JournalsjemsVol. 22, No. 12pp. 3953–3980

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

  • Sandro Bettin

    Università di Genova, Italy
  • Hung M. Bui

    The University of Manchester, UK
  • Xiannan Li

    Kansas State University, Manhattan, USA
  • Maksym Radziwiłł

    California Institute of Technology, Pasadena, USA
A quadratic divisor problem and moments of the Riemann zeta-function cover
Download PDF

A subscription is required to access this article.


We estimate asymptotically the fourth moment of the Riemann zeta-function twisted by a Dirichlet polynomial of length T14εT^{\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 T111εT^{\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 am1m2bn1n2=ham_1m_2 - bn_1n_2 = h is summed with smooth averaging on the variables m1,m2,n1,n2,hm_1, m_2, n_1, n_2, h and arbitrary weights in the average on a,ba,b. Using Watt's work allows us to exploit all averages simultaneously. It turns out that averaging over m1,m2,n1,n2,hm_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.

Cite this article

Sandro Bettin, Hung M. Bui, Xiannan Li, Maksym Radziwiłł, A quadratic divisor problem and moments of the Riemann zeta-function. J. Eur. Math. Soc. 22 (2020), no. 12, pp. 3953–3980

DOI 10.4171/JEMS/999