Paper IV: Coefficient asymptotics from the nearest zeros

13 Coefficient asymptotics from the nearest zeros

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For an entire function \(f\), let \({\mathcal Z}(f)\) denote its multiset of nonzero zeros. Let \(r\) be a positive integer. Then \({\mathsf D}_N^{-r}\) is meromorphic and a simple zero of \({\mathsf D}_N\) becomes a pole of order \(r\).

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Theorem 13.1: Nearest-pole coefficient expansion

Let \(r\in{\mathbb Z}_{>0}\). Suppose the zeros of \({\mathsf D}_N\) of least modulus \(R\) are simple and form the finite set \({\mathcal Z}_R\). Suppose the next zero modulus is \(R_2>R\). At each \(\rho\in{\mathcal Z}_R\), write the principal part

\[ {\mathsf D}_N(z)^{-r} = \sum_{\ell=1}^{r} a_{\rho,\ell} \left(1-\frac z\rho\right)^{-\ell} + O(1). \]

Then, for every \(R_*\) with \(R<R_*<R_2\) and \(m=Nn\), as \(n\to\infty\),

\[ P_{N,n}(r) = m! \sum_{\rho\in{\mathcal Z}_R} \sum_{\ell=1}^{r} a_{\rho,\ell} \binom{m+\ell-1}{\ell-1} \rho^{-m} + O_{R_*}(m!R_*^{-m}). \]

The leading Laurent coefficient is

\[ a_{\rho,r} = (-\rho{\mathsf D}_N'(\rho))^{-r}. \]

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Proof of Theorem 13.1

Subtract the complete principal parts at the zeros on \(|z|=R\). The remainder is holomorphic on \(|z|<R_2\), hence on a neighborhood of every closed disk \(|z|\leq R_*<R_2\). Cauchy's estimate gives an ordinary coefficient bound \(O_{R_*}(R_*^{-m})\). The coefficient of \(z^m\) in

\[ \left(1-\frac z\rho\right)^{-\ell} \]

\[ \binom{m+\ell-1}{\ell-1}\rho^{-m}. \]

Multiplication by \(m!\) converts the ordinary coefficient to \(P_{N,n}(r)\).

Near a simple zero,

\[ {\mathsf D}_N(z) = -\rho{\mathsf D}_N'(\rho) \left(1-\frac z\rho\right) +O\left(\left(1-\frac z\rho\right)^2\right). \]

Raising to the power \(-r\) gives the formula for \(a_{\rho,r}\).

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Corollary 13.2: Leading dominant zero term

Under the hypotheses of Theorem 13.1,

\[ \begin{aligned} P_{N,n}(r) ={}&m!\binom{m+r-1}{r-1} \sum_{\rho\in{\mathcal Z}_R} \frac{\rho^{-m}}{(-\rho{\mathsf D}_N'(\rho))^r} \\ &+O\left(m!m^{r-2}R^{-m}\right) +O_{R_*}\left(m!R_*^{-m}\right). \end{aligned} \]

For \(r=1\), the first error term is absent.

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Proof of Corollary 13.2

The terms with \(\ell<r\) in Theorem 13.1 are bounded by a constant times \(m^{r-2}R^{-m}\). The term with \(\ell=r\) gives the displayed main term.

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Corollary 13.3: Orbit reduced nearest circle term

Assume the hypotheses of Theorem 13.1. Choose one representative from each orbit of the nearest zeros under multiplication by \(\omega\), and call the resulting set \({\mathcal Z}_R^{\mathrm{orb}}\). With \(m=Nn\),

\[ \begin{aligned} P_{N,n}(r) ={}&N m!\binom{m+r-1}{r-1} \sum_{\tau\in{\mathcal Z}_R^{\mathrm{orb}}} \frac{\tau^{-m}}{(-\tau{\mathsf D}_N'(\tau))^r} \\ &+O\left(m!m^{r-2}R^{-m}\right) +O_{R_*}\left(m!R_*^{-m}\right). \end{aligned} \]

For \(r=1\), the first error term is absent.

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Proof of Corollary 13.3

If \(\rho=\omega^k\tau\), then

\[ {\mathsf D}_N'(\rho)=\omega^{-k}{\mathsf D}_N'(\tau). \]

Therefore

\[ -\rho{\mathsf D}_N'(\rho) =-\tau{\mathsf D}_N'(\tau). \]

Since \(m=Nn\), one also has \(\rho^{-m}=\tau^{-m}\). Every orbit therefore contributes \(N\) equal terms in Corollary 13.2.

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Corollary 13.4: Secant asymptotic

For \(N=2\) and \(r=1\),

\[ P_{2,n}(1) \sim 2(2n)!\left(\frac2\pi\right)^{2n+1}. \]

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Proof of Corollary 13.4

The nearest zeros of \(\cos z\) are \(\rho=\pm\pi/2\), both simple. Their contributions in Corollary 13.2 are equal.

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