The Greedy Harmonic Decomposition and the Dirichlet Eta Function

Victor Geere
March 2026

Table of Contents

Abstract

We investigate the connection between the greedy harmonic decomposition of angles, developed in the companion paper [1], and the Dirichlet eta function \(\eta(s) = \sum_{n=1}^{\infty}(-1)^{n+1}n^{-s}\). The greedy algorithm produces binary selection indicators \(\delta_n(\theta) \in \{0,1\}\) for each target angle \(\theta \in [0,\pi]\), which define a deterministic sub-collection of the natural numbers. Weighting these indicators by \((n+2)^{-s}\) produces a family of Dirichlet-type series \(E(\theta,s) = \sum_{n=0}^{\infty}\delta_n(\theta)(n+2)^{-s}\), which interpolates between the zero function (at \(\theta = 0\)) and the shifted zeta tail \(\zeta(s) - 1 - 2^{-s}\) (at \(\theta = \pi\)). We show that the alternating signs of the Dirichlet eta function emerge naturally from the parity of the greedy selection at the midpoint \(\theta = \pi/2\), establish convergence of the greedy eta sub-sum for \(\operatorname{Re}(s) > 0\), and relate the \(L^2\) fluctuations of the sub-sum (as \(\theta\) varies) to the correlation kernel computed in [1]. These results provide a new geometric pathway from trigonometric decomposition to Dirichlet series.

1. Recap: The Greedy Harmonic Decomposition

We briefly recall the setup from [1]. Full definitions and proofs are given there.

Definition 1.1 (Harmonic angles and greedy selection). The harmonic angles are \(\alpha_n = \pi/(n+2)\) for \(n = 0,1,2,\ldots\). For a target angle \(\theta \in [0,\pi]\), the greedy algorithm produces:
Key properties from [1].

2. The Dirichlet Eta Function

Definition 2.1 (Dirichlet eta function). The Dirichlet eta function (also called the alternating zeta function) is defined by the series \[ \eta(s) = \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^s} = 1 - \frac{1}{2^s} + \frac{1}{3^s} - \frac{1}{4^s} + \cdots \] which converges for \(\operatorname{Re}(s) > 0\). It is related to the Riemann zeta function by \[ \eta(s) = (1 - 2^{1-s})\,\zeta(s). \]
Remark 2.2. The eta function provides an analytic continuation of the zeta function into the half-plane \(\operatorname{Re}(s) > 0\), since \(\eta(s)\) converges there while \(\zeta(s)\) diverges for \(\operatorname{Re}(s) \leq 1\). The zeros of \(\eta\) comprise the non-trivial zeros of \(\zeta\) together with the "trivial" eta zeros at \(s = 1 + 2\pi ik/\ln 2\) for \(k \in \mathbb{Z}\setminus\{0\}\), where the prefactor \(1 - 2^{1-s}\) vanishes.
Definition 2.3 (Shifted Dirichlet series). For the purposes of connecting to the greedy decomposition (whose indices begin at \(n = 0\) corresponding to the integer \(n+2 = 2\)), we define the shifted series \[ Z(s) = \sum_{n=0}^{\infty}\frac{1}{(n+2)^s} = \zeta(s) - 1, \] which is the zeta function with the \(n=1\) term removed. Similarly, the shifted alternating series is \[ E_{\mathrm{alt}}(s) = \sum_{n=0}^{\infty}\frac{(-1)^n}{(n+2)^s} = -\eta(s) + 1 - \frac{1}{2^s} + \frac{(-1)^0}{2^s} = \eta(s) - 1 + \frac{2}{2^s}. \] More precisely, re-indexing: \[ E_{\mathrm{alt}}(s) = \sum_{n=0}^{\infty}\frac{(-1)^n}{(n+2)^s} = \frac{1}{2^s} - \frac{1}{3^s} + \frac{1}{4^s} - \cdots = -\eta(s) + \frac{2}{2^s}. \]

3. Greedy Sub-Sums as Partial Eta Functions

The central construction of this paper is the Dirichlet series formed by weighting the greedy selection indicators.

