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Dirichlet and Poincaré series

Published online by Cambridge University Press:  18 May 2009

A. Good
A. Good
Affiliation:
School of Mathematics, The Institute for Advanced Study, Princeton NJ 08540
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The study of modular forms has been deeply influenced by famous conjectures and hypotheses concerning

where T(n) denotes Ramanujan's function. The fundamental discriminant Δ is a cusp form of weight 12 with respect to the modular group. Its associated Dirichlet series

defines an entire function of s and satisfies the functional equation

The most penetrating statements that have been made on T(n) and LΔ(s)are:

Of these four problems only A1 has been established so far. This was done by Deligne [1] using methods from algebraic geometry and number theory. While B1 trivially holds with ε > 1/2, it was established in [2] for every ε>1/3. Serre [12] proved A2 for a positive proportion of the integers and Hafner [5] showed that LΔ has a positive proportion of its non-trivial zeros on the line σ=6. The proofs of the last three results are largely analytic in nature.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 27 , October 1985 , pp. 39 - 56
Copyright
Copyright © Glasgow Mathematical Journal Trust 1985

References

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