Bilinear forms with Kloosterman fractions and applications

We establish improved bounds for bilinear forms with Kloosterman fractions of the form ${\sum\sum}_{m,n} \alpha_m \beta_n e(a\overline{m}/(bn))$ with $M<m\le 2M$, $N < n \le 2N$ and $(m,n)=1$. Our approach works directly with arbitrary coefficient sequences $(\alpha_m), (\beta_n) \in \mathbb{C}$, avoiding the temporary restriction to squarefree support used in prior work. While this requires handling additional arithmetic complexity, it yields strictly stronger bounds that improve upon the estimates of Duke, Friedlander, and Iwaniec \cite{DFI} and Bettin-Chandee \cite{BC}; in the balanced case $M \approx N$, the new saving over the trivial bound is $1/12$%, compared to $1/48$ in \cite{DFI} . As an application, we prove a generalized asymptotic formula for the twisted second moment of the Riemann zeta-function with Dirichlet polynomials of length $T^{1/2+\delta}$ for $\delta = 1/46$, extending beyond the previously limiting $\theta = 1/2$ barrier established by Bettin, Chandee, and Radziwi{\l}{\l} \cite{BCR}. We also establish bounds for related Hermitian sums involving Sali\'{e}-type exponential phases and develop techniques for more general bilinear forms with Kloosterman fractions.