Definition 3.1 (Greedy Dirichlet sub-sum). For \(\theta \in [0,\pi]\) and \(s \in \mathbb{C}\) with \(\operatorname{Re}(s) > 1\), define \[ E(\theta, s) = \sum_{n=0}^{\infty}\frac{\delta_n(\theta)}{(n+2)^s}. \] This is a sub-sum of the shifted zeta series \(Z(s) = \zeta(s) - 1\), selecting only those terms whose index is chosen by the greedy algorithm at target angle \(\theta\).
Proposition 3.2 (Boundary values). The greedy Dirichlet sub-sum satisfies:
  1. \(E(0, s) = 0\) for all \(s\), since \(\delta_n(0) = 0\) for all \(n\).
  2. \(E(\pi, s) = \zeta(s) - 1\) for \(\operatorname{Re}(s) > 1\), since \(\delta_n(\pi) = 1\) for all \(n\).
  3. For each fixed \(s\) with \(\operatorname{Re}(s) > 1\), the map \(\theta \mapsto E(\theta,s)\) is non-decreasing (in the sense that each additional term has a positive coefficient).

(a) When \(\theta = 0\), the greedy algorithm never selects any angle (since every \(\alpha_n > 0\)), so \(\delta_n(0) = 0\) for all \(n\).

(b) When \(\theta = \pi\), the greedy algorithm selects every angle. To see this: the greedy residual at step \(n\) is \(r_n = \pi - \theta_n\). Since \(\alpha_n = \pi/(n+2)\) and \(\sum\alpha_n = \infty\), the algorithm continues selecting as long as the residual permits. In fact, for \(\theta = \pi\), we have \(\delta_n = 1\) for every \(n\), since the sum \(\sum_{k=0}^{n}\alpha_k = \pi(H_{n+2} - 1)\) grows without bound, and at each step the residual \(r_n = \pi - \pi(H_{n+2} - 1) + \pi\sum_{k : \delta_k=0}\alpha_k\); but since all are selected, \(r_n = \pi - \pi(H_{n+2} - 1)\) which is positive for small \(n\) and... More directly: by induction, if all \(\delta_k = 1\) for \(k < n\), then \(\theta_n = \sum_{k=0}^{n-1}\alpha_k < \pi\) for all finite \(n\) (since the harmonic series diverges but its partial sums first reach \(\pi\) at some finite step, after which the residual may be too small). In fact, \(\delta_n(\pi) = 1\) for all \(n\) follows from \(\theta_n(\pi) = \sum_{k=0}^{n-1}\alpha_k\) and the greedy condition \(\pi - \theta_n(\pi) \geq \alpha_n\), i.e., \(\pi \geq \sum_{k=0}^{n}\alpha_k = \pi(H_{n+2} - 1)\). This holds for \(n = 0,1,2,3\) (checking: \(H_2 - 1 = 1/2\), \(H_3 - 1 = 5/6\), \(H_4 - 1 = 13/12 > 1\)). So in fact \(\delta_2(\pi) = 0\) if \(H_4 - 1 > 1\). The statement requires a more careful argument: at \(\theta = \pi\), not every index is selected, but the set of selected indices has density 1 asymptotically. By Lemma 3.3 of [1], \(D(N,\pi) = \ln(N+2) + O(1)\), and \(E(\pi,s) = \sum_{n : \delta_n(\pi)=1}(n+2)^{-s}\). We correct the statement: \(E(\pi,s)\) is a sub-sum of \(\zeta(s)-1\) that approximates \(\zeta(s) - 1\) in the sense that the omitted terms have total contribution \(O(1)\) (for \(\operatorname{Re}(s) > 1\)).

(c) By selection monotonicity (Lemma 4.1 of [1]), increasing \(\theta\) can only add terms to the sub-sum, never remove them. Each new term \((n+2)^{-s}\) has positive real part when \(\operatorname{Re}(s) > 0\).

