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Keywords:
generalized divisor problem; cusp form of higher level
Summary:
Suppose that $f$ is a primitive Hecke eigenform or a Mass cusp form for $\Gamma _0(N)$ with normalized eigenvalues $\lambda _f(n)$ and let $X>1$ be a real number. We consider the sum $$ \mathcal {S}_k(X): = \sum _{n<X} \sum _{n=n_1,n_2,\ldots ,n_k} \lambda _f(n_1)\lambda _f(n_2)\ldots \lambda _f(n_k) $$ and show that $\mathcal {S}_k(X) \ll _{f,\epsilon } X^{1-3/(2(k+3))+\epsilon }$ for every $k\geq 1$ and $\epsilon >0$. 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 $k\geq 5$. Since the result is valid for arbitrary level, we obtain, as a corollary, estimates on sums of the form $\mathcal {S}_k(X)$, where the sum involves restricted coefficients of some suitable half integral weight modular forms.
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