The dyadic Cauchy-kernel identity:
several roads back to classical objects
June 10, 2026
This is an expository note. We take the dyadic Cauchy-kernel identity of Castillo–Costin–Costin [1] — a global rational/factorial decomposition built on the polylogarithm — and follow it down several specializations. In each direction it returns to a classical landmark: the polylogarithm duplication formula and Hurwitz’s Fourier-series formula (§sec:sec:sec:sine?); representations of \(\zeta\) at the special argument \(\pi\) and at rational arguments, in the neighborhood of Hurwitz’s multiplication theorem (§sec:sec:sec:zeta?); the Hasse–Sondow globally convergent series (§sec:sec:sec:mellin?); and, through its discrete scale invariance, the extra zeros of the Dirichlet eta function together with the harmonic-sum asymptotics of Flajolet–Gourdon–Dumas, with Dirichlet \(L\)-values emerging as the amplitudes of a log-periodic oscillation (§sec:sec:sec:Lvalues?). The aim is unification: to exhibit one compact identity as an organizing center from which these classical results may be read off. We claim no new theorems; where an identity may not previously have been displayed in exactly this form, we say so and explain why it is nonetheless a recombination of known ingredients.
Our starting point is the following identity, established in [1]: for \(\mathfrak{R}p>0\) and \(0<\mathfrak{R}s<1\), \[\label{GRAid} \pi p^{s-1}=\Gamma(s)\sin(\pi s)\left[\operatorname{Li}_s\!\left(e^{-p}\right)-\sum_{k=1}^\infty 2^{-k(1-s)}\operatorname{Li}_s\!\left(-e^{-2^{-k}p}\right)\right],\tag{1}\] where \(\operatorname{Li}_s(z)=\sum_{n\ge1}z^n/n^s\) is the polylogarithm. We refer to 1 as the dyadic Cauchy-kernel identity: it expresses the homogeneous function \(p^{s-1}\) through a polylogarithm at \(e^{-p}\) corrected by a geometrically weighted sum of polylogarithms at the dyadically contracted, sign-reversed points \(-e^{-2^{-k}p}\).
It is worth recording at the outset what 1 is, because it organizes everything that follows.
Lemma 1. For \(\mathfrak{R}p>0\) and \(0<\mathfrak{R}s<1\), the bracket in 1 telescopes: \[\label{telescope} \operatorname{Li}_s\!\left(e^{-p}\right)-\sum_{k=1}^\infty 2^{-k(1-s)}\operatorname{Li}_s\!\left(-e^{-2^{-k}p}\right)=\lim_{k\to\infty}2^{-k(1-s)}\operatorname{Li}_s\!\left(e^{-2^{-k}p}\right)=\Gamma(1-s)\,p^{s-1}.\qquad{(1)}\] In particular, modulo the reflection formula \(\Gamma(s)\Gamma(1-s)\sin(\pi s)=\pi\), the dyadic Cauchy-kernel identity is the iterated polylogarithm duplication formula, telescoped.