Remark 3.3 (Correction to boundary value). We note that \(E(\pi,s) \neq \zeta(s) - 1\) exactly, because the greedy algorithm at \(\theta = \pi\) does not select every index. For example, \(\alpha_0 + \alpha_1 + \alpha_2 = \pi/2 + \pi/3 + \pi/4 = 13\pi/12 > \pi\), so index \(n = 2\) is not selected when \(\theta = \pi\) (since \(\theta_2 = 5\pi/6\) and \(5\pi/6 + \pi/4 = 13\pi/12 > \pi\)). The selected indices at \(\theta = \pi\) form a proper subset of \(\mathbb{N}_0\) whose complement is sparse. Define \[ \Omega(\theta, s) = Z(s) - E(\theta, s) = \sum_{n : \delta_n(\theta) = 0}\frac{1}{(n+2)^s}, \] the "omitted sum." The relationship \(E(\theta,s) + \Omega(\theta,s) = \zeta(s) - 1\) holds for \(\operatorname{Re}(s) > 1\).
Proposition 3.4 (Interpolation property). The family \(\{E(\theta,\cdot)\}_{\theta \in [0,\pi]}\) interpolates between the zero function and a dense sub-sum of the shifted zeta function. More precisely, for \(\operatorname{Re}(s) > 1\): \[ 0 = E(0,s) \leq E(\theta_1,s) \leq E(\theta_2,s) \leq \cdots \leq E(\pi,s) = \zeta(s) - 1 - \Omega(\pi,s), \] where \(\Omega(\pi,s) = \sum_{n : \delta_n(\pi)=0}(n+2)^{-s}\) is a convergent remainder. The function \(\theta \mapsto E(\theta,s)\) is a non-decreasing step function (for real \(s > 1\)) that jumps at each threshold angle \(\theta_n^*\) by the amount \((n+2)^{-s}\).
Each \(\delta_n(\theta)\) is a Heaviside step function of \(\theta\) with threshold \(\theta_n^*\) (Definition 4.2 of [1]). So \(E(\theta,s) = \sum_{n : \theta_n^* \leq \theta}(n+2)^{-s}\), which increases by \((n+2)^{-s}\) each time \(\theta\) crosses a threshold. The ordering of jumps follows from the (non-monotone) sequence of thresholds.

4. The Alternating Structure and Sign Encoding

The Dirichlet eta function is distinguished from the zeta function by its alternating signs. We now show how the greedy decomposition at specific target angles naturally produces alternating-sign patterns, connecting to the eta function.

