A quadratic divisor problem and moments of the Riemann zeta-function

Abstract

We estimate asymptotically the fourth moment of the Riemann zeta-function twisted by a Dirichlet polynomial of length $T^{\frac{1}{4}-\epsilon }$. 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 $T^{\frac{1}{11}-\epsilon }$ 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 considerably simplifies the combinatorics of the main terms in the twisted fourth moment.