Proof. The duplication (“square”) formula for the polylogarithm [2] reads \(\operatorname{Li}_s(z)+\operatorname{Li}_s(-z)=2^{1-s}\operatorname{Li}_s(z^2)\). With \(z=e^{-2^{-k}p}\), so that \(z^2=e^{-2^{-(k-1)}p}\), \[\operatorname{Li}_s\!\left(-e^{-2^{-k}p}\right)=2^{1-s}\operatorname{Li}_s\!\left(e^{-2^{-(k-1)}p}\right)-\operatorname{Li}_s\!\left(e^{-2^{-k}p}\right).\] Writing \(b_k:=2^{-k(1-s)}\operatorname{Li}_s(e^{-2^{-k}p})\) and multiplying by \(2^{-k(1-s)}\) gives \(2^{-k(1-s)}\operatorname{Li}_s(-e^{-2^{-k}p})=b_{k-1}-b_k\), so the dyadic sum telescopes: \[\sum_{k=1}^\infty 2^{-k(1-s)}\operatorname{Li}_s\!\left(-e^{-2^{-k}p}\right)=\sum_{k=1}^\infty(b_{k-1}-b_k)=b_0-\lim_{k\to\infty}b_k=\operatorname{Li}_s\!\left(e^{-p}\right)-\lim_{k\to\infty}b_k.\] The bracket therefore equals \(\lim_k b_k\). Finally, the standard boundary expansion \(\operatorname{Li}_s(e^{-\varepsilon})=\Gamma(1-s)\varepsilon^{s-1}+\sum_{m\ge0}\zeta(s-m)(-\varepsilon)^m/m!\) as \(\varepsilon\to0^+\) gives, with \(\varepsilon=2^{-k}p\), \[\lim_{k\to\infty}b_k=\lim_{k\to\infty}2^{-k(1-s)}\Gamma(1-s)(2^{-k}p)^{s-1}=\Gamma(1-s)\,p^{s-1},\] since \(2^{-k(1-s)}2^{-k(s-1)}=1\). Combining with 1 and the reflection formula recovers \(\pi p^{s-1}=\pi p^{s-1}\). ◻
Lemma 1 is the expository thesis in miniature: the dyadic identity is built from a single classical ingredient (duplication) plus a boundary asymptotic. The sections that follow specialize 1 in various ways, and each time the same phenomenon recurs — a classical object emerges. We have verified the numerical identities reported below to high precision; we indicate this where relevant.
Setting \(p=\pm i\omega\) in 1 and symmetrizing yields a real Fourier-type series.
Proposition 2. Given \(\omega>0\) with \(\omega\notin 2\pi\mathbb{Z}\) and \(s\in\mathbb{C}\) with \(0<\mathfrak{R}s<1\), \[\label{Clausian} \omega^{s-1}=\frac{1}{\Gamma(1-s)}\sum_{n=1}^{\infty}\frac{1}{n^s}\left\{\sin\!\left(\frac{\pi s}{2}+n\omega \right) -(-1)^n\sum_{k=1}^{\infty} \frac{\sin\!\left(\frac{\pi s}{2}+\frac{n\omega}{2^k}\right)}{2^{k(1-s)}}\right\}.\qquad{(2)}\]
Proof. Recall the Clausen functions \(C(\omega,s)=\sum_n\cos(n\omega)/n^s\) and \(S(\omega,s)=\sum_n\sin(n\omega)/n^s\), so that \(\operatorname{Li}_s(e^{\mp i\omega})=C(\omega,s)\mp iS(\omega,s)\) and \(\operatorname{Li}_s(-e^{\mp i\omega2^{-k}})=\sum_n(-1)^n n^{-s}e^{\mp in\omega2^{-k}}\). Evaluating 1 at \(p=i\omega\) and at \(p=-i\omega\) and expanding the polylogarithms as power series gives the conjugate pair \[\begin{align} \pi (i\omega)^{s-1}&=\Gamma(s)\sin(\pi s)\sum_{n=1}^\infty \frac{1}{n^s}\Big\{ e^{-in\omega}-\sum_{k=1}^\infty 2^{-k(1-s)}(-1)^n e^{-in\omega 2^{-k}}\Big\},\tag{2}\\ \pi (-i\omega)^{s-1}&=\Gamma(s)\sin(\pi s)\sum_{n=1}^\infty \frac{1}{n^s}\Big\{ e^{in\omega}-\sum_{k=1}^\infty 2^{-k(1-s)}(-1)^n e^{in\omega 2^{-k}}\Big\}.\tag{3} \end{align}\] Since \((\pm i\omega)^{s-1}=e^{\pm i\pi(s-1)/2}\omega^{s-1}\), multiply 2 by \(e^{-i\pi(s-1)/2}\) and 3 by \(e^{i\pi(s-1)/2}\), so that each left-hand side becomes \(\pi\omega^{s-1}\). Using \(\Gamma(s)\sin(\pi s)=\pi/\Gamma(1-s)\) and the elementary identity \[\label{expansion} e^{\mp i\pi(s-1)/2}e^{\mp in\varphi}=\sin\!