Czechoslovak Mathematical Journal, Vol. 72, No. 1, pp. 259-263, 2022
Generalized divisor problem for new forms of higher level
Krishnarjun Krishnamoorthy
Received October 20, 2020. Published online June 21, 2021.
Abstract: 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 $k\geq1$ 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 $k\geq5$. 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.
Keywords: generalized divisor problem; cusp form of higher level
Affiliations: Krishnarjun Krishnamoorthy, Harish-Chandra Research Institute, Homi Bhabha National Institute, Chhatnag Road, Jhunsi, Allahabad, 211 019, India, e-mail: krishnarjunk@hri.res.in