Paper IV: Hadamard products and zero power sums

Part IV Products, moments, and zero zeta functions

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12 Hadamard products and zero power sums

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The central idea

The product uses every reduced zero with its multiplicity. Its logarithm turns multiplication into addition, so reciprocal-zero power sums become coefficients. This step depends on the global product theorem. A truncated zero list provides a numerical partial sum, while the identity uses the entire multiset.

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Two order-two zero-moment partial sums over one through forty retained zeros approach exact dashed levels one half and one sixth from below. — Order-two moment sums over the first forty reduced zeros \(\lambda_k=\pi^2(k+1/2)^2\). The curves are finite numerical partial sums. The dashed limits \(1/2\) and \(1/6\) follow from the exact logarithmic coefficients and the product identity.

The reduction from \(z\) to \(w=z^N\) removes the rotational repetition of the zeros.

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Theorem 12.1: Genus zero product

For \(N\geq2\), let \(\Lambda_N\) be the nonzero zero multiset of \(\Delta_N\). Then

\[ \Delta_N(w) = \prod_{\lambda\in\Lambda_N} \left(1-\frac w\lambda\right), \]

where the product converges locally uniformly. Consequently,

\[ {\mathsf D}_N(z) = \prod_{\lambda\in\Lambda_N} \left(1-\frac{z^N}{\lambda}\right). \]

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Proof of Theorem 12.1

By Proposition 11.2, the order of \(\Delta_N\) is \(1/N<1\). By Corollary 11.13, it has infinitely many zeros, and Corollary 11.15 gives

\[ \sum_{\lambda\in\Lambda_N}|\lambda|^{-1}<\infty. \]

Hadamard factorization therefore gives a canonical product of genus zero. The only possible exponential factor is a nonzero constant because the order is below one. Since \(\Delta_N(0)=1\), that constant is one. Substitution of \(w=z^N\) gives the second formula.

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Theorem 12.2: Diagonal Fredholm determinant representation

Let \(I_N\) be a multiplicity-sensitive countable index set for \(\Lambda_N\) and put \(\mathcal H_N=\ell^2(I_N)\). On its standard orthonormal basis, define the diagonal operator

\[ K_Ne_\lambda=\lambda^{-1}e_\lambda. \]

Then \(K_N\) is trace class and

\[ \Delta_N(w)={\operatorname{det}_{\!F}}(I-wK_N). \]

For every \(m\geq1\),

\[ \kappa_{N,m}={\operatorname{tr}}(K_N^m). \]

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Proof of Theorem 12.2

By Corollary 11.15,

\[ \sum_{\lambda\in\Lambda_N}|\lambda|^{-1}<\infty. \]

Thus the diagonal operator \(K_N\) is trace class. Its Fredholm determinant is the locally uniform product

\[ {\operatorname{det}_{\!F}}(I-wK_N) = \prod_{\lambda\in\Lambda_N} \left(1-\frac w\lambda\right), \]

which equals \(\Delta_N(w)\) by Theorem 12.1.

Taking the logarithm gives

\[ -\log\Delta_N(w) = \sum_{m\geq1}\frac{w^m}{m}{\operatorname{tr}}(K_N^m) \]

near the origin. Comparison with the definition of \(\kappa_{N,m}\) proves the trace identity.

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This is the standard diagonal encoding of this family's genus zero product. It is not a general trace-class resolvent or Feshbach theorem.

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Corollary 12.3: Complete symmetric function form

For a positive integer \(r\), let each eigenvalue \(\lambda^{-1}\) of \(K_N\) be repeated \(r\) times. If \(h_n\) denotes the complete homogeneous symmetric function of degree \(n\) in this absolutely summable multiset, then

\[ \frac{P_{N,n}(r)}{(Nn)!}=h_n. \]

For arbitrary complex \(r\), the Bell polynomial formula is the polynomial continuation obtained by multiplying every power sum by \(r\).

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Proof of Corollary 12.3

The generating function for the complete homogeneous symmetric functions is

\[ \prod_{\lambda\in\Lambda_N} \left(1-\frac w\lambda\right)^{-r} = \Delta_N(w)^{-r}. \]

Coefficient comparison gives the first identity. Power sums determine the complete homogeneous functions through the Newton recurrence. Replacing each power sum by \(r\) times itself gives Theorem 7.2 and Theorem 7.6.

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Proposition 12.4: Products for the normalized sections

For every \(0\leq j<N\), the entire function

\[ \Phi_{N,j}(w) = j!\sum_{n\geq0}\frac{w^n}{(Nn+j)!} \]

has order \(1/N\). If \(N\geq2\) and \(\Lambda_{N,j}\) is its nonzero zero multiset, then

\[ \Phi_{N,j}(w) = \prod_{\mu\in\Lambda_{N,j}} \left(1-\frac w\mu\right). \]

For \(m\geq1\), put

\[ \sigma_{N,j,m} = \sum_{\mu\in\Lambda_{N,j}}\mu^{-m}. \]

Then

\[ C_{N,j,n}(r) = \frac{(Nn)!}{n!} B_n\bigl( r\sigma_{N,j,1}, r1!\sigma_{N,j,2}, \ldots, r(n-1)!\sigma_{N,j,n} \bigr). \]

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Proof of Proposition 12.4

The coefficient formula for the order gives \(1/N\). The function is not a polynomial. An entire function of order below one with only finitely many zeros would be a polynomial times a constant by Hadamard factorization, which is impossible here. The genus zero product follows as in Theorem 12.1.

