The asymptotic formula of the fourth moment of Dirichlet L-functions at the central value was predicted in a conjecture by J. B. Conrey, D. W. Farmer, J. P. Keating, M. O. Rubinstein, and N. C. Snaith, and the prime moduli case was proved by M. P. Young in 2011. This work establishes this asymptotic formula for general moduli. The work relies on the study of a special divisor sum function, called \({\mathcal {D}}_q\)-function, which plays a key role in deducing the main terms. Another key ingredient is an application of new bounds for double sums in Kloosterman sums, applied in the estimate of the error terms.

JB Conrey、DW Farmer、JP Keating、MO Rubinstein 和 NC Snaith 在一个猜想中预测了Dirichlet L函数在中心值处的四阶矩的渐近公式，并且 MP Young 在 2011 年证明了素模情况. 这项工作为一般模量建立了这个渐近公式。这项工作依赖于对特殊除数和函数的研究，称为\({\mathcal {D}}_q\) -函数，该函数在推导主要项中起着关键作用。另一个关键因素是在 Kloosterman 和中应用双重和的新界限，应用于误差项的估计。