Kloosterman Sums, Fourier Coefficients of Cusp Forms and Multiplicative Number Theory

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Analytic number theory is the branch of pure mathematics in which one applies tools from analysis to study the distribution of the prime numbers and other interesting sequences such as the sequence of squarefree numbers. Tools from analysis include Fourier analysis, Fourier and Laplace transforms, automorphic forms and spectral theory, sieve methods, and character theory and exponential sums. A Kloosterman sum is a particular exponential sum that is frequently encountered in multiplicative number theory. While individual Kloosterman sums are known to satisfy general upper bounds, Kuznetsov proved a trace formula for GL_2 and used it to show that weighted sums of Kloosterman sums satisfy much sharper bounds. In 1982, Deshouillers and Iwaniec published a groundbreaking paper extending Kuznetsov's trace formula and combining it with the large sieve. Their resulting bounds on sums of Kloosterman sums is now know as 'Kloostermania" and has had a large number of applications to multiplicative number theory during the past decade. This is a proposal to re-examine the Kuznetsov trace formula and the Deshouillers-Iwaniec paper, with the goal of improving the Deshouillers-Iwaniec results.