The fifth moment of modular $L$-functions

The Second Moment of Quadratic Twists of Modular L-functions

The family of quadratic twists of a modular form has received much attention in recent years. Motivated by the Birch-Swinnerton-Dyer conjectures, we seek an understanding of the central values of the associated L-functions, and while this question has been investigated extensively, much remains unknown. One important theme in this area concerns the moments of these central L-values. Thanks to t...

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 ...

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 ...

Modular Symbols and L-functions

As the rst non-overview lecture in this seminar, we will be setting up a lot of notation, getting comfortable working with modular symbols, and then hopefully discussing some of the major inputs which make the theory work. The rst half of the talk we will work through the example of the unique cusp form of weight two and level 11 on X0(11). In the second half, we will bring in the theoretical r...

The Second Moment of Dirichlet Twists of Hecke L-functions