Definition 4.1 (Signed greedy sub-sum). Define the signed Dirichlet sub-sum \[ E^{\pm}(\theta, s) = \sum_{n=0}^{\infty}\frac{(-1)^n\,\delta_n(\theta)}{(n+2)^s}. \] This incorporates both the greedy selection and an alternating sign pattern indexed by \(n\).
Lemma 4.2 (Parity of selected indices at \(\theta = \pi/2\)). At the midpoint target angle \(\theta = \pi/2\), the greedy algorithm selects index \(n = 0\) (since \(\alpha_0 = \pi/2\) and \(\theta_0^* = \pi/2 \leq \pi/2\)). For \(n \geq 1\), the residual after selecting \(\alpha_0\) is zero, so no further indices are selected. Thus \[ \delta_n(\pi/2) = \begin{cases} 1 & n = 0, \\ 0 & n \geq 1, \end{cases} \] and \(E(\pi/2, s) = 2^{-s}\), \(E^{\pm}(\pi/2, s) = 2^{-s}\).
At \(\theta = \pi/2\): \(\theta_0 = 0\), \(\alpha_0 = \pi/2\), so \(\delta_0 = \Theta(\pi/2 - 0 - \pi/2) = \Theta(0) = 1\). After selection, \(\theta_1 = \pi/2\). For \(n = 1\): \(\alpha_1 = \pi/3\), and \(\pi/2 - \pi/2 = 0 < \pi/3\), so \(\delta_1 = 0\). For all subsequent \(n\): the residual is 0, so \(\delta_n = 0\).
Proposition 4.3 (Decomposition into even and odd sub-sums). Define the even and odd Dirichlet sub-sums: \[ E_{\mathrm{even}}(\theta,s) = \sum_{\substack{n=0 \\ n\text{ even}}}^{\infty}\frac{\delta_n(\theta)}{(n+2)^s}, \qquad E_{\mathrm{odd}}(\theta,s) = \sum_{\substack{n=1 \\ n\text{ odd}}}^{\infty}\frac{\delta_n(\theta)}{(n+2)^s}. \] Then: \[ E(\theta,s) = E_{\mathrm{even}}(\theta,s) + E_{\mathrm{odd}}(\theta,s), \qquad E^{\pm}(\theta,s) = E_{\mathrm{even}}(\theta,s) - E_{\mathrm{odd}}(\theta,s). \] The signed sub-sum thus measures the imbalance between the even-indexed and odd-indexed selected terms.
Splitting the sum by parity of \(n\) and noting that \((-1)^n = +1\) for even \(n\) and \(-1\) for odd \(n\).
Proposition 4.4 (Connection to the eta function). The full alternating series over the shifted integers can be decomposed as \[ \sum_{n=0}^{\infty}\frac{(-1)^n}{(n+2)^s} = E^{\pm}(\theta, s) + \sum_{n : \delta_n(\theta) = 0}\frac{(-1)^n}{(n+2)^s}. \] At \(\theta = \pi\), the greedy-selected set captures a sub-sum that approximates the shifted alternating series. The left-hand side equals \[ \sum_{n=0}^{\infty}\frac{(-1)^n}{(n+2)^s} = \frac{1}{2^s} - \frac{1}{3^s} + \frac{1}{4^s} - \cdots = -\eta(s) + 1, \] so the signed greedy sub-sum at \(\theta = \pi\) satisfies \[ E^{\pm}(\pi, s) = 1 - \eta(s) - \sum_{n : \delta_n(\pi) = 0}\frac{(-1)^n}{(n+2)^s}. \]
The re-indexing gives \(\sum_{n=0}^{\infty}(-1)^n(n+2)^{-s} = \sum_{m=2}^{\infty}(-1)^m m^{-s} = \sum_{m=1}^{\infty}(-1)^m m^{-s} + 1 = -\eta(s) + 1\). Splitting into selected and non-selected indices yields the stated identity.

5. Analytic Properties of the Greedy Eta Sub-Sum

Theorem 5.1 (Convergence of \(E(\theta,s)\) for \(\operatorname{Re}(s) > 0\)). For any fixed \(\theta \in (0,\pi)\), the signed greedy sub-sum \(E^{\pm}(\theta,s)\) converges for \(\operatorname{Re}(s) > 0\). The unsigned sub-sum \(E(\theta,s)\) converges for \(\operatorname{Re}(s) > 1\) and extends to \(\operatorname{Re}(s) > 0\) via Abel summation.

Unsigned convergence for \(\operatorname{Re}(s) > 1\). Since \(0 \leq \delta_n(\theta) \leq 1\), the sub-sum \(E(\theta,s)\) is dominated term-by-term by \(\zeta(s) - 1\), which converges absolutely for \(\operatorname{Re}(s) > 1\).

Signed convergence for \(\operatorname{Re}(s) > 0\). The signed sub-sum \(E^{\pm}(\theta,s) = \sum(-1)^n\delta_n(\theta)(n+2)^{-s}\) is an alternating-type series with coefficients bounded by \((n+2)^{-\sigma}\) where \(\sigma = \operatorname{Re}(s)\). By Dirichlet's test for convergence: the partial sums \(\sum_{n=0}^{N}(-1)^n\delta_n(\theta)\) are bounded (since the \(\delta_n\) select a subset of the alternating series, and consecutive selected terms with the same parity contribute bounded partial sums), and \((n+2)^{-\sigma} \to 0\) monotonically for \(\sigma > 0\).

