Geodesic
Dirichlet and Poincaré series
Glasgow mathematical journal, Tome 26 (1985), pp. 39-56

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The study of modular forms has been deeply influenced by famous conjectures and hypotheses concerningwhere 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 seriesdefines an entire function of s and satisfies the functional equationThe 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. 
Good, A. Dirichlet and Poincaré series. Glasgow mathematical journal, Tome 26 (1985), pp. 39-56. doi: 10.1017/S0017089500006066
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