In Search of Approximate Polynomial Dependencies Among the Derivatives of the Alternating Zeta Function

It is well-known that the Riemann zeta function does not satisfy any exact polynomial differential equation. Here we present numerical evidence for the existence of approximate polynomial dependencies between the values of the alternating zeta function and its initial derivatives. A number of conjectures is stated.

💡 Research Summary

The paper “In Search of Approximate Polynomial Dependencies Among the Derivatives of the Alternating Zeta Function” investigates whether the Dirichlet eta function η(s)=∑_{n≥1}(-1)^{n-1}n^{-s} and its initial derivatives satisfy any non‑trivial algebraic differential relations. It is well‑known, following work of Hilbert, Stadigh, Mordukhai‑Boltovskoi and others, that the Riemann zeta function ζ(s) does not satisfy any exact polynomial differential equation; the same negative result transfers to η(s) because η(s)=(1−2^{1−s})ζ(s). The author therefore asks a more modest question: can one find approximate polynomial relations among the values η(a) and its derivatives η^{(k)}(a) for a given complex point a?

To formalise this, the author introduces, for each integer N≥1, a pair of integer‑coefficient polynomials ⟨V_N(y_0,…,y_N), W_N(y_0,…,y_{N-1})⟩. The conjectured approximate relation is

  V_N(η(a),…,η^{(N-1)}(a)) · W_N(η(a),…,η^{(N-1)}(a))^{-1} → 1 as N→∞,

for all but countably many a∈ℂ. The paper proceeds entirely by numerical experimentation to support this conjecture.

Construction of the data.
Two families of point sets in the complex plane are used:

1. Grids A_G(a,δ₁,δ₂,N₁,N₂) = {a + kδ₁ + lδ₂ | 0≤k≤N₁, 0≤l≤N₂}.
2. Discrete circles A_C(c,r,N) = {c + r e^{2πik/N} | k=0,…,N−1}.

For a given set A={a₀,…,a_{N-1}} the author solves the linear system

 η(a_j) + c₁ η′(a_j) + … + c_{N-1} η^{(N-1)}(a_j) = b (j=0,…,N−1),

which can be written in matrix form as D_N · x = b · 1, where D_N is the N×N matrix with entries x_{j,k}=η^{(k)}(a_j). By Cramer’s rule the solution is expressed as ratios of determinants. The determinant D_N itself is denoted by a symbolic matrix whose entries are variables x_{j,k}; replacing these variables by the actual derivative values yields the “numerator” polynomial V_N(A)·λ^N+…+V₀(A). Replacing x_{j,0} by 1 (i.e., ignoring the zeroth column) gives the “denominator” polynomial W_N(A)·λ^N+…+W₀(A). The ratio Q_n(A)=V_n(A)/W_n(A) is then examined.

Numerical evidence.
Tables 2.1–2.3 present the coefficients c_k(A) and the constant term b(A) for several grids and circles. Strikingly, b(A) is consistently extremely close to 1 (differences on the order of 10^{-15}–10^{-40}) even for modest N (e.g., N≈30). Table 4.1 shows that not only Q₀(A) but also Q_n(A) for small n remain near 1. The author interprets this as empirical support for the conjectured limit (1.5).

Missing derivatives.
In Section 4.2 the author explores what happens if some derivatives are omitted. Let D={d₀=0<d₁<…<d_{N-1}} be a strictly increasing sequence of non‑negative integers. The linear system is then built only from η^{(d_k)}(a_j). The corresponding ratio Q(A,D) deviates more strongly from 1 as more derivatives are omitted; Table 4.2 quantifies this effect for a specific circle set.

Dropping η(a) itself.
Section 5 investigates the case where the zeroth derivative is excluded (l≥1). The determinant D_N, now built from higher derivatives only, often becomes numerically tiny, suggesting an approximate relation of the form

 D_N|{x{m,n}=y} ≈ 0,

where solving for y yields a value close to η^{(l+n)}(a_m). Numerical tests (5.7) confirm that the discrepancy is below 10^{-7} for a wide range of indices.

Conjectures and outlook.
The paper concludes with several conjectures:

1. For any family of points (whether widely spaced or tightly clustered) the ratios Q_n(A) converge rapidly to 1.
2. There exist explicit bounds B_G and B_C depending on the geometric parameters of the grid or circle such that |Q(A)−1|<B.
3. Analogous approximate polynomial relations might be transferable to ζ(s) itself, despite the known impossibility of exact algebraic differential equations.

Critical assessment.
The work is notable for introducing a systematic computational framework based on determinants and characteristic polynomials to probe hidden algebraic structures in η(s). The extensive tables (some reaching N≈200) demonstrate impressive numerical stability and suggest that the observed near‑unity ratios are not accidental. However, the paper remains purely experimental; no theoretical justification for why the ratios should tend to 1 is offered, nor is there an analysis of the dependence on the choice of points beyond the two regular families. The conjectures are plausible but unproven, and the paper does not address the measure‑theoretic aspect of “all but countably many a”. Moreover, the reliance on high‑precision arithmetic raises questions about the scalability of the method to even larger N or to more irregular point sets.

In summary, the paper provides compelling numerical evidence that the alternating zeta function and its derivatives obey surprisingly accurate approximate polynomial relations when evaluated on structured point sets. It opens a new direction for exploring “near‑algebraic” properties of special functions, but further work is required to develop a rigorous analytic theory, to understand the limits of the approximation, and to extend the results to the Riemann zeta function itself.