A NOTE ON THE SHIMURA CORRESPONDENCE AND THE RAMANUJAN τ(n) FUNCTION

The Shimura correspondence is a family of maps which sends modular forms of half-integral weight to forms of integral weight by a Mellin transform [9]. Prior to Shimura’s work, Selberg discovered a special case of this correspondence when the half-integral weight form is a cusp eigenform on SL2(Z) times the classical theta function. In a recent paper [1], Cipra generalizes Selberg’s method and explicitly demonstrates the image of certain Shimura maps for half-integral weight forms that are newforms of type (k, χ) on Γ0(N) times a theta series. In this note, we apply this lift twice to a newform with complex multiplication by K = Q(i), and we obtain a formula for the Ramanujan function τ(n) in terms of the arithmetic of ideals in the ring OK via a Hecke character. These observations are connected to Lehmer’s conjecture on the nonvanishing of τ(n), the representations of integers as sums of 5 squares, and affine root systems of simple Lie algebras. 1. Cipra’s theorem In this section we state Cipra’s theorem which explicitly describes the image of a particular Shimura map applied to a newform times a theta series. We borrow Cipra’s notation for its clarity. Let χ be a Dirichlet character mod 4N, and let t be a positive square-free positive integer. Let F (τ) = ∑∞ n=1 b(n)q n ∈ Sk+ 1 2 (4N,χ) where k ∈ Z . Define At(n) by the following identity: ∞ ∑ n=1 At(n) ns = L(s− k + 1, χ t ) ∞ ∑