\left(\tfrac{\pi s}{2}+n\varphi\right)\pm i\cos\!\left(\tfrac{\pi s}{2}+n\varphi\right),\tag{4}\] and averaging the two resulting identities, the cosine series — single and dyadic alike — cancel, leaving \[\label{AfRefProp1Id} \omega^{s-1}=\frac{1}{\Gamma(1-s)}\Big\{\sum_{n=1}^\infty \frac{1}{n^s}\sin\!\left(\tfrac{\pi s}{2}+n\omega\right)-\sum_{k=1}^{\infty}2^{-k(1-s)}\sum_{n=1}^\infty \frac{(-1)^n}{n^s}\sin\!\left(\tfrac{\pi s}{2}+\tfrac{n\omega}{2^k}\right)\Big\}.\tag{5}\] The hypothesis \(\omega\notin2\pi\mathbb{Z}\) guarantees convergence of the Clausen series and that \(\omega2^{-k}\not\equiv\pi\pmod{2\pi}\) for every \(k\). It remains to justify the interchange of summations, for which we invoke the Moore–Osgood theorem [3]. By Abel summation, for \(z\in\mathbb{C}\setminus\{1\}\) with \(|z|=1\), \[\label{AbelEst} \left|\sum_{n=1}^N\frac{z^n}{n^s}\right|\leq \frac{C_s}{|1-z|},\qquad C_s=2\left(1+\frac{|s|}{\mathfrak{R}s}\right),\tag{6}\] uniformly in \(N\). Writing the inner sine as exponentials with \(\theta=\omega2^{-k}\) and applying 6 at \(z=-e^{\pm i\theta}\), \[\label{InnernSumEst} \left|\sum_{n=1}^N \frac{(-1)^n}{n^s}\sin\!\left(\tfrac{\pi s}{2}+n\theta\right)\right|\leq \frac{C_s\cosh\!\left(\tfrac{\pi \mathfrak{I}s}{2}\right)}{|1+e^{i\theta}|}.\tag{7}\] With \(\theta_k=\omega2^{-k}\to0\) we have \(|1+e^{i\theta_k}|\to2\) and each is positive, so \(\inf_k|1+e^{i\theta_k}|>0\); hence \[M_s:=\sup_{(k,N)\in\mathbb{N}^2}\left|\sum_{n=1}^N \frac{(-1)^n}{n^s}\sin\!\left(\tfrac{\pi s}{2}+n\theta_k\right)\right|\leq \frac{C_s\cosh\!\left(\tfrac{\pi \mathfrak{I}s}{2}\right)}{\inf_{k}|1+e^{i\theta_k}|}<\infty.\] For each fixed \(K\) the inner limit in \(N\) exists by the Dirichlet test [4], and writing \(a_{n,k}=(-1)^n n^{-s}\sin(\tfrac{\pi s}{2}+n\theta_k)\), \(S_{N,K}=\sum_{k\le K}\sum_{n\le N}2^{-k(1-s)}a_{n,k}\), \[|S_{N,\infty}-S_{N,K}|\leq M_s \sum_{k=K+1}^\infty 2^{-k(1-\mathfrak{R}s)}=\frac{M_s\, 2^{-K(1-\mathfrak{R}s)}}{2^{\,1-\mathfrak{R}s}-1},\] uniformly in \(N\). The two iterated limits therefore agree, which is the required interchange. ◻
Remark 3 (what is classical here). The single sum in ?? is, term for term, Hurwitz’s Fourier-series formula for the Hurwitz zeta function. Indeed, with \(a=\omega/2\pi\), \[\label{singleHurwitz} \sum_{n=1}^\infty\frac{1}{n^s}\sin\!\left(\tfrac{\pi s}{2}+n\omega\right)=(2\pi)^{s-1}\Gamma(1-s)\big[\zeta(1-s,a)-\cos(\pi s)\,\zeta(1-s,1-a)\big],\qquad{(3)}\] which is Hurwitz’s formula [5], [6], valid throughout \(0<\mathfrak{R}s<1\) by the extension of Boudjelkha [7]. By Lemma 1 the dyadic correction is the telescoped duplication formula. Thus Proposition 2 combines two classical ingredients — Hurwitz’s formula and the duplication formula — and produces no transcendental content beyond them.
Specializing ?? to \(\omega=\pi\) yields a representation of \(\pi^{s-1}\).
