| Title:
|
Generalized divisor problem for new forms of higher level (English) |
| Author:
|
Krishnamoorthy, Krishnarjun |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
72 |
| Issue:
|
1 |
| Year:
|
2022 |
| Pages:
|
259-263 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
generalized divisor problem |
| Keyword:
|
cusp form of higher level |
| MSC:
|
11F30 |
| MSC:
|
11N37 |
| idZBL:
|
Zbl 07511565 |
| idMR:
|
MR4389118 |
| DOI:
|
10.21136/CMJ.2021.0451-20 |
| . |
| Date available:
|
2022-03-25T08:32:08Z |
| Last updated:
|
2024-04-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/149585 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
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| . |
