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1 \pageheading{ORBIT SPECTRA\; /\; CONTINUATION THEOREM}{Global meromorphic continuation}
2 For every integer $N\ge2$, the orbit zeta has a meromorphic continuation to
3 the whole complex plane with exactly one pole:
4 \[
5 \boxed{s=1/N,\qquad
6 \operatorname{Res}_{s=1/N}\mathcal Z_N(s)=\frac{\sin(\pi/N)}{N\pi}.}
7 \]
8 Its exact absolute-convergence abscissa is $1/N$. At nonpositive integers,
9 \[
10 \boxed{\mathcal Z_N(-m)=0\quad(m\ge0),\qquad
11 \mathcal Z_N'(0)=-\log N.}
12 \]
13 The zeta-regularized product of the orbit coordinates is consequently
14 $\exp[-\mathcal Z_N'(0)]=N$.
15 \fig{15_spectral_poles}{Pole and zero locations follow from the proved
16 continuation. The second panel shows the shifted extension, where a pole
17 ladder appears. These are exact structural locations rather than fitted roots.}
18 The continuation uses classical Mellin regularization. The contribution here is a complete local application with multiplicities,
19 principal branches, and all convergence hypotheses checked.
20 
21 \pageheading{ORBIT SPECTRA\; /\; LARGE RADIUS}{A finite exponential sum controls the tail}
22 Put $E_N(t)=d_N(-t)=\sum_{n\ge0}t^n/(Nn)!$ and
23 $\omega=e^{2\pi i/N}$. For $x\ge0$,
24 \[
25 E_N(x^N)=\frac1N\sum_{j=0}^{N-1}e^{\omega^jx}.
26 \]
27 Root filtering proves this identity term by term: only powers divisible by
28 $N$ remain. The positive real exponential is dominant, so
29 \[
30 \log E_N(t)=t^{1/N}-\log N+O(e^{-\kappa_N t^{1/N}}),
31 \quad \kappa_N=1-\cos(2\pi/N)>0,
32 \]
33 with corresponding differentiated estimates on the positive ray.
34 \fig{14_spectral_tail}{Computed logarithmic derivatives of the finite
35 exponential sum approach one in the root-radius variable. The decaying
36 remainder supplies the Mellin continuation's large-radius input.}
37 \claim{Why complex zero arguments do not break the integral}
38 In a sufficiently narrow sector around negative $w$, choose the root
39 $x=(-w)^{1/N}$ near the positive real ray. The first exponential dominates
40 the other $N-1$ terms by a uniform real-part gap at large radius, so this
41 sector is eventually zero free. The finitely many remaining zeros avoid
42 the negative real ray because $d_N(-t)>0$ for $t\ge0$.
43 Thus all zero arguments have a uniform gap from the branch cut.
44 In particular $|t+w_j|\ge c_N(t+|w_j|)$ for $t\ge0$.
45 
46 \pageheading{ORBIT SPECTRA\; /\; MELLIN PROOF}{The two ends have different jobs}
47 For $1/N<\Re s<1$, the genus-zero log derivative and the angular gap give
48 absolute interchange in
49 \[
50 \mathcal Z_N(s)=\frac{\sin\pi s}{\pi}
51 \int_0^\infty t^{-s}\frac{d}{dt}\log E_N(t)\dd t.
52 \]
53 Each term uses the principal-branch beta integral
54 $\int_0^\infty t^{-s}(t+w)^{-1}\dd t=\pi w^{-s}/\sin\pi s$.
55 The denominator estimate bounds the absolute integral by a constant times
56 $|w|^{-\Re s}$, whose sum converges in this strip.
57 
58 Near zero write $f(t)=(\log E_N)'(t)=\sum_{k\ge1}\alpha_kt^{k-1}$,
59 where $\alpha_k=(-1)^{k+1}\mathcal Z_N(k)$.
60 For any integer $K\ge1$, subtract $K$ terms on $[0,1]$:
61 \[
62 \begin{aligned}
63 I_K(s)={}&\int_0^1t^{-s}\left[f(t)-\sum_{k=1}^K\alpha_kt^{k-1}\right]\dd t
64 +\sum_{k=1}^K\frac{\alpha_k}{k-s}\\
65 &+\int_1^\infty t^{-s}\left[f(t)-\frac1N t^{1/N-1}\right]\dd t
66 +\frac{1}{N(s-1/N)}.
67 \end{aligned}
68 \]
69 The first integral is holomorphic for $\Re s<K+1$. The second is entire because
70 its remainder decays exponentially in $t^{1/N}$. The factor $\sin\pi s$ cancels
71 the positive-integer poles. Increasing $K$ covers the plane, leaving only
72 $1/N$. At nonpositive integers $I_K$ is regular, so the sine factor gives zero.
73 
74 At zero the finite parts give
75 \(I_K(0)=\lim_{T\to\infty}[\log E_N(T)-T^{1/N}]=-\log N.\)
76 Since $(\sin\pi s/\pi)'_{s=0}=1$, this equals $\mathcal Z_N'(0)$.
77 Absolute convergence at a smaller real exponent would give a holomorphic
78 series across $1/N$, contradicting the pole and proving the exact abscissa.
79 
80 \pageheading{ORBIT SPECTRA\; /\; SHIFTS}{A Hurwitz-type deformation and its determinant}
81 For real $a\ge0$, define initially
82 \[
83 \mathcal Z_N(s;a)=\sum_j(w_j+a)^{-s},\qquad\Re s>1/N.
84 \]
85 The rightward shift preserves avoidance of the negative branch cut.
86 Normalize the product with $E_N(a)$ and use
87 $\log E_N(a+t)$ in the Mellin argument.
88 For $a>0$ there are genuine simple poles at $s=1/N-k$, $k\ge0$:
89 \[
90 \boxed{\operatorname{Res}_{s=1/N-k}\mathcal Z_N(s;a)
91 =\frac{\sin(\pi/N)}{N\pi}\frac{(1-1/N)_k}{k!}a^k.}
92 \]
93 Indeed the large-radius derivative expands as
94 \[
95 \frac1N(a+t)^{1/N-1}
96 =\frac1N\sum_{k\ge0}\frac{(-1)^k(1-1/N)_k}{k!}a^kt^{1/N-1-k}
97 \]
98 as an asymptotic series. The alternating sign is canceled by
99 $\sin(\pi(1/N-k))=(-1)^k\sin(\pi/N)$.
100 At $a=0$ all secondary residues vanish.
101 
102 The negative-integer values remain zero, and
103 \[
104 \mathcal Z_N'(0;a)=-\log[N E_N(a)],\qquad
105 \boxed{\prod_j^{\rm reg}(w_j+a)=N E_{N,1}(a).}
106 \]
107 The finite part $-\log N$ and normalization $-\log E_N(a)$ give a
108 zeta-regularized product, not an ordinary product. At $N=2$ it is
109 $2\cosh\sqrt a$, the classical half-integer spectral determinant reproduced
110 by the four shifted benchmarks. Functional equations and derivative-weighted
111 zero sums remain open here.

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