Notation and glossary
Notation is defined locally when each family is introduced. The same letter can serve different roles in separate chapters. In particular, \(a\) is an arithmetic weight parameter in Chapter 2, a contraction in Chapter 6, and a spectral shift in Appendix D.
- Contracted feedback \(T_{q,r}\).
The normalized odd entire solution of \(T'_{q,r}(z)=1+T_{q,r}(qz)T_{q,r}(rz)\), \(T_{q,r}(0)=0\), for \(0<q,r<1\). The diagonal notation is \(f_q=T_{q,q}\).
- Feedback bank \(F_\Theta\).
A finite collection of channels with \(h_j\ge0\), \(\sum_jh_j=1\) and \(0<q_j,r_j<1\), entering \(F_\Theta'=1+\sum_jh_jF_\Theta(q_jz)F_\Theta(r_jz)\). Zero-weight channels are removed before taking an active-channel maximum.
- Maximum modulus and entire order.
\(M_f(R)=\max_{|z|=R}|f(z)|\). For a nonpolynomial entire function,
\[ \rho(f)=\limsup_{R\to\infty}\frac{\log\log M_f(R)}{\log R}. \]At positive finite order \(\rho\), the lower and upper types used here are the liminf and limsup of \(\log M_f(R)/R^\rho\). Their equality is not assumed.
- Coupled pair \((U,V)\).
The normalized entire solution of \(U'=1+U(qz)V(rz)\), \(V'=1+U(az)V(bz)\) and \(U(0)=V(0)=0\), with \(q,r,a,b\in(0,1)\).
- Perron balance.
\(\operatorname{spr}P\) denotes the spectral radius of a matrix. The characteristic exponent satisfies \(\operatorname{spr}\begin{pmatrix}q^\rho&r^\rho\\a^\rho&b^\rho\end{pmatrix}=1\). A positive right eigenvector gives the proportions of the two type vectors.
- Arithmetic depth \(A_{a,\beta,Q}\).
The entire sum \(\sum_{n\ge2,\ (n,Q)=1}\mu(n)^2n^{-2a}(1-n^{-\beta})^{2s}\), with \(a>1/2\), \(\beta>0\), \(Q\ge1\) integer and the real logarithm in the complex power. The atom \(n=1\) is excluded at every depth.
- Möbius function and modulus radical.
\(\mu(1)=1\). For a product of \(k\) distinct primes, \(\mu(n)=(-1)^k\). If a prime square divides \(n\), then \(\mu(n)=0\). The radical \(\operatorname{rad}(Q)=\prod_{p\mid Q}p\) records the excluded prime set, with \(\operatorname{rad}(1)=1\).
- Squarefree Dirichlet sum.
For \(v>1\), \(Z_Q(v)=\zeta(v)\zeta(2v)^{-1}\prod_{p\mid Q}(1+p^{-v})^{-1}\). The unrestricted notation is \(D(v)=Z_1(v)\). Thus \(A_{a,\beta,Q}(0)=Z_Q(2a)-1\).
- Prime-sieve measure.
\(\nu_{F,\boldsymbol\lambda}\) is the arithmetic atomic measure obtained by multiplying the mass at \(x_n=(1-n^{-2})^2\) by \(\prod_{p\in F}(1+\lambda_p\mathbf1_{p\mid n})\). It is positive exactly when every real strength satisfies \(\lambda_p\ge-1\).
- Hankel order and polynomial degree.
For moments \(M_j=\int x^j{\,d}\nu\), the order-\(m\) matrix is \(H_m=[M_{i+j}]_{0\le i,j<m}\). It tests quadratic forms of real polynomials of degree at most \(m-1\). At depth \(k\), the matrix is \([M_{k+i+j}]_{0\le i,j<m}\). An order-five test therefore covers degree four, not degree five.
- Signed quadratic form.
For real polynomials, \(Q_\nu(f)=\int f(x)^2{\,d}\nu(x)\). On a positive measure it is a squared norm. On a signed measure it may be negative and is not a norm.
- Orbit spectral zeta \(\mathcal Z_N\).
Let \(d_N(w)=\sum_{n\ge0}(-1)^nw^n/(Nn)!\) and \(D_N(z)=d_N(z^N)\), \(N\ge2\). The series \(\mathcal Z_N(s)=\sum_jw_j^{-s}\) initially runs over the zeros \(w_j\) of \(d_N\), counted with multiplicity, for \(\Re s>1/N\). Each \(w_j\) represents one \(N\)-fold rotation orbit of zeros of \(D_N\). Powers use \(\operatorname{Arg}w\in(-\pi,\pi]\).
- Rising factorial.
\((s)_k=s(s+1)\cdots(s+k-1)\) for \(k\ge1\), and \((s)_0=1\). It occurs in the shifted spectral residues.
- Coefficient conventions.
An ordinary generating function uses \(\sum_na_nz^n\). An exponential generating function uses \(\sum_na_nz^n/n!\). Their product and shift rules differ.
- Polyphase section and finite taps.
Polyphase splits a coefficient sequence by residue class. Finite impulse response (FIR) taps act by finite ordinary convolution. A phase matrix must retain the carry delay when a tap crosses a residue-class boundary. This is not the Hurwitz product of exponential generating functions.
- Resolution certificate.
An explicit bound or rational witness for a structural distinction at a specified radius, precision or polynomial degree. A sufficient threshold need not be optimal.
The word packet describes several analytic, arithmetic and spectral representations. Its meaning is fixed by the objects and maps in each construction, not by an assumed equivalence among the families.