Corollary 4. For \(0<\mathfrak{R}s<1\), \[\label{piRepEq} \pi^{s-1} = \frac{1}{\Gamma(1-s)}\left\{- \sin\!\left(\tfrac{\pi s}{2}\right)(1-2^{1-s})\zeta(s) -\sum_{n=1}^{\infty}\sum_{k=1}^{\infty} \frac{(-1)^n}{n^s} \frac{\sin\!\left(\tfrac{\pi s}{2}+\tfrac{n\pi}{2^k}\right)}{2^{k(1-s)}}\right\}.\qquad{(4)}\]
Proof. At \(\omega=\pi\), the angle-addition formula gives \(\sin(\tfrac{\pi s}{2}+n\pi)=(-1)^n\sin(\tfrac{\pi s}{2})\), so the single sum in ?? factors as \(\sin(\tfrac{\pi s}{2})\sum_n(-1)^n n^{-s}\). The alternating Dirichlet series satisfies \(\sum_n(-1)^{n-1}n^{-s}=(1-2^{1-s})\zeta(s)\) (immediate for \(\mathfrak{R}s>1\) from \(\zeta(s)-2\cdot2^{-s}\zeta(s)\), and valid in the strip by continuation), whence \(\sum_n(-1)^n n^{-s}=-(1-2^{1-s})\zeta(s)\). The double sum is carried over unchanged. No appeal to the functional equation is required. ◻
Write \(\eta(s)=(1-2^{1-s})\zeta(s)\) for the Dirichlet eta function and \[\mathcal{D}(s):=\sum_{n=1}^{\infty}\sum_{k=1}^{\infty} \frac{(-1)^n}{n^s} \frac{\sin\!\left(\tfrac{\pi s}{2}+\tfrac{n\pi}{2^k}\right)}{2^{k(1-s)}}\] for the dyadic double sum. Combining Corollary 4 with Riemann’s functional equation gives a representation of the ratio of \(\zeta\) to its reflection.
Proposition 5. For \(0<\mathfrak{R}s<1\), away from zeros of \(\zeta(1-s)\), \[\frac{\zeta(s)}{\zeta(1-s)}=-2^s \sin\!\left(\tfrac{\pi s}{2}\right)\left[\sin\!\left(\tfrac{\pi s}{2}\right) \eta(s) +\mathcal{D}(s)\right].\]
Proof. By Corollary 4, \(\Gamma(1-s)\pi^{s-1}=-[\sin(\tfrac{\pi s}{2})\eta(s)+\mathcal{D}(s)]\). Riemann’s functional equation in the form \(\zeta(s)=2^s\pi^{s-1}\sin(\tfrac{\pi s}{2})\Gamma(1-s)\zeta(1-s)\) gives \(\zeta(s)/\zeta(1-s)=2^s\sin(\tfrac{\pi s}{2})(\pi^{s-1}\Gamma(1-s))\); substituting the previous display yields the assertion as an identity of meromorphic functions. ◻
Remark 6. Proposition 5 is equivalent to Corollary 4: substituting \(\mathcal{D}(s)=-\pi^{s-1}\Gamma(1-s)-\sin(\tfrac{\pi s}{2})\eta(s)\) collapses it identically to Riemann’s functional equation. It is recorded only because the ratio form is suggestive; it carries no information beyond Corollary 4.
Write \(\mathcal{D}(s;p/q)\) for the dyadic double sum at \(\omega=2\pi p/q\), \[\mathcal{D}(s;p/q):=\sum_{n=1}^{\infty}\sum_{k=1}^{\infty} \frac{(-1)^n\sin\!\left(\tfrac{\pi s}{2}+\tfrac{2 n\pi p}{q 2^k }\right)}{n^s 2^{k(1-s)}}.\] Specializing ?? to \(\omega=2\pi p/q\) and invoking Hurwitz’s formula gives a representation at rational arguments.
