  We estimate asymptotically the fourth moment of the Riemann zeta-function
twisted by a Dirichlet polynomial of length $T^{\frac14 - \varepsilon}$. Our
work relies crucially on Watt's theorem on averages of Kloosterman fractions.
In the context of the twisted fourth moment, Watt's result is an optimal
replacement for Selberg's eigenvalue conjecture.
  Our work extends the previous result of Hughes and Young, where Dirichlet
polynomials of length $T^{\frac{1}{11}-\varepsilon}$ were considered. Our
result has several applications, among others to the proportion of critical
zeros of the Riemann zeta-function, zero spacing and lower bounds for moments.
  Along the way we obtain an asymptotic formula for a quadratic divisor
problem, where the condition $a m_1 m_2 - b n_1 n_2 = h$ is summed with smooth
averaging on the variables $m_1, m_2, n_1, n_2, h$ and arbitrary weights in the
average on $a,b$. Using Watt's work allows us to exploit all averages
simultaneously. It turns out that averaging over $m_1, m_2, n_1, n_2, h$ right
away in the quadratic divisor problem simplifies considerably the combinatorics
of the main terms in the twisted fourth moment.