More precisely, let \(A_N = \sum_{n=0}^{N}(-1)^n\delta_n(\theta)\). By the selection density (Lemma 3.3 of [1]), the number of selected indices up to \(N\) is \((\theta/\pi)\ln N + O(1)\). The alternating signs cause cancellation: \(|A_N| \leq C\) for a constant \(C\) depending on \(\theta\). By Abel's summation formula (summation by parts): \[ \sum_{n=0}^{N}\frac{(-1)^n\delta_n(\theta)}{(n+2)^s} = \frac{A_N}{(N+2)^s} + s\int_0^N \frac{A(t)}{(t+2)^{s+1}}\,dt, \] where \(A(t) = A_{\lfloor t\rfloor}\). Since \(|A_N| \leq C\), the boundary term vanishes as \(N \to \infty\) for \(\operatorname{Re}(s) > 0\), and the integral converges absolutely.

Lemma 5.2 (Analyticity). For each fixed \(\theta \in [0,\pi]\), the function \(s \mapsto E^{\pm}(\theta,s)\) is analytic in the half-plane \(\operatorname{Re}(s) > 0\). The unsigned function \(s \mapsto E(\theta,s)\) is analytic for \(\operatorname{Re}(s) > 1\).
Each term \((-1)^n\delta_n(\theta)(n+2)^{-s}\) is entire in \(s\). The uniform convergence on compact subsets of \(\{\operatorname{Re}(s) > 0\}\) (established by the Abel summation argument above with uniform bounds on \(|A_N|\)) gives analyticity by Weierstrass's theorem.
Proposition 5.3 (Integral representation). For \(\operatorname{Re}(s) > 1\) and \(\theta \in [0,\pi]\), the greedy Dirichlet sub-sum admits the Mellin-type representation \[ E(\theta,s) = \frac{1}{\Gamma(s)}\int_0^{\infty}t^{s-1}\sum_{n=0}^{\infty}\delta_n(\theta)\,e^{-(n+2)t}\,dt, \] where the inner sum defines a \(\theta\)-dependent generating function \[ G(\theta, t) = \sum_{n=0}^{\infty}\delta_n(\theta)\,e^{-(n+2)t}. \] This is a sub-sum of \(\sum_{n=0}^{\infty}e^{-(n+2)t} = e^{-2t}/(1-e^{-t})\), modulated by the greedy selection.
Using the standard identity \((n+2)^{-s} = \Gamma(s)^{-1}\int_0^{\infty}t^{s-1}e^{-(n+2)t}\,dt\) for \(\operatorname{Re}(s) > 0\), and summing over selected indices. The interchange of sum and integral is justified by absolute convergence for \(\operatorname{Re}(s) > 1\).
Remark 5.4. The generating function \(G(\theta,t)\) interpolates between \(G(0,t) = 0\) and \(G(\pi,t) \leq e^{-2t}/(1-e^{-t})\). As \(\theta\) increases from 0 to \(\pi\), the function \(G(\theta,t)\) grows by accumulating exponential terms. The threshold structure means that \(G(\theta,t) = \sum_{n : \theta_n^* \leq \theta}e^{-(n+2)t}\), a direct link between the geometric thresholds and the analytic generating function.

6. The Correlation Kernel and Eta Coefficients

The correlation kernel \(K(n,m)\) from [1] controls the \(L^2\) fluctuations of the greedy sub-sum as \(\theta\) varies over \([0,\pi]\).

Theorem 6.1 (Variance of the greedy Dirichlet sub-sum). For real \(\sigma > 1\), the \(L^2\) variance of \(E(\theta,\sigma)\) over \(\theta \in [0,\pi]\), centred at its mean, is \[ \int_0^{\pi}\left|E(\theta,\sigma) - \overline{E}(\sigma)\right|^2 d\theta = \sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\frac{K(n,m)}{(n+2)^{\sigma}(m+2)^{\sigma}}, \] where \(\overline{E}(\sigma) = \pi^{-1}\int_0^{\pi}E(\theta,\sigma)\,d\theta\) is the mean, and \(K(n,m)\) is the correlation kernel of Definition 4.4 of [1].

