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 result much further by computing the fourth moment for prime moduli q with a power saving in the error term. The problem essentially reduces to the analysis of a particular divisor sum. To this end, Young used various techniques to estimate the off-diagonal terms. In the case that the t-aspect is also included, a result of Montgomery [2] states that