Fourier coefficients of Eisenstein series of degree 3
The paper gives explicit formulas for the Fourier coefficients \(a_ k(T)\) of Eisenstein series of degree 3 and even weight k. By Siegel's works on Eisenstein series, \(a_ k(T)\) can be expressed as a product \(a_ k(T)=c\prod_{p}\alpha_ p(T,S),\) where c is a certain constant including the values of the gamma function, \(\alpha_ p(T,S)\) is the so- called p-adic density, and S is a certain matrix. The author gives explicit formulas for \(\alpha_ p(T,S)\) for any odd prime p and non-primitive T. One may note that the explicit formulas for \(\alpha_ p(T,S)\), (i) for odd prime p and primitive T, and (ii) for \(p=2\) and primitive T, have already been given by the reviewer and \textit{T. Washio} [J. Reine Angew. Math. 345, 148-171 (1983; Zbl 0512.10022)]. The case, when \(p=2\) and T is non-primitive, remains open.
scientific article; zbMATH DE number 3877279