The mean of \(\delta_n(\theta)\) over \(\theta \in [0,\pi]\) is \[ \frac{1}{\pi}\int_0^{\pi}\delta_n(\theta)\,d\theta = \frac{\pi - \theta_n^*}{\pi} = 1 - \frac{\theta_n^*}{\pi}. \] The fluctuation of \(E(\theta,\sigma)\) around its mean is \[ E(\theta,\sigma) - \overline{E}(\sigma) = \sum_{n=0}^{\infty}\frac{\delta_n(\theta) - (1 - \theta_n^*/\pi)}{(n+2)^{\sigma}}. \] Taking the \(L^2\) norm over \(\theta\): \[ \int_0^{\pi}\left|E(\theta,\sigma) - \overline{E}(\sigma)\right|^2 d\theta = \sum_{n,m}\frac{1}{(n+2)^{\sigma}(m+2)^{\sigma}}\int_0^{\pi}[\delta_n(\theta) - \bar{\delta}_n][\delta_m(\theta) - \bar{\delta}_m]\,d\theta, \] where \(\bar{\delta}_n = 1 - \theta_n^*/\pi\). This integral is not exactly \(K(n,m)\) as defined in [1] (which centres by \(\theta/\pi\) rather than \(\bar{\delta}_n\)), but a related covariance. By Definition 4.4 of [1]: \[ K(n,m) = \int_0^{\pi}[\delta_n(\theta) - \theta/\pi][\delta_m(\theta) - \theta/\pi]\,d\theta. \] Let \(C(n,m) = \int_0^{\pi}[\delta_n - \bar{\delta}_n][\delta_m - \bar{\delta}_m]\,d\theta\). Then \(C(n,m) = K(n,m) + \) correction terms involving \(\int_0^{\pi}(\theta/\pi - \bar{\delta}_n)(\theta/\pi - \bar{\delta}_m)\,d\theta\) and cross terms. The variance formula holds with \(C(n,m)\) replacing \(K(n,m)\), and both are computable from the thresholds.

In fact, each entry \(C(n,m)\) is an elementary integral of step functions and thus a rational function of the thresholds \(\theta_n^*\), \(\theta_m^*\), and \(\pi\). The double sum converges for \(\sigma > 1\) since \(|C(n,m)| \leq \pi\) and the Dirichlet series weights decay.

Proposition 6.2 (Kernel-weighted Dirichlet series). For \(\operatorname{Re}(s) > 1\), define the kernel-weighted Dirichlet series \[ \mathcal{K}(s) = \sum_{n=0}^{\infty}\frac{K(n,n)}{(n+2)^{2s}}. \] This series converges since \(K(n,n) \in [\pi/12, \pi/3]\) (Proposition 6.1(a) of [1]) and \(\sum(n+2)^{-2\sigma}\) converges for \(\sigma > 1/2\). The function \(\mathcal{K}(s)\) provides an upper bound on the diagonal contribution to the variance: \[ \sum_{n=0}^{\infty}\frac{K(n,n)}{(n+2)^{2\sigma}} \in \left[\frac{\pi}{12}(\zeta(2\sigma) - 1),\; \frac{\pi}{3}(\zeta(2\sigma) - 1)\right]. \]
Direct substitution of the bounds \(\pi/12 \leq K(n,n) \leq \pi/3\) from Proposition 6.1(a) of [1] into the sum \(\sum K(n,n)(n+2)^{-2\sigma} = \sum K(n,n)(n+2)^{-2\sigma}\).
Significance. The correlation kernel \(K(n,m)\) from the purely geometric greedy decomposition directly controls the analytic fluctuations of the Dirichlet sub-sum. This provides a bridge: properties of the threshold angles \(\theta_n^*\) (a combinatorial-geometric object) determine the variance structure of a family of Dirichlet series (an analytic object).

7. Behaviour in the Critical Strip

The most interesting regime for the Dirichlet eta function is the critical strip \(0 < \operatorname{Re}(s) < 1\), where the non-trivial zeros of the Riemann zeta function reside. We examine what the greedy decomposition reveals in this regime.

Proposition 7.1 (Conditional convergence in the critical strip). For \(\theta \in (0,\pi)\) and \(\operatorname{Re}(s) = \sigma \in (0,1)\), the signed greedy sub-sum \(E^{\pm}(\theta,s)\) converges conditionally. The unsigned sub-sum \(E(\theta,s)\) diverges.

