We prove an asymptotic formula with four main terms for the fourth moment of quadratic Dirichlet L-functions unconditionally. Our proof is based on the work of Li [15], Soundararajan [20], and Soundararajan-Young [21]. Our proof requires several new ingredients. These include a modified large sieve estimate for quadratic characters where we consider a fourth moment, rather than a second, as well as observing cross cancellations between diagonal and off-diagonal terms, which involve somewhat delicate combinatorial arguments.

We study the fourth moment of quadratic Dirichlet L-functions at s = 1 2 . We show an asymptotic formula under the generalized Riemann hypothesis, and obtain a precise lower bound unconditionally.For simplicity, we only focus on quadratic characters of the form χ 8d , and the result for χ d may be established similarly. We note that Soundararajan and Young [18] have established the

We obtain the asymptotic formula with an error term O(X 1 2 +ε ) for the smoothed first moment of quadratic twists of modular L-functions. We also give a similar result for the smoothed first moment of the first derivative of quadratic twists of modular L-functions. The argument is largely based on Young's recursive method [19,20].

No abstract

Let P denote the set of all primes. P1, P2, P3 are three subsets of P. Let δ(Pi) (i = 1, 2, 3) denote the lower density of Pi in P, respectively. It is proved that if δ(P1) > 5/8, δ(P2) ≥ 5/8, and δ(P3) ≥ 5/8, then for every sufficiently large odd integer n, there exist pi ∈ Pi such that n = p1 + p2 + p3. The condition is the best possible.