Proposition 7. Let \(p,q\in\mathbb{N}\) with \(0<p<q\) and \(0<\mathfrak{R}s<1\). Then \[\left(\frac{p}{q}\right)^{s-1}=\zeta\!\left(1-s,\tfrac{p}{q}\right)-\cos(\pi s)\,\zeta\!\left(1-s,1-\tfrac{p}{q}\right)-\frac{\mathcal{D}(s;p/q)}{(2\pi)^{s-1}\Gamma(1-s)}.\]
Proof. Set \(a:=p/q\in(0,1)\); then \(\omega=2\pi a\notin2\pi\mathbb{Z}\), so Proposition 2 applies. Its double sum is \(\mathcal{D}(s;a)\), and writing \(\omega^{s-1}=(2\pi)^{s-1}(p/q)^{s-1}\), \[\label{ratStart} \left(\tfrac{p}{q}\right)^{s-1}=\frac{\mathcal{T}(s;a)-\mathcal{D}(s;a)}{(2\pi)^{s-1}\Gamma(1-s)},\qquad \mathcal{T}(s;a):=\sum_{n=1}^\infty\frac{1}{n^s}\sin\!\left(\tfrac{\pi s}{2}+2\pi n a\right).\tag{8}\] With the periodic zeta function \(F(x,s):=\sum_n e^{2\pi inx}/n^s\) and \(c:=e^{i\pi s/2}\), expanding the sine gives \(\mathcal{T}(s;a)=\tfrac{1}{2i}(c\,F(a,s)-c^{-1}F(-a,s))\). Hurwitz’s formula [5], established for \(\mathfrak{R}s>1\), \(0<a\le1\) and continued to \(0<\mathfrak{R}s<1\), \(0<a<1\) via the conditional convergence of \(F(\pm a,s)\) (Dirichlet’s test, \(a\notin\mathbb{Z}\)) and the continuation of the Hurwitz zeta, reads \[\label{Hurwitz} \zeta(1-s,a)=\frac{\Gamma(s)}{(2\pi)^s}\big(c^{-1}F(a,s)+c\,F(-a,s)\big).\tag{9}\] Applying 9 at \(a\) and \(1-a\) (using \(F(-a,s)=F(1-a,s)\) and \(F(a-1,s)=F(a,s)\)) yields the system \(c^{-1}A+cB=Z_1\), \(cA+c^{-1}B=Z_2\), with \(A=F(a,s)\), \(B=F(-a,s)\), \(Z_1=\tfrac{(2\pi)^s}{\Gamma(s)}\zeta(1-s,a)\), \(Z_2=\tfrac{(2\pi)^s}{\Gamma(s)}\zeta(1-s,1-a)\). Solving and using \(c^2\pm c^{-2}=2\cos\pi s,\,2i\sin\pi s\), \[\mathcal{T}(s;a)=\tfrac{1}{2i}(cA-c^{-1}B)=\frac{Z_1-\cos(\pi s)Z_2}{2\sin\pi s}=\frac{(2\pi)^s}{2\sin(\pi s)\Gamma(s)}\big[\zeta(1-s,a)-\cos(\pi s)\zeta(1-s,1-a)\big].\] Dividing by \((2\pi)^{s-1}\Gamma(1-s)\) and using \(\Gamma(s)\Gamma(1-s)=\pi/\sin(\pi s)\) collapses the prefactor to \(1\), so \(\mathcal{T}(s;a)/[(2\pi)^{s-1}\Gamma(1-s)]=\zeta(1-s,a)-\cos(\pi s)\zeta(1-s,1-a)\). Substituting into 8 gives the claim. ◻
Remark 8. Proposition 7 is Hurwitz’s formula at rational arguments, with the dyadic term as a determined remainder. It lies adjacent to the Hurwitz multiplication theorem \(k^s\zeta(s)=\sum_{n=1}^k\zeta(s,n/k)\), which already supplies rational-argument relations such as \(\zeta(s,\tfrac12)=(2^s-1)\zeta(s)\).
