This dissertation is concerned with the zeros of holomorphic Hecke cusp forms in the space of newforms. We estimate a lower bound for the number of zeros on the imaginary axis and on the vertical line R(z)=1/2 in the upper half plane, both of which are outside the unit circle centered at the origin, and we denote these by δ1 and δ2 respectively. Ghosh and Sarnak call those zeros that lie on the rays 'real' including the arc z=exp (iθ), π/3 ≤ θ ≤ π/2, and they showed that a lower bound for the zeros on those geodesic lines is C log k for all sufficiently large weight k for the level 1 case. We extend their results to the newforms with levels N which are positive integers not divisible by 4 on δ2, and N which are positive integers on δ1. On δ2 we have C log k zeros if the weight k is sufficiently large and on δ1 we assume a nonnegativity result on the first negative Hecke eigenvalue and get a conditional result C log k zeros as the weight k goes to infinity. The analysis is closely related to the knowledge of Hecke eigenvalues λf (n). Most importantly it requires Deligne's bound λf (n) n^e (for every e > 0) with which we look into the proof of Theorem 3.1 in Ghosh and Sarnak cite[1], and get the same the approximation theorem for any level in Chapter 2. The estimation of zeros on δ1 also requires a `good' upper bound for the first negative Hecke eigenvalue for which we investigate an upper bound for central values of Hecke L-functions and a nonnegativity result on those values. Those will be studied in Chapters 3 and 4. In Chapter 5 we estimate lower bounds for the number of zeros on δi , i = 1, 2.



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Physical and Mathematical Sciences; Mathematics



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newforms real zeros