Generalized divisor problem for new forms of higher level

Abstract

Suppose that f is a primitive Hecke eigenform or a Mass cusp form for Γ₀(N) with normalized eigenvalues λf (n) and let X > 1 be a real number. We consider the sum ${{\cal S}_k}(X): = \sum\limits_{n < X} {\sum\limits_{n = {n_1},{n_2}, \ldots ,{n_k}} {{\lambda _f}({n_1}){\lambda _f}({n_2}) \ldots {\lambda _f}({n_k})}}$ and show that ${{\cal S}_k}(X){\ll _{f,\varepsilon }}{X^{1 - 3/(2(k + 3)) + \varepsilon}}$ for every k ⩾ 1 and ε > 0. The same problem was considered for the case N = 1, that is for the full modular group in Lü (2012) and Kanemitsu et al. (2002). We consider the problem in a more general setting and obtain bounds which are better than those obtained by the classical result of Landau (1915) for k ⩾ 5. Since the result is valid for arbitrary level, we obtain, as a corollary, estimates on sums of the form ${{\cal S}_k}(X)$, where the sum involves restricted coefficients of some suitable half integral weight modular forms.