Mollification of the fourth moment of Dirichlet $L$-functions

The Fourth Moment of Dirichlet L-functions

for certain explicitly computable constants ai. The difficult part of extending Ingham’s result to include the lower-order terms is asymptotically evaluating the off-diagonal terms. The family of all primitive Dirichlet L-functions of modulus q is similar in some ways to the Riemann zeta function in t-aspect, but is more difficult to analyze. In 1981, Heath-Brown obtained an asymptotic formula ...

The Fourth Moment of Dirichlet L-functions

Here ∑∗ denotes summation over primitive characters χ (mod q), φ(q) denotes the number of primitive characters (mod q), and ω(q) denotes the number of distinct prime factors of q. Note that φ(q) is a multiplicative function given by φ(p) = p − 2 for primes p, and φ(p) = p(1 − 1/p) for k ≥ 2 (see Lemma 1 below). Also note that when q ≡ 2 (mod 4) there are no primitive characters (mod q), and so ...

A Note on the Fourth Moment of Dirichlet L-functions

For χ a Dirichlet character (mod q), the moments of L(s, χ) have many applications, for example to the distribution of primes in the arithmetic progressions to modulus q. The asymptotic formula of the fourth power moment in the q-aspect has been obtained by Heath-Brown [1], for q prime, and more recently by Soundararajan [5] for general q. Following Soundararajan’s work, Young [7] pushed the re...

Mollification of the fourth moment of automorphic L-functions and arithmetic applications

In this paper, we compute the twisted fourth power moment of modular L-functions of prime level near the critical line. This allows us to prove some new non-vanishing results on the central values of automorphic L-functions, in particular whose obtained by base change from GL2(Q) to GL2(K) for K a cyclic field of low degree.

The Second Moment of Dirichlet Twists of Hecke L-functions