Taking the logarithm gives

\[ -\log\Phi_{N,j}(w) = \sum_{m\geq1}\sigma_{N,j,m}\frac{w^m}{m}. \]

Since \({\mathsf C}_{N,j}(z)^r=\Phi_{N,j}(z^N)^{-r}\), the Bell polynomial calculation from Theorem 7.2 gives the coefficient formula.

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Definition 12.5

For \(m\geq1\), define

\[ \sigma_{N,m} = \sum_{\lambda\in\Lambda_N}\lambda^{-m}, \]

with multiplicities.

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The series converges absolutely because the exponent of convergence of \(\Lambda_N\) is \(1/N<1\).

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Theorem 12.6: Newton sums for the reduced zeros

For \(|w|\) below the least zero modulus, with the analytic logarithm normalized to zero at the origin,

\[ -\log\Delta_N(w) = \sum_{m\geq1}\sigma_{N,m}\frac{w^m}{m}. \]

Thus

\[ \kappa_{N,m}=\sigma_{N,m}. \]

\[ a_m=\frac{(-1)^m}{(Nm)!}, \qquad a_0=1, \]

then

\[ \sigma_{N,m} = -ma_m- \sum_{k=1}^{m-1}a_k\sigma_{N,m-k}. \]

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Proof of Theorem 12.6

Take the logarithm of the product in Theorem 12.1. Absolute convergence in a disk below the first zero permits termwise expansion

\[ -\log\left(1-\frac w\lambda\right) = \sum_{m\geq1}\frac{w^m}{m\lambda^m}. \]

This gives the logarithmic formula and identifies \(\kappa_{N,m}\) with \(\sigma_{N,m}\).

Differentiate the product identity in logarithmic form to obtain

\[ \Delta_N'(w) = -\Delta_N(w) \sum_{m\geq1}\sigma_{N,m}w^{m-1}. \]

Comparison of the coefficient of \(w^{m-1}\) gives

\[ ma_m = -\sum_{k=0}^{m-1}a_k\sigma_{N,m-k}. \]

Solving for \(\sigma_{N,m}\) gives the recurrence.

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Corollary 12.7: Convergent Mittag–Leffler partial-fraction form

Away from the zeros of \({\mathsf D}_N\),

\[ {\mathsf T}_N(z) = Nz^{N-1} \sum_{\lambda\in\Lambda_N} \frac1{\lambda-z^N}. \]

The series converges absolutely and locally uniformly on compact sets away from the poles. Thus no regularizing terms are required.

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Proof of Corollary 12.7

Logarithmic differentiation of the genus zero product gives

\[ -\frac{\Delta_N'(w)}{\Delta_N(w)} = \sum_{\lambda\in\Lambda_N}\frac1{\lambda-w}. \]

The series is locally uniform because \(\sum_{\lambda}|\lambda|^{-1}<\infty\). Substitute \(w=z^N\) and use the chain rule.

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Proposition 12.8: Theta Bell polynomial formula

Put

\[ \widetilde{\sigma}_{N,m}=(-1)^m\sigma_{N,m}. \]

Then

\[ {\mathrm{\Theta}}_{N,n}(u) = \frac{(Nn)!}{n!} B_n\bigl( \widetilde{\sigma}_{N,1}(u^N+u), 1!\widetilde{\sigma}_{N,2}(u^{2N}+u), \ldots, (n-1)!\widetilde{\sigma}_{N,n}(u^{nN}+u) \bigr). \]

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Proof of Proposition 12.8

The reduced unsigned denominator is

\[ \Psi_N(w)=\sum_{n\geq0}\frac{w^n}{(Nn)!}=\Delta_N(-w). \]

By Theorem 12.6,

\[ -\log\Psi_N(w) = \sum_{m\geq1}\widetilde{\sigma}_{N,m}\frac{w^m}{m}. \]

With \(w=z^N\), the logarithm of the theta generating function is

\[ \sum_{m\geq1} \widetilde{\sigma}_{N,m}(u^{Nm}+u)\frac{w^m}{m}. \]

The complete Bell polynomial formula follows.

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Corollary 12.9: Tangent coefficients as zero sums

For every \(n\geq0\),

\[ T_N(n) = N(Nn+N-1)!\sigma_{N,n+1}. \]

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Proof of Corollary 12.9

Combine Proposition 8.2 and Theorem 12.6.

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Corollary 12.10: Reciprocal coefficients from zero data

For every \(n\geq0\),

\[ P_{N,n}(r) = \frac{(Nn)!}{n!} B_n\bigl( r\sigma_{N,1}, r1!\sigma_{N,2}, \ldots, r(n-1)!\sigma_{N,n} \bigr). \]

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Proof of Corollary 12.10

Use Theorem 7.2 and Theorem 12.6.

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Example 12.11: The cosine orbit sums

For \(N=2\),

\[ \Delta_2(w)=\cos\sqrt w. \]

Its zeros are

\[ \lambda_m=\pi^2\left(m+\frac12\right)^2, \qquad m\geq0. \]

Hence

\[ \sigma_{2,q} = \frac{2^{2q}-1}{\pi^{2q}}\zeta(2q). \]

The identity in Corollary 12.9 recovers the classical tangent coefficients from these zero power sums.

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