Generalized divisor problem for new forms of higher level

Krishnamoorthy, Krishnarjun. "Generalized divisor problem for new forms of higher level." Czechoslovak Mathematical Journal 72.1 (2022): 259-263. <http://eudml.org/doc/297968>.

@article{Krishnamoorthy2022,
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 $k\ge 1$ 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 $k\ge 5$. 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.},
author = {Krishnamoorthy, Krishnarjun},
journal = {Czechoslovak Mathematical Journal},
keywords = {generalized divisor problem; cusp form of higher level},
language = {eng},
number = {1},
pages = {259-263},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Generalized divisor problem for new forms of higher level},
url = {http://eudml.org/doc/297968},
volume = {72},
year = {2022},
}

TY - JOUR
AU - Krishnamoorthy, Krishnarjun
TI - Generalized divisor problem for new forms of higher level
JO - Czechoslovak Mathematical Journal
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 72
IS - 1
SP - 259
EP - 263
AB - 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 $k\ge 1$ 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 $k\ge 5$. 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.
LA - eng
KW - generalized divisor problem; cusp form of higher level
UR - http://eudml.org/doc/297968
ER -