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Explicit Estimates for the Zeros of Hecke L-functions | | Asif Zaman
; | | Date: |
19 Feb 2015 | | Abstract: | Let $K$ be a number field of degree $n_K$ with absolute discriminant $d_K =
|mathrm{disc}(K/mathbb{Q})|$ and absolute norm $mathbb{N} =
mathbb{N}^K_{mathbb{Q}}$. For an integral ideal $mathfrak{q}$, let
$mathrm{Cl}(mathfrak{q})$ be the ray class group modulo $mathfrak{q}$ of
Hecke characters $chi$ and let $L(s,chi)$ be the Hecke $L$-function attached
to $chi$ with $s = sigma + it in mathbb{C}$. Consider the functions [
Z_1(s) := prod_{substack{ chi in mathrm{Cl}(mathfrak{q}) \ chi
eq
chi_0} } L(s,chi) qquad ext{ and} qquad Z_0(s) := prod_{substack{ chi
in mathrm{Cl}(mathfrak{q}) } } L(s,chi). ] We establish various results on
$Z_1(s)$ and $Z_0(s)$, with explicit constants, for zero-free regions, zero
repulsion, Deuring-Heilbronn phenomenon, and zero density estimates. For
example, we prove $Z_1(s)$ and $Z_0(s)$ each have at most 1 zero, counting with
multiplicity, in the rectangle [ sigma geq 1 - frac{c}{ log d_K +
frac{3}{4} log mathbb{N}mathfrak{q} + n_K cdot
u(n_K)} qquad |t| leq
1, ] with $c = 0.1227$ for $Z_1(s)$ and $c=0.0875$ for $Z_0(s)$, provided
$d_K(mathbb{N}mathfrak{q}) e^{n_K}$ is sufficiently large and where $
u(x)$
is any fixed positive increasing function $geq 4$ satisfying $
u(x)
ightarrow infty$ as $x
ightarrow infty$. Further, if such an exceptional
zero $
ho_1$ exists, it is real and attached to a real character $chi_1$.
This zero-free region improves upon [Kad12, AK14] unless $n_K$ is unusually
large compared to the discriminant $d_K$. | | Source: | arXiv, 1502.5679 | | Services: | Forum | Review | PDF | Favorites | |
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