Zero distribution of Dirichlet L-functions

Abstract

The series Lχ converges absolutely and uniformly in the region Re(s) ≥ 1 + e, for any e > 0. It therefore represents a holomorphic function on the half-plane Re(s) > 1, which further extends to a meromorphic function in the complex plane C. In particular, for the principal character χ = 1, we get back the Riemann zeta function ζ(s). For a ∈ C, the zeros of L χ − a which we denote by ρa = βa + iγa, are called the a-points of Lχ, and their distribution has long been an object of study (see [17, 18, 20]). The function Lχ has only real zeros in the half plane Re(s) < 0, these zeros are called the trivial zeros. If χ(−1) = 1, the trivial zeros of Lχ are s = −2n for all non-negative integers n. If χ(−1) = −1, the trivial zeros of Lχ are s = −2n − 1 for all non-negative integers n. Beside the trivial zeros of Lχ, there are infinitely many non-trivial zeros lying in the strip 0 < Re(s) < 1. For a 6= 0, it can be shown that there is always a a-point in some neighbourhood of any trivial zero of Lχ with sufficiently large negative real part, and with finitely many exceptions there are no other in the left half-plane, thus the number of these a-points having real part in [−R, 0] is asymptotically 1 2 R. The remaining a-points all lie in a strip 0 < Re(s) < A, where A depends on a, and we call these non-trivial a-points (see [16, 19]). For a positive number T , let N χ(T ) denote the number of non-trivial a-points ρa = βa+ iγa of Lχ with |γa| ≤ T . We have the following formula