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a.mod c/ ad 1.mod c/ maCnd 2 i c e: Here the numbers m and n are any integers, and the modulus c is a positive integer. Ac.m; n / is a \genuine " Kloosterman sum if mn ¤ 0, Ac.m; 0 / D Ac.0; m / is a Ramanujan sum if m ¤ 0, and Ac.0; 0 / is simply Euler's totient.c/. The significance of Kloosterman sums to the theory of modular forms dates back a century to an astonishingly little-known work of Poincare [10]. In 1926 Kloosterman [4] published his seminal paper regarding Ramanujan's problem of representing sufficiently large integers by quaternary quadratic forms. Since then these sums have surfaced with an almost unreasonable ubiquity throughout arithmetic. It is plain that Ac.m; n / D Ac. m; n / and hence Ac.m; n / is real. As such, it is natural to ask whether the sequence fAc.m; n/g1 cD1 is oscillatory for fixed integers m and n. That is, are there infinitely many c such that Ac.m; n /> 0 and infinitely many c such that Ac.m; n / < 0? Obviously, fAc.0; 0/g1 cD1 is positive. And it is clear that fAc.m; 0/g1 cD1 is oscillatory for m ¤ 0 because of the familiar formula (see, for example, [2, p. 238]

Topics:
Kloosterman sums, oscillatory sequences, Dirichlet series

Year: 2013

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oai:CiteSeerX.psu:10.1.1.352.7353

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