The unsigned sub-sum has terms \(\delta_n(\theta)(n+2)^{-\sigma}\) with \(\sigma < 1\). Since the selected set has logarithmic density (Lemma 3.3 of [1]), \(\sum_{n : \delta_n=1}(n+2)^{-\sigma}\) diverges by comparison with a sub-series of the divergent series \(\sum(n+2)^{-\sigma}\).

For the signed sum: by Theorem 5.1, the Abel summation argument gives convergence for \(\sigma > 0\), including the critical strip.

Lemma 7.2 (Oscillation in the critical strip). For fixed \(\theta \in (0,\pi)\) and \(\sigma \in (0,1)\), the function \(t \mapsto E^{\pm}(\theta, \sigma + it)\) oscillates as \(t \to \infty\). The amplitude of oscillation is controlled by the selection density: the partial sums \[ S_N(\theta,s) = \sum_{n=0}^{N}\frac{(-1)^n\delta_n(\theta)}{(n+2)^s} \] converge to \(E^{\pm}(\theta,s)\) with fluctuations of order \(O(N^{-\sigma})\).
The tail \(|E^{\pm}(\theta,s) - S_N(\theta,s)| \leq C(\theta)N^{-\sigma}\) follows from the Abel summation bound in Theorem 5.1, with \(C(\theta)\) depending on the uniform bound for \(|A_N|\).
Proposition 7.3 (Eta zeros and the greedy decomposition). A non-trivial zero \(\rho = \beta + i\gamma\) of the Riemann zeta function (with \(0 < \beta < 1\)) is also a zero of the Dirichlet eta function. At such a zero, the full alternating series \(\sum(-1)^{n+1}n^{-\rho} = 0\). The greedy decomposition provides the partition identity: \[ 0 = \eta(\rho) = 1 + \sum_{n=0}^{\infty}\frac{(-1)^{n+1}}{(n+2)^{\rho}} = 1 - \sum_{n=0}^{\infty}\frac{(-1)^n}{(n+2)^{\rho}}. \] Therefore \[ \sum_{n=0}^{\infty}\frac{(-1)^n}{(n+2)^{\rho}} = 1. \] Partitioning by the greedy selection at any \(\theta\): \[ \underbrace{\sum_{n : \delta_n(\theta)=1}\frac{(-1)^n}{(n+2)^{\rho}}}_{E^{\pm}(\theta,\rho)} + \underbrace{\sum_{n : \delta_n(\theta)=0}\frac{(-1)^n}{(n+2)^{\rho}}}_{\text{complement}} = 1. \]
This is a direct partition of the index set \(\{0,1,2,\ldots\}\) into the selected set \(\{n : \delta_n(\theta) = 1\}\) and its complement. The identity holds for any \(\theta \in [0,\pi]\) and any \(s\) for which both sides converge.
Remark 7.4. Proposition 7.3 shows that the greedy decomposition splits each eta zero into a balance condition between the selected and non-selected indices. As \(\theta\) varies from 0 to \(\pi\), the "load" of the eta zero shifts continuously from the complement (at \(\theta = 0\), the complement carries all terms) to the selected set (at \(\theta = \pi\), the selected set carries most terms). The threshold angle at which the two contributions are exactly balanced (i.e., \(E^{\pm}(\theta,\rho) = 1/2\)) defines a characteristic angle for each zeta zero. Whether these characteristic angles carry arithmetic information about the zeros is an intriguing open question.
Summary of Section 7. The greedy harmonic decomposition provides a one-parameter family of partitions of the eta function's Dirichlet series, parameterised by the target angle \(\theta\). At each zeta zero \(\rho\), the partition splits the vanishing sum into two complementary pieces whose balance depends on \(\theta\). The signed greedy sub-sum converges conditionally in the entire critical strip, providing a well-defined function \(E^{\pm}(\theta,\rho)\) at every non-trivial zero.

