Abstract of My Philosphiae Doctor Thesis

This thesis was completed under the tutelage of Professor Henryk Iwaniec of Rutgers University from September of 2001 to April of 2003. The dissertation defense took place on 15 April 2003, with Professors Stephen David Miller and Jerrold Bates Tunnell of Rutgers University and Professor Patrick X. Gallagher of Columbia University on the defense committee.

The central theme that runs through this thesis is the large sieve type inequalities. We first extend the classical theory additive character to square moduli. One expects the result should be weaker than that of the classical inequalities, but, conjecturally at least, not by much. It is believed that our result is the best possible by the means of exponential sums.

We also developed large sieve type inequalities for special Dirichlet characters to prime square moduli. Our result yields non-trivial result in certain ranges and the method is generalizable to composite and higher power moduli. We also developed some large sieve type inequalities for quadratic amplitudes.

The thesis is concluded by considering the cancellations of Fourier coefficents at shifted primes, where we obtained a non-trivial upper bound for average of Fourier coefficients of cusp forms for the full modular group at primes twisted with an exponential function whose amplitude is the square root function. We also believe that this result is the best possible with the present technology.

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