There is a second, independent way the dyadic machinery meets the zeta function. Applying the global factorial expansion of [1] to the Mellin representation \[\zeta(s)=\frac{1}{s-1}+\frac{\sin(\pi s)}{\pi}\int_0^\infty\big(\ln(1+x)-\Psi(1+x)\big)x^{-s}\,dx \qquad(0<\mathfrak{R}s<1)\] produces a convergent double series of Mellin transforms of rational functions. Evaluating those integrals in closed form and summing the resulting geometric series in the dyadic index collapses the expansion to \[\label{HS} \zeta(s)=-\frac{2^{s-1}}{1-2^{s-1}}\,\eta_{\mathrm{HS}}(s),\qquad \eta_{\mathrm{HS}}(s)=\sum_{n=0}^\infty\frac{1}{2^{n+1}}\sum_{m=0}^n(-1)^m\binom{n}{m}(m+1)^{-s},\tag{10}\] in which \(\eta_{\mathrm{HS}}\) is precisely the Hasse–Sondow globally convergent series for \(\eta\) [8], [9]; one checks directly that \(-2^{s-1}/(1-2^{s-1})\) times \((1-2^{1-s})\zeta(s)=\eta(s)\) equals \(\zeta(s)\). We have verified 10 numerically, including on the critical line and at the first nontrivial zero. Thus the Mellin route, like the Fourier route of §2, recovers a classical convergent series; the finite-difference inner sum is the Euler-transformation (Nörlund–Rice) structure underlying Hasse–Sondow.
The dyadic weight \(2^{-k(1-s)}\) carries a scaling structure that we now isolate, following the analytic theory of harmonic sums [10]. For a profile \(\phi:(0,\infty)\to\mathbb{C}\) of suitable decay, set \[\label{harmonic} F_\phi(\omega):=\sum_{k=1}^\infty 2^{-k(1-s)}\phi\!\left(\omega 2^{-k}\right);\tag{11}\] the dyadic correction of Proposition 2 is \(F_\phi\) with profile \(\phi(\theta)=\sum_n(-1)^n n^{-s}\sin(\tfrac{\pi s}{2}+n\theta)\). Two structural facts govern \(F_\phi\).
Renormalization. A reindexing gives the exact functional equation \[\label{RG} F_\phi(2\omega)=2^{s-1}\big[\phi(\omega)+F_\phi(\omega)\big],\tag{12}\] the statement of dyadic discrete scale invariance: up to the boundary term, doubling \(\omega\) multiplies \(F_\phi\) by \(2^{s-1}\).
Complex dimensions. Taking the Mellin transform \(\widetilde{F}_\phi(\sigma)=\int_0^\infty F_\phi(\omega)\omega^{\sigma-1}d\omega\) and using \(\int_0^\infty\phi(\omega2^{-k})\omega^{\sigma-1}d\omega=2^{k\sigma}\widetilde{\phi}(\sigma)\), \[\label{MellinF} \widetilde{F}_\phi(\sigma)=\frac{\widetilde{\phi}(\sigma)}{2^{(1-s)-\sigma}-1},\tag{13}\] whose denominator vanishes exactly at the tower \[\label{tower} \sigma_n=(1-s)-\frac{2\pi i n}{\ln 2},\qquad n\in\mathbb{Z}.\tag{14}\] These are the complex dimensions of the dyadic structure in the sense of Lapidus–van Frankenhuijsen [11]: equally spaced on \(\mathfrak{R}\sigma=1-\mathfrak{R}s\) with period \(2\pi/\ln 2\). Equivalently, in the \(s\)-variable the resummed weight \(\sum_k 2^{-k(1-s)}=2^{s-1}/(1-2^{s-1})\) has poles at \(s=1+2\pi i n/\ln 2\) — the zeros of the dyadic Dirichlet polynomial \(1-2^{1-s}\), which are exactly the extra zeros of the Dirichlet eta function \(\eta(s)=(1-2^{1-s})\zeta(s)\) on the line \(\mathfrak{R}s=1\) [9].
By the Mellin–asymptotic correspondence for harmonic sums [10], transferring 13 across the tower gives, as \(\omega\to\infty\), \[\label{logperiodic} F_\phi(\omega)\sim\frac{\omega^{s-1}}{\ln 2}\sum_{n\in\mathbb{Z}}\widetilde{\phi}(\sigma_n)\,e^{2\pi i n\log_2\omega},\tag{15}\] so that \(F_\phi(\omega)/\omega^{s-1}\) is asymptotically periodic in \(\log_2\omega\) — a log-periodic oscillation whose Fourier coefficients are the values of the profile’s Mellin transform at the complex dimensions.
