Bilinear forms in the Fourier coefficients of half-integral weight cusp forms and sums over primes
Let \(\hat f_ n\) be the n-th Fourier coefficient of a holomorphic cusp form of weight \(k=1/2+\ell \geq 5/2\) for \(\Gamma_ 0(N)\), \(N\equiv 0(mod 4)\) and \(f_ n=\hat f_ nn^{-(k-1)/2}.\) For arbitrary complex numbers \(a_ m\) supported on square-free integers it is shown that \[ \sum_{n\leq X}\quad | \sum_{m\leq Y}a_ mf_{mn}|^ 2\quad \ll \quad (X+X^{1/2}Y^ 2)(XY)^{\epsilon} \| a\|^ 2 \] for any X,Y\(\geq 1\) and \(\epsilon >0\), the implied constant depending only on f and \(\epsilon\). Stronger estimates for bilinear forms with special coefficients are also given. These are combined to yield the following estimate for the sum over primes \[ \sum_{p<X}f_ p\quad \ll \quad X^{155/156+\epsilon}, \] where again the implied constant depends only on f and \(\epsilon\). Proofs are based on an exponential sums alternative to the Rankin-Selberg method.
scientific article; zbMATH DE number 4097478