We propose a new approach to the value distribution theory of entire holomorphic curves. We define a packing density of an entire holomorphic curve, and show that it has various non-trivial properties. We prove a gap theorem for holomorphic maps from elliptic curves to the complex projective space, and study the relation between theta functions and our packing problem. Applying the Nevanlinna t...

This paper studies non-vanishing of L-functions associated to holomorphic cusp forms of weight k and level N at various points inside the critical strip. We will establish lower bounds in terms of level for the number of holomorphic cusp forms whose twisted L-functions with a fixed Dirichlet character do not vanish on a certain disc inside the critical strip.

In this paper we continue the study of free holomorphic functions on the noncommutative ball [B(H)]1 := n (X1, . . . , Xn) ∈ B(H) n : ‖X1X ∗ 1 + · · ·+ XnX ∗ n‖ 1/2 < 1 o , where B(H) is the algebra of all bounded linear operators on a Hilbert space H, and n = 1, 2, . . . or n = ∞. Several classical results from complex analysis have free analogues in our noncommutative setting. We prove a maxi...