The Sixth Power Moment of Dirichlet L-functions

Understanding moments of families of L-functions has long been an important subject with many number theoretic applications. Quite often, the application is to bound the error term in an asymptotic expression of the average of an arithmetic function. Also, a good bound for a moments can be used to obtain a point-wise bound for an individual L-function in the family; strong enough bounds of this type, known as subconvexity bounds, can often be used to obtain the equidistribution of an arithmetically defined set. Indeed, if one could accurately bound all moments one would obtain the Generalized Lindelöf Hypothesis (which is known to follow from the Generalize Riemann Hypothesis; see the recent work of Soundararajan [Sou1] for a very precise statement). Precise asymptotic formulas for moments are the tools to obtain refined information about L-functions, such as approximations to the Riemann Hypothesis (zero density estimates and results about the frequency of zeros on the critical line) and detailed information about the value distribution of the L-functions. It is only in the last ten years that enough of an understanding of the structure of moments has begun to emerge to gain insight into the last of these applications. The new vision began with the work of Keating and Snaith, for their recognition that L-values can be modeled by characteristic polynomials from classical compact groups, and to Katz and Sarnak for their realization that families of L-functions have symmetry types associated with them that reveal which of the classical groups to use to model the family. See [KaSa], [KS1], [KS2], and [CKRS]. Prior to these works, Conrey and Ghosh predicted, on number theoretic grounds, that