Article
Full entry |
PDF
(0.1 MB)
Feedback
PDF
(0.1 MB)
Feedback
Keywords:
generalized divisor problem; cusp form of higher level
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.
References:
[1] Aggarwal, K.: Weyl bound for GL(2) in $t$-aspect via a simple delta method. J. Number Theory 208 (2020), 72-100. DOI 10.1016/j.jnt.2019.07.018 | MR 4032289 | Zbl 1446.11092
[2] Booker, A. R., Milinovich, M. B., Ng, N.: Subconvexity for modular form $L$-functions in the $t$ aspect. Adv. Math. 341 (2019), 299-335. DOI 10.1016/j.aim.2018.10.037 | MR 3872849 | Zbl 06988563
[3] Fomenko, O. M.: On summatory functions for automorphic $L$-functions. J. Math. Sci., New York 184 (2012), 776-785. DOI 10.1007/s10958-012-0899-8 | MR 2870227 | Zbl 1266.11070
[4] Good, A.: The square mean of Dirichlet series associated to cusp forms. Mathematika 29 (1982), 278-295. DOI 10.1112/S0025579300012377 | MR 0696884 | Zbl 0497.10016
[5] Iwaniec, H., Kowalski, E.: Analytic Number Theory. Colloquium Publications 53. American Mathematical Society, Providence (2004). DOI 10.1090/coll/053 | MR 2061214 | Zbl 1059.11001
[6] Kanemitsu, S., Sankaranarayanan, A., Tanigawa, Y.: A mean value theorem for Dirichlet series and a general divisor problem. Monatsh. Math. 136 (2002), 17-34. DOI 10.1007/s006050200031 | MR 1908078 | Zbl 1022.11047
[7] Landau, E.: Über die Anzahl der Gitterpunkte in gewissen Bereichen. Gött. Nachr. 1915 (1915), 209-243 German \99999JFM99999 45.0312.02.
[8] Lü, G.: On general divisor problems involving Hecke eigenvalues. Acta. Math. Hung. 135 (2012), 148-159. DOI 10.1007/s10474-011-0150-y | MR 2898795 | Zbl 1265.11095
[9] Munshi, R.: Sub-Weyl bounds for $GL(2)$ $L$-functions. Available at https://arxiv.org/abs/1806.07352 (2018), 30 pages.
[10] Shimura, G.: On modular forms of half integral weight. Ann. Math. (2) 97 (1973), 440-481. DOI 10.2307/1970831 | MR 0332663 | Zbl 0266.10022
[11] Zhang, W.: Some results on divisor problems related to cusp forms. Ramanujan J. 53 (2020), 75-83. DOI 10.1007/s11139-019-00199-0 | MR 4148459 | Zbl 07343715
Search
Partner of
