On a sum involving Fourier coefficients of cusp forms. (English) Zbl 1162.11337
Lith. Math. J. 46, No. 4, 459-474 (2006) and Liet. Mat. Rink. 46, No. 4, 565-583 (2006).
Let \(x \geq x_0\) (\(x_0\) is a sufficiently large real number), and, for integers \(j \geq 1\), define
\[
S_j(x)=\sum_{n \leq x} a(n^j),
\]
where \(a(n^j)\) is the \(n^j\)-th Fourier coefficient of the normalized Hecke eigenform \(f(z)\) defined over the full modular group \(\text{SL}(2,\mathbb Z)\). Then the author improves the upper bound for the quantity \(| \sum_{n \leq x}a(n^2)| \), i.e., he obtains that the estimate
\[
S_2(x) \ll x^{3/4}(\log x)^{19/2}\log\log x
\]
holds uniformly for any holomorphic cusp form \(f^*(z)\) of even integral weight \(k\) (with a Hecke eigenform \(f(z)\)) for the full modular group satisfying \(k \ll x^{1/3}(\log x)^{22/3}\), and the implied constant is effective.
Reviewer: Roma Kačinskaitė (Siauliai)
MSC:
| 11F30 | Fourier coefficients of automorphic forms |
| 11F66 | Langlands \(L\)-functions; one variable Dirichlet series and functional equations |
| 11F11 | Holomorphic modular forms of integral weight |
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