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36 Chapter 2. Global aspects
types of meromorphic functions (see for instance [Boes89], [Lang31], and the references cited therein).
The complex plane is divided into 2h sectors
Sk ={t∈C|φk−1 ≤argt≤φk}, 1≤k≤2h
where the following holds true:
(i) φ0 ≤0<φ1 <φ2 <···<φ2h =φ0 +2π≤2π.
(ii) For 1 ≤ k ≤ 2h there is a permutation σk of {1,...,N} with R(tζ(aσk(1))) ≤ R(tζ(aσk(2))) ≤ ··· ≤ R(tζ(aσk(N))) for t ∈ Sk.
(2.31)
(iii) The sectors Sk are the maximal sectors (3.3.4) with properties (i) and (ii).
We note that the permutations σk are in general not yet uniquely determined; this is not necessary. In addition here, for convenience, we assume σ1 = id.
Notice that the eigenvalues λ1,...,λm+n and consequently also ζ(aσk(1)), ..., ζ(aσk(N)), 1 ≤ k ≤ 2h, of a real matrix M are either real or appear in conjugate complex pairs; therefore the arguments in (i) fulfill the symmetry relation φk+h = φk + π, or in other words t ∈ Sk, 1 ≤ k ≤ h, if and only if −t ∈ Sk+h. Replacing −t ∈ Sk+h in (ii) shows that the permutation σk+h just reverses the order of the real parts, i.e.,
R(−tζ(aσk(N))) = R(tζ(aσh+k(1))) ≤ ··· ≤ R(−tζ(aσk(1))) = R(tζ(aσh+k(N))),
for t ∈ Sh+k, 1 ≤ k ≤ h. From the results on the asymptotic behavior of the
zeros of exponential sums [Lang31] (see also [Boes89], Satz 2.1.1 for further de-
tails) and from formula (2.14) we infer in the next lemma that lim W (reiφ, WC ) r→∞
exists for φ ̸= φj, j = 1,...,2h.
Lemma 2.4.1. Let ε > 0 and let C ∈ C(m+n)×n be such that Q0 in Theorem 2.3.1 is regular. Then almost all zeros (i.e., except at most finitely many) of det Q(t), as given in (2.18), are contained in the set
2h
{t∈C|φk −ε<argt<φk +ε}.
k=1
Additionally, for φ ∈ [0,2π] \ {φ1,...,φ2h}, 1 ≤ l ≤ m and 1 ≤ α ≤ n there
exists
wlα(eiφ · ∞; WC ) := lim wlα(reiφ; WC ) ∈ Cˆ (2.32) r→∞
with respect to the chordal metric on the Riemann sphere Cˆ = C ∪ {∞}. The limits are easily determined from (2.30).