Mean Values of Quadratic Twists of AutomorphicL-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...

Non-vanishing of Quadratic Twists of Modular L-functions

Mean values of multiplicative functions

Let f(n) be a totally multiplicative function such that |f(n)| ≤ 1 for all n, and let F (s) = ∑∞ n=1 f(n)n−s be the associated Dirichlet series. A variant of Halász’s method is developed, by means of which estimates for ∑N n=1 f(n)/n are obtained in terms of the size of |F (s)| for s near 1 with 1. The result obtained has a number of consequences, particularly concerning the zeros of the p...

Decay of Mean-values of Multiplicative Functions

p 1−f(p) p diverges then the limit in (1.1) exists, and equals 0 = Θ(f,∞). Wirsing’s result settled an old conjecture of P. Erdős and Wintner that every multiplicative function f with −1 ≤ f(n) ≤ 1 had a mean-value. The situation for complex valued multiplicative functions is more delicate. For example, the function f(n) = n (0 6= α ∈ R) does not have a mean-value because 1 x ∑ n≤x n iα ∼ x 1+i...

ON p-ADIC PROPERTIES OF CENTRAL VALUES OF QUADRATIC TWISTS OF AN ELLIPTIC CURVE

We study p-indivisibility of the central values L(1, Ed) of quadratic twists Ed of a semi-stable elliptic curve E of conductor N . A consideration of the conjecture of Birch and Swinnerton-Dyer shows that the set of quadratic discriminants d splits naturally into several families FS , indexed by subsets S of the primes dividing N . Let δS = gcdd∈FS L(1, Ed) alg, where L(1, Ed) denotes the algeb...