On a sum involving Fourier coefficients of cusp forms (Q2471652)

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.