecture. Let NK,C be the smallest prime for which Frobp belongs to C. For S3-fields, we obtain an unconditional result for the average of NK,C for C = [(12)].Pollack [23] also computed the average of the least inert primes over cyclic number fields of prime degree.We generalize this problem to the setting of general number fields. We call a number field K of degree n, an S n -field if its Galois closure K over Q is an S n Galois extension. Let C be a conjugacy class of S n . For an unramified prime p, denote by Frob p , a Frobenius automorphism

Let K K be an S n S_n -field. For a nonidentity conjugacy class C C , define N K , C N_{K,C} to be the smallest prime p p such that Frob p ∈ C _p\in C . By using the observation that N K , C N_{K,C} is interpreted as the first prime sign change of the Dirichlet coefficients of automorphic L L -functions, we improve the known bound on N K , C N_{K,C} for n = 3 , 4 , 5 n=3,4,5 . (For n = 5 n=5 , we need to assume the strong Artin conjecture.)