An asymptotic formula with power-saving error is established for the twisted fourth moment of Dirichlet L-functions to a fixed prime power modulus q=q0^n0 with n0 at least 50.
The fourth moment of Dirichlet $L$-functions at the central value
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abstract
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.
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Twisted fourth moment of Dirichlet $L$-functions to a fixed modulus
An asymptotic formula with power-saving error is established for the twisted fourth moment of Dirichlet L-functions to a fixed prime power modulus q=q0^n0 with n0 at least 50.