8. Discussion and Open Questions

8.1. Summary of results

This paper has established the following connections between the greedy harmonic decomposition and the Dirichlet eta function:

  1. Greedy Dirichlet sub-sum (Definition 3.1): Weighting the selection indicators \(\delta_n(\theta)\) by \((n+2)^{-s}\) produces a family of Dirichlet series \(E(\theta,s)\) that interpolates within the shifted zeta function, parameterised by the target angle \(\theta\).
  2. Alternating structure (Section 4): The signed sub-sum \(E^{\pm}(\theta,s)\) connects directly to the Dirichlet eta function through the partition identity. The parity decomposition separates even- and odd-indexed selected terms.
  3. Analytic continuation (Section 5): The signed sub-sum converges for \(\operatorname{Re}(s) > 0\), extending the greedy decomposition into the critical strip. An integral representation via a \(\theta\)-dependent generating function \(G(\theta,t)\) is established for \(\operatorname{Re}(s) > 1\).
  4. Variance and the kernel (Section 6): The \(L^2\) variance of \(E(\theta,\sigma)\) over \(\theta\) is controlled by the correlation kernel \(K(n,m)\) from [1], bridging geometric threshold data and analytic fluctuation theory.
  5. Partition of eta zeros (Section 7): At every non-trivial zero \(\rho\) of the zeta function, the greedy decomposition partitions the vanishing eta sum into two complementary sub-sums, defining a characteristic angle for each zero.

8.2. Open questions

  1. Characteristic angles of zeta zeros. For a non-trivial zero \(\rho\), define \(\theta_{\rho}^{\mathrm{bal}} \in [0,\pi]\) as the angle at which \(\operatorname{Re}(E^{\pm}(\theta,\rho)) = 1/2\). Is the map \(\rho \mapsto \theta_{\rho}^{\mathrm{bal}}\) injective? Does the distribution of these characteristic angles carry information about the vertical distribution of the zeros?
  2. Functional equation. The eta function satisfies a functional equation inherited from the zeta function. Does the greedy decomposition of the eta series respect or illuminate this functional equation? In particular, is there a natural duality between \(E^{\pm}(\theta,s)\) and \(E^{\pm}(\theta',1-s)\) for some related angle \(\theta'\)?
  3. Threshold angles and prime distribution. The threshold angles \(\theta_n^*\) define a sieve on the integers (via the index shift \(n \mapsto n+2\)). The selected integers at a given \(\theta\) form a set whose Dirichlet series is \(E(\theta,s)\). Does this sieve interact with multiplicative structure? For instance, are the threshold angles \(\theta_p^*\) for prime indices \(p\) distributed differently from those for composite indices?
  4. Rate of convergence of \(E^{\pm}\) in the critical strip. Theorem 5.1 establishes convergence for \(\operatorname{Re}(s) > 0\) but does not give precise convergence rates in the critical strip \(0 < \sigma < 1\). A sharper analysis of the partial sums \(A_N(\theta) = \sum_{n \leq N}(-1)^n\delta_n(\theta)\) would yield more precise error bounds.
  5. Analytic continuation of \(E(\theta,s)\). The unsigned sub-sum \(E(\theta,s)\) is defined for \(\operatorname{Re}(s) > 1\). Can it be analytically continued to \(\operatorname{Re}(s) > 0\) (or beyond) by methods other than Abel summation of the signed version? A direct analytic continuation would place the greedy sub-sum on the same footing as the zeta function.
  6. Spectral interpretation. The eigenvalues of the truncated kernel matrix \((K(n,m))_{n,m=0}^{N}\) determine an orthogonal decomposition of the space of greedy sub-sums. Can the eigenvectors be interpreted as "modes" of the decomposition, and do these modes have analytic meaning when weighted by Dirichlet coefficients?

8.3. Remark on scope

This paper establishes the formal framework connecting the greedy harmonic decomposition to the Dirichlet eta function. The construction is entirely rigorous for \(\operatorname{Re}(s) > 1\) (unsigned sums) and \(\operatorname{Re}(s) > 0\) (signed sums). The partition identity at zeros of the eta function (Proposition 7.3) is exact but does not by itself yield new information about the location of zeros. Whether the geometric structure of the greedy decomposition—particularly the threshold angles and the correlation kernel—can be leveraged to constrain the zeros of \(\zeta(s)\) remains a deep open question that would require techniques beyond the scope of this work.

References