Fluctuations in the distribution of hecke eigenvalues

Abstract

A famous conjecture of Sato and Tate now a celebrated theorem of Taylor etal predicts that the normalised p th Fourier coeffcients of a non CM Heckeeigenform follow the Sato Tate distribution as we vary the primes p In 1997 Serre obtained a distribution law for the vertical analogue of the Sato Tatefamily where one fixes a prime p and considers the family of p th coefficientsof Hecke eigenforms In this thesis we address a situation in which we varythe primes as well as families of Hecke eigenforms In the same year Conrey Duke and Farmer obtained distribution measures for Fourier coefficients ofHecke eigenforms in these families Later in 2006 Nagoshi obtained similarresults under weaker conditions We consider another quantity namely thenumber of primes p for which the p th Fourier coefficient of a Hecke eigenformlies in a fixed interval I On averaging over families of Hecke eigenforms wederive an expression for the uctuations in the distribution of these eigenvaluesabout the Sato Tate measure Further the uctuations are shown tofollow a Gaussian distribution In this way we obtain a conditional CentralLimit Theorem for the quantity in question Similar results are also provedin the setting of Maass forms This extends a result of Wang 2014 whoproved that the Sato Tate theorem holds on average in the context of Maassforms In a separate project we consider a classical result in number theory Dirichlet stheorem on the density of primes in an arithmetic progression We proveia similar result for numbers with exactly k prime factors for k gt 1 Buildingupon a proof by E M Wright in 1954 we compute the asymptotic densityof such numbers where each prime satisfies a congruence condition As anapplication we obtain the density of squarefree n 8804 x with k prime factorssuch that a fixed quadratic equation has exactly 2k solutions modulo newline newline