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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.

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Acknowledgements

The author is grateful to Dr. Kalyan Chakraborty for introducing the author to this problem and to the Kerala School of Mathematics for their generous hospitality. The author also thanks the referee for carefully reading the manuscript and suggesting corrections.

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Correspondence to Krishnarjun Krishnamoorthy.

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Krishnamoorthy, K. Generalized divisor problem for new forms of higher level. Czech Math J 72, 259–263 (2022). https://doi.org/10.21136/CMJ.2021.0451-20

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  • DOI: https://doi.org/10.21136/CMJ.2021.0451-20

Keywords

  • generalized divisor problem
  • cusp form of higher level

MSC 2020

  • 11N37
  • 11F30