ON THE GENERAL TRIPLE CORRELATION SUMS FOR GL2×GL2×GL2

Abstract

Fix $X\ge 2$. Let $f$ be a Hecke newform of prime level $p$. We investigate the general triple correlation sum

$$\sum _{h\ge 1}\sum _{l\ge 1}\sum _{n\ge 1}{\lambda }_f(n){\lambda }_f(n+h){\lambda }_f(n+l)U\left(\frac{n}{X}\right)V\left(\frac{h}{H}\right)R\left(\frac{l}{L}\right)$$

for $H,L\ge 1$ in the level aspect. As a result, we prove a nontrivial bound for any $H,L$ satisfying that and . It can be shown that there exist certain newforms such the nontrivial bound for the triple sum can be achieved, so long as $\text{max }\{H,L\}\ge X^{1∕4+𝜀}$. Particularly, whenever $L=H$, we present a nontrivial estimate for any $p$ such that , and further obtain the more significant cancellations for these sums in the different segments of $H$.