Master's Thesis presentation in matematics, Alf Söderberg

Abstract: We study the family of L-functions attached to Hecke newforms of even weight k and squarefree level N and their low-lying zeroes by means of the 1-level density. We follow Iwaniec, Luo and Sarnak in verifying the Katz-Sarnak Density Conjecture for limited support, conditional on the GRH. Then, we investigate a lower order term found by Miller when k is fixed and N tends to infinity through the primes. Finally, we study the 1-level density through the Ratios Conjecture. We verify Miller's conclusion that it accurately predicts the shape of the 1-level density.