Shimura sums related to imaginary quadratic fields
Motivated by the relation of the Fourier coefficients of modular forms and their Shimura lift, the author defines Shimura sums as follows: Let \(g(m)\) be an arithmetical function and \(c\) a positive integer. Then for \(m \geq 1\) the Shimura sum \(Sh(m,g,c)\) is defined by: \[ Sh(m,g,c) = \sum^{m-1}_{k=1} g \left( {m^ 2 - k^ 2 \over c} \right). \] In the special case when \(K = \mathbb{Q} (\sqrt {-2})\), the author proves that: \(P = \sqrt {{-Sh(4p,a,4) \over 2}}\), if \(p\) is inert in \(K\), and \(P = \sqrt {a^ 2(p) + {Sh(4p,a,4) \over 2}}\), if \(p\) splits or ramifies in \(K\). Here \(a(n)\) is the \(n\)-th coefficient of the Fourier expansion of some modular form of weight \(k = 3\) on \(\Gamma_ 0 (8)\) with Nebentypus \(\chi (d) = \left( {-2 \over d} \right)\).
scientific article; zbMATH DE number 701470