THE TWISTED FOURTH MOMENT OF THE RIEMANN ZETA FUNCTION

We compute the asymptotics of the fourth moment of the Riemann zeta func- tion times an arbitrary Dirichlet polynomial of length T 1 11 " . The study of the moments of the Riemann zeta function has a long and distinguished history, starting with the work of Hardy and Littlewood in 1918 and continuing to the present day. One motivation for understanding moments is that they yield information about the maximum size of the zeta function (the Lindelof Hypothesis); another application is to zero density estimates which in turn have consequences for primes in short intervals. However they have become an interesting topic in their own right. Very few rigorous results are known, just the second and fourth power moments. Indeed, it is only recently that a believable conjecture for higher powers has been made. The twisted moments (that is, moments of the Riemann zeta function times an arbitrary Dirichlet polynomial) are important too, for example Levinson's method of detecting zeros of the zeta function lying on the critical line requires knowing the asymptotics of the mollified second moment. In a series of papers, Duke, Friedlander, and Iwaniec used estimates for amplified moments of a family of L-functions in order to deduce a subconvexity bound for an individual member of the family. Of course, there are far easier methods to give a subconvexity bound for zeta, but there are close analogies between different families and it is desirable to understand the structure of these amplified moments in general. In this paper, we prove an asymptotic formula for the twisted fourth moment of the Riemann zeta function, where we may take a Dirichlet polynomial of length up to T 1 11 −e .