We study fluctuations in the distribution of families of p-th Fourier coefficients a f (p) of normalised holomorphic Hecke eigenforms f of weight k with respect to SL 2 (Z) as k → ∞ and primes p → ∞. These families are known to be equidistributed with respect to the Sato-Tate measure. We consider a fixed interval I ⊂ [−2, 2] and derive the variance of the number of a f (p)'s lying in I as p → ∞ and k → ∞ (at a suitably fast rate). The number of a f (p)'s lying in I is shown to asymptotically follow a Gaussian distribution when appropriately normalised. A similar theorem is obtained for primitive Maass cusp forms.
In 1997, Serre proved an equidistribution theorem for eigenvalues of Hecke operators on the space S(N, k) of cusp forms of weight k and level N. In this paper, we derive an effective version of Serre's theorem. As a consequence, we estimate, for a given d and prime p coprime to N, the number of eigenvalues of the pth Hecke operator T p acting on S(N, k) of degree less than or equal to d. This allows us to determine an effectively computable constant B d such that if J 0 (N) is isogenous to a product of Q-simple abelian varieties of dimensions less than or equal to d, then N B d . We also study the effective equidistribution of eigenvalues of Frobenius acting on a family of curves over a fixed finite field as well as the eigenvalue distribution of adjacency matrices of families of regular graphs. These results are derived from a general "all-purpose" equidistribution theorem.
We study the distribution of the zeroes of the zeta functions of the family of Artin-Schreier covers of the projective line over Fq when q is fixed and the genus goes to infinity. We consider both the global and the mesoscopic regimes, proving that when the genus goes to infinity, the number of zeroes with angles in a prescribed non-trivial subinterval of [−π, π) has a standard Gaussian distribution (when properly normalized).
Abstract. After giving a brief overview of the theory of multiple zeta functions, we derive the analytic continuation of the multiple Hurwitz zeta functionusing the binomial theorem and Hartogs' theorem. We also consider the cognate multiple L-functions,where χ 1 , ..., χ r are Dirichlet characters of the same modulus.
The k-higher Mahler measure of a non-zero polynomial P is the integral of log k |P | on the unit circle. In this note, we consider Lehmer's question (which is a long-standing open problem for k = 1) for k > 1 and find some interesting formulas for 2-and 3-higher Mahler measure of cyclotomic polynomials.
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