The arithmetic specialization is now immediate. For a Dirichlet character \(\chi\) take the profile \[\phi_\chi(\theta)=\sum_{m=1}^\infty\chi(m)e^{-m\theta},\qquad \widetilde{\phi}_\chi(\sigma)=\Gamma(\sigma)L(\sigma,\chi),\] so that the oscillation amplitudes in 15 are Dirichlet \(L\)-values at the tower: \[\label{Lamplitudes} c_n=\frac{\Gamma(\sigma_n)\,L(\sigma_n,\chi)}{\ln 2},\qquad \sigma_n=(1-s)-\frac{2\pi i n}{\ln 2}.\tag{16}\] For the character \(\chi_{-4}\) modulo \(4\) — conductor a power of two, so the character period matches the dyadic scale — the profile has the closed form \(\phi_{\chi_{-4}}(\theta)=\tfrac{1}{2}\operatorname{sech}\theta\), with \(L(\sigma,\chi_{-4})=\beta(\sigma)\) the Dirichlet beta function, giving \(c_n=\Gamma(\sigma_n)\beta(\sigma_n)/\ln 2\). A direct numerical extraction of the oscillation confirms 16 : the mean and first harmonic agree with \(\Gamma(\sigma_n)\beta(\sigma_n)/\ln2\), the higher harmonics being correct but below the numerical floor.
Remark 9 (suppression). The amplitudes decay extremely fast. Since \(|\mathfrak{I}\sigma_n|=2\pi|n|/\ln2\approx 9.06\,|n|\) and \(|\Gamma(\sigma+i\tau)|\sim e^{-\pi|\tau|/2}\) as \(|\tau|\to\infty\), the first harmonic is already of relative size \(\sim10^{-6}\) and the remainder is essentially invisible. The log-periodic structure is genuine but, for a single dyadic scale, exponentially small.
Remark 10 (what is classical here). Identity 15 is an instance of the harmonic-sum Mellin asymptotics of Flajolet–Gourdon–Dumas [10]; the appearance of \(L\)-values in 16 is the statement that the Mellin transform of an arithmetic profile is an \(L\)-function. The complex dimensions coincide with the eta extra-zeros. The \(L\)-values occur at generic points of the line \(\mathfrak{R}\sigma=1-\mathfrak{R}s\), not at distinguished arguments. So this route too recovers classical objects — the analytic theory of harmonic sums and the eta function — rather than producing new ones.
It is worth collecting what the preceding sections show, because one pattern runs through all of them. The dyadic Cauchy-kernel identity is a single compact object, and each specialization returns to a classical landmark:
On the unit circle, symmetrized, it is the iterated polylogarithm duplication formula telescoped against Hurwitz’s Fourier-series formula for \(\zeta(s,x)\) (Lemma 1, §2).
At \(\omega=\pi\) it gives a representation of \(\pi^{s-1}\) through the eta–zeta relation, and with the functional equation, the ratio \(\zeta(s)/\zeta(1-s)\) (§3).
At \(\omega=2\pi p/q\) it gives Hurwitz’s formula at rational arguments, in the neighborhood of the multiplication theorem (§3).
Through its Mellin transform it reproduces the Hasse–Sondow globally convergent series (§4).
Through its discrete scale invariance its complex dimensions are the eta extra-zeros, and the log-periodic amplitudes of the associated harmonic sums are Dirichlet \(L\)-values (§5).
The recurrence is not accidental. By Lemma 1 the identity is assembled from the duplication formula and, through Hurwitz’s formula and the Mellin apparatus, from the classical analytic theory of \(\zeta\) and \(L\); any specialization inherits that content. What this note offers, then, is not new theorems but a single vantage point — a compact dyadic identity from which several familiar results may be read off, together with a map of how they connect. Where a particular display above may not previously have appeared in exactly this form, it is in each case a recombination of the ingredients named, and we make no claim of novelty for it. The value, if any, is organizational: the duplication formula, Hurwitz’s formula, the Hasse–Sondow series, the eta extra-zeros, and the harmonic-sum asymptotics are not usually seen as facets of one identity, and here they are.